hard ball systems and the lorentz gas

Encyclopaedia of Mathematical SciencesVolume 101

Mathematical Physics II

Subseries Editors:J. Frohlich S.P. Novikov D. Ruelle

Springer-Verlag Berlin Heidelberg GmbH

1. A. Bunimovich D. Burago N. ChernovE. G. D. Cohen C. P. Dettmann J. R. Dorfman

S. Ferleger R. Hirschl A. Kononenko J.1. LebowitzC. Liverani T. J. Murphy J. Piasecki H. A. Posch

N. Simanyi Ya. Sinai D. Szasz T.Tel H. van BeijerenR. van Zon J. Vollmer 1. S.Young

Hard Ball Systemsand the Lorentz Gas

Edited byD. Szasz

With 75 Figures

Including One Colour Figure

Springer

Subseries Editors

Prof. Dr. J. FrohlichTheoretische Physik

Dept. Physik (D-PHYS)HPZ G 17

ETH Honggerberg8093 Zurich, Switzerland

e-mail : [email protected]

Prof. S. P. NovikovDepartment of Mathematics

University of Maryland at College Park-IPSTCollege Park, MD 20742-2431, USA

e-mail: [email protected]

Prof. D. RuelleIHES, Le Bois-Marie35, Route de Chartres

91440 Bures-sur-Yvette, Francee-mail: ruelle @ihes.fr

Founding Editor of the Encyclopaedia of Mathematical Sciences:R. V. Gamkrelidze

Mathematics Subject Classificat ion (2000):37-XX,82-XX

ISSN 0938 -0 396

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@ Springer.Veriag Berlin Heidelberg 2000Originally published by Springer-Verlag Berlin Heidelberg New York in 2000.

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List of Editor and Authors

Consult ing Editor

D. Szasz, Budapest University of Technology and Economics,Institute of Mathematics, P.O. Box 91, 1521 Budapest, Hungary;e-mail : szasz@math .bme .hu

Authors

L. A. Bunimovich, Southeast Applied Analy sis Center, Georgia Instituteof Technology, Atlanta, GA 30332, USA ; e-mail : [email protected]

D. Burago , Department of Mathematics, The Penn sylvania State University,University Park, PA 16802, USA; e-mail : burago@math .psu.edu

N. Chernov, Department of Mathematics, University of Alabama at Birmingham,Birmingham, AL 35294, USA ; e-mail : [email protected] .uab.edu

E. G. D. Cohen, Laboratory of Theoretical Physics, The Rockefeller University,1230 York Ave., New York, NY 10021, USA ; e-mail: [email protected]

C. P. Dettmann, University of Bristol, Department of Mathematics,University Walk, Bristol BS 8 ITW, UK

J. R. Dorfman , Institute for Physical Science and Technology,Department of Physics, University of Maryland, College Park, MD 20742 , USA;e-mail: [email protected] .edu

S. Ferleger, Department of Mathematics, SUNY at Stony Brook, Stony Brook ,NY 11794-3651 , USA ; e-mail : [email protected]

R. HirschI, Institute for Experimental Physics, University of Vienna,Boltzmanngasse 5, 1090 Vienna , Austria

A. Kononenko, Renaissance Tech. Corp , 600 Rt. 25-A E. Setanket, NY 11787,USA ; e-mail: kononena@yahoo .com

J. L. Lebowitz, Center for Mathematical Sciences Research,110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA;e-mail: [email protected]

C. Liverani, Dipartimento di Matematica, Universita di Roma II (Tor Vergata),Via delIa Ricerca Scientifica, 00133 Roma, Italy ;e-mail: [email protected]

VI List of Editors and Authors

T. J. Murphy, Department of Chemistry, University of Maryland,College Park, MD 20742, USA ; e-mail : [email protected]

J. Piasecki, Institute of Theoretical Physics, Warsaw University,Hoza 69, 00 681 Warsaw, Poland; e-mail: [email protected]

H. A. Posch, Institute for Experimental Physics, University of Vienna,Boltzmanngasse 5, 1090 Vienna, Austria; e-mail: [email protected]

N. Simanyi, Univers ity of Alabama at Birmingham, Department of Mathematics,Campbell Hall , Birmingham, AL 35294, USA ; e-mail: [email protected]

Ya. Sinai, Princeton University, Dept. of Mathematics,708 Fine Hall, Washington Road, Princeton, NJ 085-44-1000, USA ;e-mail: [email protected]

D. Szasz , Budapest University of Technology and Economics,Institute of Mathematics, P.O. Box 91, 1521 Budapest, Hungary;e-mail: [email protected] .hu

T. Tel , Institute for Theoretical Physics, Eotvos University,P.O. Box 32, 1518 Budape st, Hungary; e-mail: [email protected]

H. van Beijeren, Institute for Theoretical Physics, University of Utrecht,Princetonplein 5, 3584 CC Utrecht, The Netherlands;e-mail: [email protected]

R. van Zon, Institute for Theoretical Physic s, University of Utrecht,Princetonplein 5, 3584 CC Utrecht, The Netherlands; e-mail : [email protected]

1. Vollmer, Fachbereich Physik, Univ.-GH Essen, 45117 Essen , Germany andMax-Planck-Institute for Polymer Research, Ackermannweg 10, 55128 Mainz,Germany; e-mail: [email protected]

L.-S. Young, Courant Institute of Mathematical Sciences, 251 Mercer St.,New York, NY 10012-1110, USA ; e-mail: [email protected]

Contents

IntroductionD. Szds:

I

I. Mathematics

A Geometric Approach to Semi-Dispersing BilliardsD. Burago, S. Fer/eger and A. Kononenko

9

On the Sequences of Collisions Among Hard Spheres in Infinite SpaceT. J. Murphy and E. G. D. Cohen

29

Hard Ball Systems and Semi-Dispersive Billiards:Hyperbolicity and Ergodicity

N. Simdnyi51

Decay of Correlations for Lorentz Gases and Hard BallsN. Chemov and L.-S. Young

89Entropy Values and Entropy Bounds

N. Chemov121

Existence of Transport CoefficientsL.A. Bunimovich

145

Interacting ParticlesC. Liverani

179

Scaling Dynamics of a Massive Piston in an Ideal GasJ. L. Lebowitz, J. Piasecki and fa. Sinai

217

VIII Contents

II. Physics

Kinetic Theory Estimates for the Kolmogorov-Sinai Entropy,and the Largest Lyapunov Exponents for Dilute, Hard Ball Gases

and for Dilute, Random Lorentz GasesR. van Zon. H. van Beijeren and J. R. Dorfman

231Simulation of Billiards and of Hard Body Fluids

H. A. Posch and R. Hirschi279

The Lorentz Gas: A Paradigm for Nonequilibrium Stationary StatesC. P. Dettmann

315

Entropy Balance, Multibaker Maps, and the Dynamics of the Lorentz GasT. Tel and J. Vollmer

367

Appendix

Boltzmann's Ergodic Hypothesis, a Conjecture for Centuries?D. Szasz

421

Author Index447

Subject Index449

Introduction

Our naive picture of a gas in a vessel is that it is a Hamiltonian system consistingof a lot of small ball-like molecules moving around and colliding with eachother. However , as finer and finer computer simulations also show, the phaseportrait of a typi cal Hamiltonian syst em reflects an extraordinarily complicatedcoexistence of integrable and non-integrable behaviours. In general, there isa positive measure set of invariant tori , a picture supported by both KAMtheory and by computations. Moreover , we can observe one or several chaot icdomains. At present , science is far from describ ing the mixture of these tori andof the chaot ic domains. Another , similarly hard question, where our knowledgeis perhaps the same far from an understanding the situation, is connected withergodic hypothesis; we know next to nothing about the expected overwhelminglynon-integrable behaviour of Hamiltonian systems when the number of degreesof freedom is high.

Consequent ly, one should be more modest and be satisfied with the study ofthe "simplest" syst ems. In these, solely one of the pure, extreme behavioursoccurs : either the completely integrable one where the phase space is fullyfoliat ed by invariant tori, or the purely non-integrable one, where it consistsof just one ergodic component. Both cases open up highly beautiful mathematical problems, ideas and theories. Beside their mathematical attractivity,non-integrable systems also play a basic role in statistical physics. The simplest- and so far essent ially the only - Hamiltonian system where various forms ofnon-integrability (hyperbolicity, ergodicity, strong decay of correlat ions, .. . )have been established are hard ball systems (HBS), i. e. collect ions of hardbilliard balls without rot ation al motion, interacting via elast ic collisions.

These models deserve prime concern not only for, beyond geodesic flowson compact man ifolds of negative curvature, they represent essent ially the soleHamiltonian dynamics where the mechanism of the appearance of non-integrability can be - and has, indeed, been - studied. Beyond that , these systems- or relat ed ones - can serve as the simplest truly physical models wherevarious fundamental laws or phenomena of statistical physics can be analyt icallytested.

2 Introduction

Before giving an idea of the structure of this collection let us present theunderlying models to be treated here. HBSs are isomorphic to point-particlebilliard systems, i. e. to a model when a point particle moves with a uniformmotion and has specular (optical) reflection at some fixed scatterers. Mostoften we consider HBS's or billiards in a (subdomain of) a torus. In general,HBSs are actually semi-dispersing, i. e. the scatterers of the isomorphic pointparticle billiard are convex but not strictly convex. A point-particle billiard inthe (noncompact) space is called the Lorentz process, whereas an ideal gas ofsuch particles is the Lorentz gas. For technical reasons , mainly those Lorentzprocesses or gases are investigated where the scatterer configuration is periodic;their behaviour might be most different from those with a general scattererconfiguration. Moreover, in general, dispersing Lorentz models are studied, i. e.those where the scatterers are strictly convex. Two historic remarks: HBSs werefirst studied by the excellent Russian physicist, N. S. Krylov (d. N. S. Krylov :The Processes of Relaxation of Statistical Systems and the Criterion of Mechanical Instability, Thesis (1942), reprinted in Development of Krylov's Ideas , ed.Ya. G. Sinai, Princeton University Press, 1977), whereas the Lorentz processwas introduced by the Dutch Nobel Prize winner physicist, H. A. Lorentz todescribe the diffusion of conduction electrons in metals. (Pleasant readings onLorentz processes are two classical surveys: E. H. Hauge: What can we learn formLorentz models?, in Transport Phenomena, Lecture Notes in Physics, Vol. 31(1974), p. 337 and H. van Beijeren: Transport properties of stochastic Lorentzmodels, Rev. in Modern Physics , 54 (1982), p. 195.)

For more than a decade Andras Kramli, Nandor Simanyi and myself havebeen planning to write a monograph on billiards and HBSs. Beside havingprepared several variants of its synopsis , we have never really started to writethe book and this happened for good reason. It was simply too hard. Thoughits scope would have been much narrower than that of the first part of thepresent collection, it always appeared to be hopeless to get a good catch onthe material. Therefore, having received the honorable invitation of the serieseditors of Encyclopaedia of Mathematical Sciences to put together a collection ofsurveys related to billiards, I got immediately attracted to the idea. Beside ourfailure in writing a monograph, my first motivation for that was the following:several areas of related mathematical research have in the last decade been instrong progress leading to an essential simplification of the methods, to increasing mathematical clarity of the ideas, to spectacular new results, and, of course,even further challenging questions which could be attacked with a realistic hopefor a solution.

As an illustration of the progress, let me mention the problem of correlationdecay for 2-D dispersing or Sinai-billiards (i. e. those with strictly convex scatterers). The first catch on the problem was the quite involved Markov partitionapproach of Bunimovich and Sinai in 1980. It provided stretched exponentialcorrelation decay. Later, in 1990, the Moscow school again, Bunimovich, Chernov and Sinai suggested an essential improvement of the 1980 method. They

Introduction 3

could find a comprehensive proof of the same result under less restrictive assumptions by also introducing the more flexible tool of Markov sieves. Then by1998, Lai Sang Young worked out the method of Markov returns . It not only ledto exponential correlation decay for 2-D Sinai-billiards, but ~ very importantly~ also made possible for her to initiate a general method. It was applicableto a wide class of hyperbolic maps : smooth ones or those with singularities,like hyperbolic logistic transformations, hyperbolic Henon-maps, smooth Anosovmaps, or those with singularities in 2-D.

This evolution of the mathematical theory was the guiding line in composingthe first part of this collection . More concretely, this book is divided into twoparts according to whether the overwhelming majority of the results is mathematically rigorous or obtained by methods of physics , i. e. they are sort ofanalytic or computer supported ones.

The first part starts with papers of Burago, Ferleger and Kononenko (A geometric approach to semi-dispersing billiards) and of Cohen and Murphy (Onthe sequences of collisions among hard spheres in infinite space) on a beautifulgeometric question which is strongly related to the definition of the dynamics forHBS: one should prove that there is only a finite number of collisions in a finitetime interval. If so, one can even try to give bounds for this number. The firstpaper, whose authors obtained not a long time ago a beautiful intuitive proof,provides a general survey of this research, while the second paper reaches sharpbounds in a particular case. Simanyi's survey (Hard ball systems and semidispersive billiards: hyperbolicity and ergodicity) treats the progress and thepresent state of our knowledge in establishing hyperbolicity, and beyond thatergodicity for HBSs. Here the recent progress has much relation to the algebraicfeature of the dynamics. The contribution of Chernov and Lai Sang Young(Decay of correlations for Lorentz gases and hard balls) discusses the state ofaffairs in relation to correlation decay, the direction mentioned before. At thispoint we only add that once one has sufficiently strong correlation decay, thenit also implies the central limit theorem. When applied to the Lorentz process ,this leads to a Brownian approximation, an important fact for the dynamicaltheory of Brownian motion.

For chaotic systems, entropy is a numerical characteristic measuring theamount of stochasticity or complexity of the dynamics. Entropy formulas andentropy bounds for both finite (Kolmogorov-Sinai entropy) and infinite systems(space-time entropy) are discussed in Chernov's article (Entropy values and entropy bounds). Moreover, it also treats questions about the Lyapunov spectrumof infinite models .

Though the previous results, too, have had much to say to physics, thelast three papers of the first part are even more directly related to questionsfrom physics. Bunimovich' survey (Existence of transport coefficients) coversdevelopments on the existence of and formulae for transport coefficients: thediffusion constant, Green-Kubo formula, viscosity in HBS and the Lorentz gas.Liverani's survey (Interacting particles) treats a variety of models where beside

4 Introduction

the hard core interaction there is also a potential in the system: the particlesinteract via a potential or they are moving in a potential field. The first par tends with the research work of Lebowitz , Piasecki and Sinai (Scaling dynamicsof a massive piston in an ideal gas). It contains a brand new result on a verynice problem deeply related to the foundations of st atis tical mechanics, a perspectivistic subj ect of future surveys.

The second part of thi s collect ion corresponds to my second motivation forputting the surveys together: when the mathematical tools are developing fastthen it is even more appropriate to confront them with problems of physics .The first pair of works is devoted to the calculat ion of the Kolmogorov-Sinaient ropy and the Lyapunov exponents by analytic methods on one hand and andby computer simulat ions on the other hand . The first survey is written by vanBeijeren, van Zon and Dorfman (Kinetic theory est imates for the KolmogorovSinai entro py, and the largest Lyapunov exponents for dilute, hard-ball gasasand for dilute, random Lorentz gases) while the second one by Posch and Hirschi(Simulation of billiards and of hard-body fluids) .

Finally, the second pair of art icles in this part introduces the reader intoa new direction of research, into the theory of nonequilibrium stationary states .Here the central concepts are thermodynamic ent ropy, entropy production andirreversibility. In the the survey by Dettmann (The Lorentz gas: A paradigmfor nonequilibrium steady states) the main models are relat ed to Lorentz gases:random, periodic or thermostat ed ones. In that of Tel and Vollmer (Ent ropybalance, multi baker maps , and the dynamics of the Lorent z gas) calculat ionsare done for multib aker chains . The relation of these models to the Lorentz gasis also explained.

The appendix of the volume contains the reprint of a lecture of mine givenon the occasion of the 150th birthday of Ludwig Boltzmann. The reason forincluding it here was that its goal was to give a hist oric account of the problemof ergodicity from the point of view of both statist ical physics and mathemat ics,and, moreover, tha t it appeared in a less accessible periodical.

When a proj ect is accomplished, it is a pleasant duty to thank those whocontributed to the work. Though indirect ly, a fundamental role has been playedby Yasha Sinai who has not only been playing a determining, leading role in thewhole theory, but he and his group have helped a lot myself and my colleaguesin Budapest to be able to get an orient ation in the subj ect and to be ableto start thinking on its questions. I have already mentioned the series editors,whose idea this collection was. My special thanks are due to David Ruelle forhis permanent attention to the preparation of this volume. I am highly indebtedto Ruth Allewelt and Martin Peters for their editorial support in all editorialaspects of the job . The help and advice of my student and friend , Imre PeterToth (alias Mogyor6) , was indispensable in managing the manuscripts: he keptproducing readable texts supplied with the correc t figures from all kind of zip-,tar- , arj-, ... files. He also kept a home page for the manuscripts to facilitat e theinteraction among the authors and with the editor. Finally but, of course, not

Introduction 5

at the last I express my sincere thanks to the authors, who took their task mostseriously. I am convinced that their work will make it easier to start with the fieldfor students or even researchers who are motivated to learn this attractive topic .I also hope that it will further stimulate the interaction between mathematiciansand physicists interested in this area.

Budapest, July 4, 2000 Domokos SZ8sZ

1. Mathematics

A Geometric Approachto Semi-Dispersing Billiards!

D. Burago"; S. Ferleger" and A. Kononenko"

Abstract. This section contains a survey of a few results obt ained by a particular realization of V. Arnold 's old idea that hard ball models of st atisticalphysics can be "considered as the limit case of geodesic flows on negativelycurved manifolds (the curvature being concentrat ed on the collisions hypersurface)". The approach is based on representing billiard trajectori es as geodesicsin appropriate spaces. These spaces are not even topological manifolds: they arelengths spaces of curvat ure bounded above in the sense of A. D. Alexandrov.Nevertheless, this method allows to transforms a certain type of problems aboutbilliards into purely geometric stateme nts; and a problem looking difficult in itsbilliard clothing somet imes turns into a relatively easy statement (by the modernst andards of metric geomet ry). In particular , this approach helped to solve anold problem of whether the numb er of collisions in a hard ball model is boundedfrom above by a quantity depending only on the system (and thus uniform forall initial conditions).

1 We would like to express our sincere gratitude to M.Brin, N.Chernov, G.Galperin,M.Gromov, A.Katok and Ya.Pesin forvery helpfulcomments and fruitful discussions.

2 partially supported by a SloanFoundation Fellowship and NSFgrant DMS-98-051753 partially supported by NSF DMS-99-715874 partially supported by NSF DMS-98-03092

10 D. Burago, S. Ferleger and A. Kononenko

While the first ideas of hyperboli city of certain billiard systems go back toKrylov ([Kri]), the mathematical theory of semi-dispersing billiards originatedwith the works ofYa.Sinai ([Si-4], [Si-5], [Si-6]) in connection with the foundationsof st atisti cal physics and the study of the hyperbolicity and ergodicity propertie s ofsuch billiards. Since then the theory of semi-dispersing and dispersing (also calledscattering or Sinai) billiards has grown in various directions, including the st udy oftheir ergodicity prop erties ([Bu-Si-I] , [Bu-Li-Pe-Su], [Kr-Sim-Sz-I], [Kr-Sim-Sz-2],[Reh], [Si-5], [Si-Ch-I], [Sim-I], [Sim-2], [Sim-Wo]), the existe nce of stable andunstable manifolds, Markov partitions and other properti es relat ed to hyperbol icity ([Bu-Si-2], [Bu-Si-Ch-2], [Ch-3], [E~ , [Ka-St], [Le]), ent ropy and periodicorbits ([Bu-I]' [Ch-Ma], [Ch-I], [Ch-2] , [Ch-4], [Mo], [Si-I ]' [Si-Ch-2], [St-I ]' [St-2],[Wo]), various st at ist ical and symbolic prop erties, and limit theorems ([Bi], [Bu-2],[Bu-Si-Ch-I]' [Ch-5], [Gal-Or]' [Tr], [Yo]) ,quantum and other generalizat ions ([Be],[Do], [Do-Li]'[Dor-Sm], [Ha-Sh], [CdV], [Ve]),and manyothers (seealso [Si-3], [Si-7],[Ta], and [Ko-Tr] for reviews and more references) .

An informal idea that a semi-dispersing billiard system is somewhat similarto a geodesic flow on a negatively curved manifold has been around for quitea while. Perhaps it was first explicitly mentioned by V. Arnold . The purposeof this pap er is to give an informal and elementary account of a par ticularmethod of formalizing this idea. This pap er is a modified version of the survey[B-F-K-4] and the lecture [B]. More details can be found in [B-F-K-I],[B-F-K-2].The emphasis of thi s pap er is not on the rigorous proofs, which an interestedreader may find in the articles mentioned above, but rather on the demonstrationof the method, its power, and its limit ations.

The approac h is based on representing billiard t ra jectories as geodesics ina certain length space. T his represent ation is similar to turning billiard t rajectories in a square billiard table into st raight lines in a plane tiled by copies of thesquare. It is important to und erst and that this const ruct ion by itself does notprovide new information regarding the billiard system in question; it only converts a dynamical problem into a geomet ric one. Nevert heless, while a problemmay seem rather difficult in its billiard clothing, its geometric counterpart mayturn out to be relat ively easy by the standards of the modern metric geometry.For the geometry of non-positively curved length spaces we refer to [Ba], [Gr-I ]and [Re].

Apparently, one of the motivations to study semi-dispersing billiard syst emscomes from gas models in stat ist ical physics. For inst ance, the hard ball modelis a syst em of round balls moving freely and colliding elast ically in a box orin empty space. Physical considerations naturally lead to several mathemat icalproblems regarding the dynamics of such systems. The problem that served asthe starting point for the research discussed in this paper asks whether thenumber of collisions in tim e one can be est imated from above. Another wellknown and st ill unsolved problem asks whether such dynamical systems areergodic. A "physical" version of both problems goes back to Boltzman , whiletheir first mathematical formulation is probably due to Ya. Sinai .

A Geometric Approach to Semi-Dispersing Billiards 11

Making a short digression here, we would mention that , in our opinion, theadequacy of these model problems for physical reality is quite questionable.In par ticular , these problems are ext remely sensitive to slight changes of theirformulat ions. Introducing par t icles that are arbit rarily close in shape to theround balls and that are allowed to rotate, one can produce unbound ed numberof collisions in unit time [Va]. It is plausible that int rodu cing even a symmet ricaland arbit rarily steep potential of interaction between particles instead of discontinuous collision "potent ial", one can destroy the ergodicity ([Do]). The result ofSinmani and Szasz ([Si-Sz])(seemingly, the best one can prove in support of theergodicity of the hard balls model in the present state of the art) assert s thatthe ergodicity does take place ... for almost all combinations of radii and massesof the balls. Such a result should be less than satisfacto ry for a physicist, sincea statement that is valid only for "balls of irrational radii" does not make anyphysical sense at all. Perhaps, one would rather hope that the existence of anergodic component whose complement is negligibly small (at least for a system ofvery many balls) is a more stable property. On the other hand, hard ball gas (ofeven very many small balls) in a spherical or a cylindrical vessel is obviously verynon-ergodic since it possesses a first integral coming from rotational symmetriesof the syste m. This happ ens regardless of a good deal of hyperbolicity producedby the dynamics of colliding balls, and it is not at all clear what happ ens if thesymmet rical shape of the vessel is slight ly perturbed.

Regardless of this minor crit icism of the physical meaning of mathematicalproblems involving gas models, the aut hors believe that these problems are quiteinterest ing on their own, and from now on we st ick to their mathematical set-up.It is well known that , by passing to the configuration space , the dynamics ofN balls can be substitute d by the dynamics of one (zero-size) particle moving inthe complement of several cylinders in R 3N and experiencing elast ic collisionswith the cylinders. These cylinders correspond to the prohibited configurat ionswhere two of the balls intersect . Another gas model, the Lorentz gas, just beginswith a dynamical system of one particle moving in the complement of a regularlatt ice of round scat terers; its dynamics can be studied on the quotient space,which is a torus with a scatterer in it. All these example fit in the followinggeneral scheme.

Let M be a complete Riemannian manifold M together with a (finite or atleas t locally-finite) collection of smooth convex subsets Bi , These convex sets B,are bounded by (smooth, convex) hypersurfaces Wi, which (together with B j's)will be referred to as walls. In most physical models, M is just a flat torus orEuclidean space (whose Euclidean st ructure given by the kinetic energy of thesyste m.) Throughout this paper we assume that M has non-positive curvatureand positive injectivi ty radius; however , local uniform bound s on the numb erof collisions remain valid without these restrictions. T he dynamics takes placein the (semi-dispersing) billiard table, which is the complement of UB, in M .More precisely, the phase space is (a subset of) the unit tangent bundle to thiscomplement. A point moves along a geodesic unt il it reaches one of the walls Wi,


and then it gets reflected so that both the magnitude and the proj ection of itsvelocity on the plane tangent to the wall are conserved. For simplicity, we excludethe t rajectories that ever experience a collision with two walls simultaneously.Systematic mathematical study of such systems, called semi-dispersing billiards,was init iated by Ya. Sinai and cont inued by many other mathematicians andphysicists.

For inst ance, consider a system of hard balls moving in a non-positivelycurved Riemannian manifold Q (with Q being a Euclid ean space as the leadingexample). Its dynamics is isomorphic to the dyn amics of a cert ain billiard in theconfigurat ion space QN (in which every ball is represent ed by its center) whichis endowed with a Riemannian metri c p,

N

p((X1 , '" ,Xn)' (Y1" " ,Yn )) = (L m iP(Xi ,Yi )2)1/2.i=l

Notice that , providing that P is a metric of non-positive curvature, p is a metricof non-positive curvature as well. The corresponding billiard is defined in thecomplement B of N(~- l ) bodies Bm ,l , each of which corresponds to a pair ofballs. Namely, for every m ,1= 1, . . . , N ,m i- I :

Every such body Bm ,l is isometric to a product of Q N-2 with a convex set inQ2 and, thus, is convex too.

Our discussion will be concentra ted around the idea of gluing several copiesof M together and then developing billiard trajectories into this new space. Thisidea is very old and its simplest versions arise even in elementary high-schoolmathematical puzzles. For inst ance, if the billiard table is a square, one canconsider a tiling of Euclidean plane by such squares, and billiard traje ctoriesturn into straight lines. Although thi s idea is rather naive, it already providesvaluable information. For inst ance, if one wonders how close a non-periodictrajectory comes to vert ices of the square, the answer is given in te rms ofrational approximat ions to the slope of the corresponding line. In this instance,a dyn amical problem is t ransformed into a question in the arithmet ic of realnumbers. It has been known for a long t ime (see, for example, [Ze-Ka]) that thedynamics of a billiard in a rational polygon may be viewed as the geometry ofits unfolding surface. Even in this simple situation the unfoldin g is not quitea Riemannian surface since in all but few cases its metric is bound to havesingularities (and away from the singularities the metric is flat) . This object ,however, is not at all pathological from the point of view of non-regular Riemannian geomet ry.

We are concerned with semi-disp ersing billiard syst ems. In the early sixti esV. Arnold "speculated" that "such systems can be considered as the limit case ofgeodesic flows on negatively curved manifolds (the curvature being concent rated


on the collisions hypersurface)" [Ar]. Indeed, this is nowadays well known (dueto the works of Sinai, Bunimovich, Chernov , Katok, St relcyn, Szasz, Sinmaniand many ot hers) that a large portion of the results in the smoot h theory of(semi-)hyperbolic systems can be generalized (with appropr iate modifications)to (semi-)dispersing billiards. In spite of this, the const ruction suggested byArnold has never been used. It also caused several serious objections; in particular , A. Kat ok pointed out that such approximations by geodesic flows onmanifolds necessarily produce geodesics that bend around collision hypersurfaces and therefore have no analogs in the billiard syste m.

To st udy the billiard flow for a fixed tim e and in a small neighborhood ofa fixed point , one can use doubling by taking two copies of M , removing theinteriors of the walls and then gluing the copies along the boundaries of thewalls. One can approximate the singular metric resulting from this procedureby smooth metrics (analogously one subst itutes hard collisions by a very steeprepelling potential) . The geodesic flows of the resulting metri cs will naturallyconverge to the billiard flow on a fixed tim e interval in a small neighborhood ofeach point.Although the approximating metrics will have certain directions withpositive sect ional curvat ure , one easily sees that these directions never containvelocity vectors of geodesics originating from the neighborhood in question .Even though this const ruct ion does not seem very useful, it already can delivercertain information. For instance, the Liouville theorem for billiard flows followsimmediat ely from the Liouville theorem for geodesic flows, as well as the factthat the billiard flow is symplectic in a sufficiently small neighborhood of everypoint where it is defined.

To illustrate both Arnold 's suggest ion and the difficulty not iced by Katok,let us consider a simple example of the billiard in the complement of a disc ina two-torus (or Euclidean plane). Taking two copies of the torus with (open)discs removed and gluing them along the bound ary circles of the discs, oneobtains a Riemannian manifold (a surface of genus 2) with a metri c singularityalong the gluing circle. Thi s manifold is flat everywhere except at this circle. Onecan think of this circle as carrying singular negative curvature . Smoothing thismetr ic by changing it in an (arbit rarily small) collar around the circle of gluing,one can obtain a non-positively curved metric, which is flat everywhere exceptin this collar. To every segment of a billiard trajectory, one can (canonically)assign a geodesic in this metric. Collisions with the disc would correspond tointersections with the circle of gluing, where the geodesic leaves one copy of thetorus and goes to the other one.

Unfortunat ely, many geodesics do not correspond to billiard trajectories.They can be described as coming from "fake" trajectories hit tin g the disc atzero angle, following an arc of its boundary circle (possibly even making severalrounds around it ) and then leaving it along a tangent line. Dynamically, suchgeodesics carry "the main portion of ent ropy" and they cannot be disregarded.On the other hand, it is difficult to tell act ual trajectories from the fake oneswhen analyzing the geodesic flow on this surface.


There is another difficulty arising in higher dimension. If one tries to repeatthe same construction for a three-torus with a ball removed, then after gluingtwo copies of this torus the gluing locus defines a totally geodesic subspace. Itcarries positive curvature, and this positive curvature persists under smoothingof the metric in a small collar of the sphere. Thus, in this case we do not geta negatively curved manifold at all.

We will (partially) avoid these difficulties by subst itut ing a non-p osit ivelycurved manifold by a length space of non-positive curvat ure in the sense ofA.D. Alexandrov. The foundations of the theory of length spaces of boundedcurvat ure were developed by A.D.Alexandrov and his collaborat ors (see [AI],[Al-B]' [AI-St], [AI-Za], [Re]) in the mid 60's. Since then it has att ractedthe attention of many leading geometers. The main object in the theory isa length space, that is a metric space where the dist ance between two points isprovided by the length of a shortest path connect ing the points . The main ideaof the definition of non-positive curvat ure is the following observation. The wellknown comparison theorems of Alexandrov and Toponogov show that there isa way to est imate the sectional curvature of a Riemannian manifold from abovesimply by comparing the geodesic t riangles on the manifold and in a modelspace (a complete simply connected surface of constant curvature ). However ,since the procedure involves measurement of certain dist ances only, it may beconsidered a definition of a space, whose curvature is bounded from above. Thisdefinition coincides with the usual one in the category of smoot h Riemannianmanifolds , but in fact makes sense for an arbitrary geodesic space. For furtherdetails see, for instance, [Ba], [Gr-1] and [Re].

Unfortuna te ly, a construction that would allow us to represent all billiardt raj ectori es as geodesics in one compact space is unknown in dimensions higherthan three. Attempts to do this lead to a striking open question: Is it possibleto glue finitely many copies of a regular 4-simplex to obtain a (bound ary-less)non-positive pseudo-manifold (d . [B-F-Kl-K])?

We introduce a construct ion that represents t raj ectori es from a certain combinatori al class, where by a combinatorial class of (a segment of) a billiardtrajectory we mean a sequence of walls that it hits.

Fix such a sequence of walls K = {Wn ; , i = 1,2, . .. N }. Consider a sequence{Mi , i = 0,1 , . .. N } of isometric copies of M. For each i, glue M, and MH 1along En;. Since each En; is a convex set, the resulting space MK has the sameupp er curvature bound as M due to Reshetnyak 's theorem ([Re]).

There is an obvious proj ection MK -+ M , and M can be isometricallyembedded into MK by identifying it with one of Mi's (regarded as subsetsof MK ) . Thus every curve in M can be lifted to MK in many ways. A billiardtrajectory whose combinatorial class is K admits a canonical lifting to M K : welift its segment till the first collision to Mo C M K , the next segment betweencollisions to M 1 C M K and so on. Such lift ing will be called developing of thetrajectory. It is easy to see that a development of a trajectory is a geodesicin M K .


Note that, in addition to several copies of the billiard table, M K containsother redundant parts formed by identified copies of Bi 's. For example, if westudy a billiard in a curved triangle with concave walls, Bi's are not the boundary curves. Instead, we choose as Bi's some convex ovals bounded by extensionsof these walls. (One may think of a billiard in a compact component of thecomplement to three discs.) In this case, these additional parts look like "fins"attached to our space (the term "fin" has been used by S. Alexander and R.Bishop in an analogous situation) . In case of the billiard in the complement ofa disc in a two-torus (see discussion above), the difference is that we do notremove the disc when we glue together two copies of the torus. Now a geodesiccannot follow an arc of the disc boundary, as the latter can be shorten by pushinginside the disc. Still, there are "fake" geodesics, which go through the disc.However, there are fewer of them than before and it is easier to separate them.

It might seem more natural to glue along the boundaries of Wn ; rather thanalong the whole Bn ; . For instance, one would do so thinking of this gluing as"reflect ing in a mirror" or by analogy with the usual development of a polygonalbilliard. However, gluing along the boundaries will not give us a non-positivelycurved space in any dimension higher than 2.

One may wonder how the interiors of Bi's may play any role here, as they are"behind the walls" and billiard trajectories never get there. For instance, insteadof convex walls in a manifold without boundary, one could begin with a manifoldwith several boundary components, each with a non-negative definite secondfundamental form (w.r.t. the inner normal). Even for one boundary component,this new set-up cannot be reduced to the initial formulation by "filling in" theboundary by a non-positively curved manifold. Such an example was pointedout to us by J . Hass ([Ha]), and our main dynamical result does fail for thisexample . Thus, it is indeed important that the walls are not only locally convexsurfaces, and we essentially use the fact that they are filled by convex bodies .

Let us demonstrate how the construction of M K can be used by first reproving (and slightly generalizing) a known result . 1. Stoyanov has shown thateach combinatorial class of trajectories in a strictly dispersing billiard (in Euclidean space or a flat torus) contains no more than one periodic trajectory.By a strictly dispersing property we mean that all walls have positive definitefundamental forms. For a semi-dispersing billiard, L. Stoyanov proved that allperiodic trajectories in the same combinatorial class form a family of paralleltrajectories of the same length. Together with local bounds on the number ofcollisions (which were known in dimension 2, and the general case is discussedbelow), these results imply exponential upper bound on the growth of the number of (parallel classes of) periodic trajectories. These estimates are analogousto the estimates on the number of periodic geodesic in non-positively curvedmanifolds.

Assume that we have two periodic trajectories in the same combinatorialclass K . Choose a point on each trajectory and connect the points by a geodesicsegment [xy]. Let us develop one period of each trajectory into M K , obtaining


two geodesics [x'x"] and [y'y"] connected by two lifts [x'y'J and [x"y"] of thesegment [xy]. Thus, M K contains a geodesic quadrangle with the sum of angles equal to 27r. It is well known that, in a non-positively curved space, sucha quadrangle bounds a flat totally-geodesic surface; in our case it has to bea parallelogram since it has equal opposite angles . Thus, Ix'x"l = ly'y"l andthe family of lines parallel to [x'x"] and connecting the sides [x'y'J and [x"y"]projects to a family of periodic trajectories. Moreover, this parallelogram hasto intersect the walls in segments, and thus it is degenerate if the fundamentalforms of the walls are positive definite . This just means that the two periodictrajectories coincide. The same is true if the sectional curvature of M is strictlynegative, as it is equal to zero for any plane tangent to the parallelogram.

This argument is ideologically very close to the proof of the following result :the topological entropy of the time-one map T of the billiard flow for a compactsemi-dispersing non-degenerate billiard table is finite . Note that the differentialof the time-one map T is unbounded, and therefore the finiteness of the topological entropy is not obvious . Moreover, it is quite plausible that the followingproblem has an affirmative solution: if one drops the curvature restriction for M ,can the topological entropy of the time-one map be infinite? Is the topologicalentropy of the billiard in a smooth convex curve in Euclidean plane always finite?

To estimate the topological entropy by h, it is enough to show that , givena positive E, there is a constant C(E) with the following property: for each N ,the space of trajectories Ti(v), i = 0,1, . . . , N can be partitioned into no morethan C(f) . exp(hN) classes in such a way that every two trajectories from thesame class stay e-close to each other.

At first glance, such a partition seems rather evident in our situation. Indeed,first let us subdivide M into several regions of diameter less than e (the numberof these regions is independent of N) . If M is simply connected, we can just saythat two trajectories belong to the same class if they have the same combinatorial class and both trajectories start from the same region and land in the sameregion of the subdivision of M. If M is not simply-connected, one also requiresthat the trajectories have the same homotopy type (formally, lifting two corresponding segments of the flow trajectories of the same combinatorial class K toM K and connecting their endpoints by two shortest path, one gets a rectangle;this rectangle should be contractible) . Since both the number of combinatorialclasses and the fundamental group of M grow at most exponentially, is rathereasy to give an exponential (in N) upper bound on the number of such classes(using again the local uniform estimates on the number of collisions, see below) .On the other hand, for two trajectories from the same class, their developmentsinto the appropriate M K have e-close endpoints and the quadrangles formed bythe geodesics and the shortest paths connecting their endpoints is contractible.For a non-positively curved space , this implies that these geodesics are e-closeeverywhere between their endpoints.

There is, however, a little hidden difficulty, which the reader should be awareof. The previous argument proves the closeness between the projections of two


trajectories onto M, while we need to establish this closeness in the phase space.Thus, some ext ra work has to be done to show that if two geodesics in M K staysufficiently close, then so do the directions of their tangent vectors (in somenatural sense) . Thi s is a compactness-type argument, which we will not dwellupon here.

Let us come back to the example used above to illustrate Arnold 's suggestion.This is 2-dimensional Lorent z gas, tha t is the billiard in the complement ofa disc in a flat two-torus. To count the numb er of classes in the above sketchof the argument, one can pass to an Abelian cover of MK (since this billiardtable has just one wall, there is no ambiguity in choosing K) . The latter istwo copies of Euclidean plane glued together along a lattice of discs centered atinteger points. A (class of) billiard t rajectories naturally determines a brokenline with integer vertices. While not every broken line with integer verticesarises from a billiard trajectory, the portion of such lines coming from "fake"trajectories approaches zero for small radii of scatterers. Counting such brokenlines is a purely combinatorial problem, and one sees that the topological entropyof Lorentz gas converges to a number between I and 2 as the radius of therepeller approaches zero. Thi s result is stable: the "limit ent ropy" is the samefor a convex repeller of any shape. The author has no idea whether this numb erhas any physical meaning.

Now we pass to the main problem of estimating the number of collisions.This problem was first posed by Sinai, who also gave a solut ion [Si-2] for billiards in polyhedral angles. The existence of such est imates is related to variousprop erties of a billiard syst em. For example, Sinai-Chernov formulas [Ch-2],[Si-I] for the metric entropy of billiards are proved und er the assumpt ion thatsuch an est imate exists.

For the hard ball system, one asks whether the number of collisions thatmay occur in this syst em can be est imated from above by a bound dependingonly on the numb er of balls and their masses. If we consider the balls moving inunbounded Euclidean space, we count the total numb er of collisions in infinitetim e. For a system of balls in a box, we mean the numb er of collisions in unittim e (for a fixed value of kinetic energy). As far as we know, these problems havebeen resolved only for syst ems of three balls (see [Sa-Th] which st at ed that therecould be four collisions, [Co-Mu] for a proof that four is the maximum numberof collisions in two or more dimensions) .

It is relatively easy to establish such upp er bounds on the number of "essential" collisions, opposed to collisions when two balls barely tou ch each other.While such "non-essential" collisions indeed do not lead to a significant exchangeby energy or momentum, they nevertheless cannot be disregarded from a "physical viewpoint" . Indeed, they may serve as the main cause of instability in thesystem: the norm of differential of the flow does not admit an upp er boundjust at such trajectori es. In a general semi-dispersing billiard it is also easier toestimate the numb er of collisions that occur at an angle separated from zero.Such arguments are based on introducing a bounded function on the phase space


so that the function does not decrease along each trajectory and increases by anamount separated from zero after each "essential" collision. For some cases, suchas 2-dimensional and polyhedral billiard tables, one can estimate the fractionof "essential collisions" among all collisions and thus get uniform bounds onthe total number of collisions (see [Va], [Ga-l]' [Ga-2]' [Si-2]) . The simplest casethat is unclear how to treat by such methods is a particle shot almost along theintersection line of two convex surfaces in 3-dimensional Euclidean spaces andhitting the surfaces at very small angles .

Contrary to dynamical arguments indicated above, we use a geometric approach based on some length comparisons. Let us first prepare the necessarynotation and formulations. When one wants to obtain uniform bounds on thenumber of collisions for a general semi-dispersing billiard table, it is clear that anadditional assumption is needed. Indeed, already for a two-dimensional billiardtable bounded by several concave walls, a trajectory may experience an arbitrarily large number of collisions (in time one) in a neighborhood of a vertex if twoboundary curves are tangent to each other. Thus, a non-degeneracy conditionis needed.

Let us give a formal definition :A billiard B is non-degenerate in a subset U c M (with constant C > 0),

if for every I C {I, .. . ,n} and for every y E (UnB)\(njEI B j) ,

dist(y, njEI Bj )-----"--=-,---..,.. 5: C,maxkEI dist(y, B k )

whenever n j E1 Bj is non-empty.A billiard B is called non-degenerate at a point x E B with constant

C if it is non-degenerate in a neighborhood of x with the same constant, andlocally non-degenerate with constant C if it is non-degenerate at every pointwith constant C.

We will say that B is non-degenerate if there exist IS > 0 and C > 0 suchthat B is non-degenerate, with constant C, in any IS-ball.

Roughly speaking, the condition means that if a point is d-close to all thewalls in I then it is Cd-close to their intersection. Formulated this way, it is veryeasy to verify in many important cases, including the hard-ball gas models. Thiscondition is always satisfied for a system of hard balls in empty space (whereas,other natural conditions are known to fail, for example, the condition that thenormals to the walls be in general position) . For a system of balls in a jarwith concave walls our non-degeneracy condition is satisfied except for somespecial sets of radii, when it is possible to "squeeze the balls tightly betweenthe walls." Actually, it is known that in those situations the system may havearbitrarily many collisions locally. In order to acquire some geometric insight ,we notice that the condition is equivalent to the following geometric property:there exists a positive r such that, at every point, the unit tangent cone toB (which is a subset of the unit sphere in the tangent space to M) contains

A Geometr ic Approach to Semi-Dispersing Billiards 19

a ball of radius T. For flat M this means that every point of B is a vertex ofa round cone of radius T which ent irely belongs to B in some neighborhood of itsvertex. As far as we know, "the cone condit ion" was first formulated by Sinai.For compact billiard tables, these definitions can also be reformulat ed in thefollowing way: the operations of taking tangent cone and intersection commutefor any collection of the complements to the walls Bi . For non-compact tables,however , this definition guarantees the non-degeneracy at all points, but theconstant C may deteriorate and have no positive lower bound .

The main local result reads as follows: if a semi-dispersing billiard tablesatisfies the non-degeneracy assumpt ion, then there exists a finite number Psuch tha t every point p in the billiard table possesses a neighborhood U(p) suchth at every trajectory segment contained in U(p) experiences no more than Pcollisions.

Passing to estimating th e global numb er of collisions (for infinite time) wewant to stay away from situations such as a particle infinitely bouncing betweentwo disjoint walls. The result for this case reads as follows: if a semi-dispersingbilliard table sat isfies th e non-degeneracy assumption, M is simply-connectedand the intersection nB, of Bi's is non-empty, then there exists a finite numberP such that every trajectory experiences no more than P collisions.

Applying this result for hard ball gas system (together with calculat ingthe corresponding constants , including checking the non-degeneracy condit ionand finding its constant) one gets the following result: The maximal numb erof collisions that may occur in a syste m of N hard elast ic balls (of arbit rarymasses and radi i) moving freely in a simply connected Riemanni an space M ofnon-posit ive sectional curvature never exceeds

where m m ax and m m i n are, correspondingly, the maximal and the minimalmasses in the system.

Thi s result was first established for JRk in [B-F-K-l]. The results of [B-F-K-3]allowed us to extend it to manifolds of non-positive curvature, and to get rid ofthe dependence on the radii that was present in [B-F-K-l] .

Let us demonstrat e how the non-degeneracy constant can be est imated fora hard ball system. We use notations introduced for this system in the beginingof the paper , where its dynamics is represented by a billiard table B in theproduct QN ,

Now we will check the uniform non-degenera cy condit ion for B .Fix a set of walls I , and let 10 = [rn](m,I) E I} . Consider an arbitrary point

Xo = (Cl,' .. ,CN) E Q N\(U(m,I)EI B m,L) and let d = maX(m,I)EI p(Xo, B m,L). Ourgoal is to est imate p(Xo, n (m,I )EI B m,L) via d.

In order to do that, let us apply the following procedure: pick some m l E

10 and move all the balls Bm , m E 10\ {mIl simultaneously and with equal

velocit ies along the geodesics in Q, connecting the centers of B m with the cent erof Bm!, until every pair of balls Bm!, Bm such that (m! ,m) E 1 intersect (if thecenter of one of the balls Bm reaches the center of Bm ! , we stop moving it anyfurther , and continue to move the other balls) . As a result, we obt ain a pointXl E M N . Since we never have to move any ball in M more than J, we havep(XO,X1 ) ::; MNJ. On the other hand , for every two geodesics 1'1 ,1'2 in thesimply connected space Q of non-positive curvature the function p(1'1(t), 1'2(t))is convex. Therefore, dist ances between any pair of the balls will not increase,so that we still have maX(m,i)EI p(X1 , Bm,i) ::; J.

Next , we apply the same procedure to some m 2 E 10\ {m1} , obtaining a pointX 2 E QN such that p(Xl, X 2) < M N J, etc . By construction, the last point

X IIol E n (m,l)EI Bm,i and p(Xo, X IIol) ::; I:l~~-l p(Xi , Xi+d < M N 2J. Therefore, it is shown that B is non-degenerate in the whole QN, with the const antMN2

.

To outline the idea of the proofs of uniform est imates on the numb er ofcollisions we rest rict ourselves to the case of two walls W 1 and W2 bounding twoconvex sets B 1 and B 2 • Thus we avoid inessential combinatorial complicationsand cumbersome indices.

We begin by discussing the local bound. Let us assume that M is simplyconnected; otherwise, one can pass to its universal cover. Consider a billiardtrajectory T connect ing two points x and y and pick any point z E B 1 nB2 .

Denote by K = {W1 , W2 , W1 , W2 . . . } the combinatorial class ofT, and considerthe development T' of T in MK . Thi s is a geodesic between two points x' and y' .By Alexandrov's theorem, every geodesic in a simply-connected non-positivelycurved space is the shortest path between its endpoints . Note that z canonicallylifts to M K since all copies of z in different copies of M got identified. Denotingthi s lift by z', we see that Izxl = Iz'x' i and Izyl = Iz'y'l. Thus we conclude thatthe lengths of T between x and y is less that Ixzl + Izyl for all z E B 1 nB 2 .

In other words, any path in M connect ing x and y and visiting the intersect ionB 1 nB2 is longer than the segment of T between x and y.

The following argument is the core of the proof. It shows that if a t raj ectory mad e too many collisions then it can be modified into a shorte r curvewith the same endpoints and passing through the inters ection B 1 nB2 . Thiscont radicts the previous assertion and thus gives a bound on the number ofcollisions.

Assume that T is contained in a neighborhood U(p) and it collided withW1 at points a1,a2, . .. aN alt ernating with collisions with W2 at b1,b2, .. · bN·Let z, be the point in B 1 nB2 closest to b, and hi be the distance from b,to the shortest geodesic [aiai+l]' By the non-degeneracy assumpt ion, IZibil ::;C ·dist(bi , B1) ::; hi' Thus the dist ance Hi from z; to the shortest geodesic aiaiHis at most (C + l)hi .

Plugging this inequality between the heights of the tri angles aibiai+l andaiziai+1 into a routine argument which develops these tri angles on both Eu-


clidean plan e and k-plane, one concludes that di :::; C1 . Di, where di = laibil +Ibiaa+ll - laia i+ll , D, = laizi I+ IZi aa+l 1- laiai+ll · Here k is the infinum of thesectional curvature in U(p) , and a constant C1 can be chosen dependin g on Calone provided that U(p) is sufficiently small.

Let dj be the smallest of di's. Let us modify the trajectory T into a curvewith the same endpoints : substitute its pieces aibiai+l by the shortest segmentsaiai+l for all i's excluding i = j . This new curve is shor ter than T by at least(N - l)dp • Let us make a final modification by replacing the piece ajbja j+lby aj zjaj+l ' It makes the path longer by D j , which is at most C1dj . Hence,N :::; C1 + 1 because otherwise we would have a curve with the same endpointsas T , passing through Zj E B 1nB 2 and shorter than T . This proves the localbound on the numb er of collisions.

Now we are ready to est imate the global numb er of collisions, and heregeometry works in its full power. Consider a trajectory T making N collisionswith the walls K = {I , 2, 1, .. . , 2, I} . Reasoning by cont radiction, assume thatN > 3P +1, where P is the local bound on the numb er of collisions. Consider thespace MK and "close it up" by gluing Mo E M K and M N E MK along the copiesof B 1 • Denote the resulting space by M. We cannot use Reshetnyak 's theoremto conclude that M is a non-positively curved space any more , since we ident ifypoints in the same space and we do not glue two spaces along a convex set .

We recall that a space has non-positive curvat ure iff every point possessesa neighborhood such that , for every tri angle contained in the neighborhood, itsangles are no bigger than the corresponding angles of the comparison tri anglein Euclidean plane. However , using the correspondence between geodesics andbilliard trajectori es, one can conclude (reasoning exact ly as in the proof of thelocal estimates on the numb er of collisions), that each side of a small t rianglecannot inters ect interiors of more than P copies of the billiard table. SinceN > 3P + 1, for every small tri angle for which we want to verify the anglecomparison prop erty, we can undo one of the gluings without tearing the sidesof the tri angle. Thi s ungluing may only increase tri angle's angles, but now wefind ourselves in a non-p ositively curved space (which is actually just M K ) , andthu s we get the desired comparison for the angles of the tri angle.

To conclude the proof, it remains to notice that the development of T in M isa geodesic connect ing two points in the same copy of B 1 . This is a contradict ionsince every geodesic in a simply-connected non-positively curved space is theonly shortest path between its endpoints; on the other hand, there is a shortestpath between the same points going inside this copy of B1 .

Let us notice that all our methods and result s remain valid even if we dropthe assumpt ion that the boundaries of B, are hyper-surfaces. Of course, in thiscase we have to change the definit ion of the outcome of a collision appropriately:it would not be uniquely defined any more , and we would require only the conservation of the tangential component of the velocity. In particular, est imates onthe numb er of collisions hold for singular trajectori es as well (i.e., the trajectori esthat enter the intersections of several bodies and reflect in arbit rary directions

22 D. Burago , S. Ferleger and A. Kononenko

preserving the component parallel to the tangent space of the intersection ofthe bodies at the point of collision). Thi s also allows us to apply our resultsto particle syst ems, i.e., billiard systems of several balls of various masses andradi i where some (mixed system) or all (pure system) of the balls may have zeroradii (particles). In such syst ems multiple simultaneous collisions are allowed,as well as collisions with the intersections of several boundary components (fordetailed definitions see [Se-Va], which also generalizes estimat es of [Ga-2] forpure particle syst ems from the I-d imensional case to higher dimensions.) Inparticular, the est imate on the numb er of collisions in a hard ball syst em holdsfor arbit rary particle systems (with exact ly the same estim at e) .

We conclude the paper with formulat ions of two open question, which areclosely relat ed to the results mentioned above. It would be desirable if onecould begin with finitely many copies of M and glue them together along wallsB, to obt ain a non-positively curved space if so that each wall participat esin at least one gluing. In par ticular , such a const ruction would immediat elyprovide an alte rnat ive proof for both local and global estimates on the numb erof collisions. For instance, for global est ima tes it is enough to not ice that everybilliard trajectory lifts to a shortest path and hence it cannot intersect a copyof one wall in if more than once. Hence the numb er of collisions is bounded bythe tot al numb er of copies of walls in if . As it is ment ioned above, it is howeverunclear whether such gluing exists even for a regular 4-simplex. Thi s leads tothe following problem:

Question 1. Is it possible to construct a compact CAT(O) boundarylesspseudo-manifold by gluing together a fin ite number of copies of a given polyhedron S along the isome tric faces? It is quite possible that the answer to thequestion is negati ve for sufficiently high dimensions .

Our est imates for topological entropy lead to the following problem:Question 2. What can be said about the topological, or even the metric

entropy of degenerate compact semi -dispersing billiards , or of the billiards onmanifolds without the non-positive curvature restric tion? In parti cular, can itbe infini te? The question is open even for degenerat e semi-dispersing billiardsin Euclid ean space. We strongly suspect that the introduction of even arbitrarily small amounts of posit ive curvature into a billiard on the Euclidean planemay produce a billiard with infinite topolo gical ent ropy, which would be a nicedemonstration how posit ive curvature can force the entropy to become infinite.

A Geometric Approach to Semi-Dispersing Billiards

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23

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[B-F-K-2]

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A. Zemlyakov, A. Katok. Topological tr ansi tivity of billiards in polygons. Math . Notes 18 , 760-764, 1975.

On the Sequences of CollisionsAmong Hard Spheres in Infinite Space

T . J. Murphy and E. G. D. Cohen

Contents

§1. Historical Int roduct ion and Motivation 311.1 "Noninte racting" Collisions. . . . 311.2 "Divergent" Terms. . . . . . . . . 33

§2. The Dynamical Problem of n Hard Spheres . 332.1 Dynamics of Three Identical Hard Spheres 342.2 The Special Case of Motion of n Part icles in One Dimension 40

§3. Open Questions . . . . . . . . . . . . . . . 45Appendices . . . . . . . . . . . . . . . . . . . . 46

A. Proof That for Any n Hard Spheres ,n(n - I}/ 2 Collisions Can Occur . . . . . . . . . . . . . . . . .. 46

B. Initial Condit ions Leading to Four CollisionsAmong Three Identical Par ticles . . . . . . .. 46

C. Proof That for n Hard Rods of Equal Mass,No More Than n(n - 1}/ 2 Collisions Can Occur 47

References 48

A bstract . Ludwig Boltzmann's kinetic equat ion for dilute gases involves knowledge of the dynamics of an isolated pair of part icles. Attempts to generalize thisequation to higher densities necessarily involve knowledge of the collective dynamics of groups of more than two particles. These attempts therefore give riseto the following mathematical quest ion: For the par ticularly simple case of hardspheres, where only two-part icle collisions occur , what is the nature of the sequences of those collisions that can occur in infinite space? In par ticular , is therea maximum number of collisions among a given number n of hard spheres?

30 T . J . Murphy and E.G. D. Cohen

A survey is given of the main results obtained so far:1) The maximum number of collisions among n hard spheres is bounded.2) The maximum numb er of collisions among three identical hard spheres is

four , except in the one-dimensional case. The possible sequences are given andthe proof outli ned.

3) T he maximum number of collisions among any three hard spheres constrained to move in one dimension is given in terms of their masses. The collisionsequences and the ult imate velocit ies of the par t icles are explicit ly given in termsof their initial velocit ies.

On th e Sequ ences of Collisions Among Hard Spheres in Infin ite Space 31

§l. Historical Introduction and Motivation

Since World War II a great number of attempts have been made to systematically generalize the Boltzmann equation for dilute gases to higher densit ies [1 , 21 .While the Boltzmann equation derived in 1872 requires knowledge only of thedynamics of an isolated pair of par ticles in infinite space, th e attempted generalizat ion tried to incorporat e in a systemat ic way the contri butions of isolatedgroups of three, four, ... par ticles. Thi s was motivat ed in par t by analogy tothe equilibrium case. For a gas in thermal equilibrium, the successive terms inexpansions of the thermodynamic prop erties in powers of the density (virial expansions) are obtained using cluster expansions [3]. Th ese expansions depend oncombinations of the (stat ic) Boltzmann factors of isolated groups of an increas ingnumb er of particles. Thi s could be formally car ried over to the nonequilibriumcase by generalizing the equilibrium cluster expansions in te rms of Boltzmannfactors to cluster expansions in te rms of dynam ical evolut ion operators of isolat ed groups of an increasing numb er of particles. The dynamical nature of theseevolution operators then leads naturally to the quest ion: what is the collisionalbehavior in the course of t ime of isolated groups of identical par ticles in infinite space? The question makes mathemati cal sense only when collisions amongparticles can be defined; thi s requires that the interactions among any group ofparticles have a finite range.

The simplest case is that of hard sphere interactions, where the only inte ract ion is an infinitely large repulsion between any two par ticles at distances lessth an the sum of their radii . In the present paper, we will restrict ourselves tothis hardsphere case.

Before we formulat e more precisely the mathemati cal problem we will treat,we have to say a few words about the general nature of the above ment ionedclust er expansions. For all detail s, we refer to the literature 12, 41.

In th e examination of many-part icle problems, the cluster expansions attempt to express th e macroscopic propert ies of an N-particle system in a systematic way in terms of the properties of small groups of an increasing numb erof particles. T he hope is then th at the first few terms in such an expansion,involving a relat ively small number n « N of particles, will give a good approximat ion to the physical behavior of the many particle system.

1.1 "Noninteracting" Collisions. The nature of the various terms inthese cluster expansions is such that the n-p article clust er term contains notonly th e contribut ions associat ed with the dynamics of n particles, but alsocorrections for contribut ions in the cluster terms for fewer than n particles,which erroneously ignore true n-p article effects.

A simple illustration of this is the well-known expression for the second virialcoefficient , which gives the first correct ion to the ideal gas law for the pressureof a dilute gas in equilibrium: [4]

32 T . J . Murphy and E. G. D. Cohen

(2)

Here the pressure of the gas p is in first approximat ion given by the idealgas Boyle-Gay-Lussac law nkBT, where n is the numb er density (i.e., the average number of gas particles per unit volume) , kB Boltzmann's constant andT the absolute temperature of the gas. The ideal gas law assumes there is nointeraction among any of the gas part icles. The first correct ion to this law, asthe density n of the gas increases, is the second virial coefficient B2(T) , givenby: [2, 41

B2(T) = -21T100

drr2[e- q,Cr J( k n T - 1] (1)

where ¢(r ) is the interparticle interaction potential energy of two sphericallysymmetric par ticles at a distance r . T he two terms inside the square brackets inEq. (1) represent the correct ion to the ideal gas ter m, which complete ly neglectsthe interparticle pot ential, due to the fact that it contains an (incorrect) contribution, with Boltzmann factor unity (as though the par t icles did not interact) ,for those interpar ticle distances r for which ¢(r) is not in fact zero. The te rm- 1 subt racts out this incorrect cont ribut ion and the exponent ial te rm replacesit with the correct Boltzmann factor.

Thus for inst ance, for three-dimensional hard spheres of diameter (J,

¢(r) = 0 for r 2: (J

= 00 for r < (J

the integral in Eq. (1) has non-vanishing contributions only for r ::; (J , so thatthen B2(T) = 21T(J3 / 3 .

A similar but increasingly complicated st ructure appea rs in the higher terms ,which are the coefficients of increasing powers of n , leading to the density expansion of the pressure, which has been shown to converge for sufficient ly lowdensities [51.

In the nonequilibrium case, the clust er expansion contains a series of terms,involving the dynamics of isolated groups of an increasing number of particles,the st ructure of which is similar to that of the corresponding terms in the equilibrium expansion [61. Let us use a dynamical language to describ e the twoterms in the integrand of Eq. (1). We say that the first term involves a rea l twopar ticle collision, while the second term involves a "noninteract ing" collision. By"noninteract ing" we mean that there is no inte raction between the par t icles, sothat they can "go through each other", where actually a collision should havetaken place. Then one can transfer this nomenclature to the case that obtainsfor the dynamical te rms appearing in the clust er expansion of the generalizedBoltzmann equation. This implies that the dynamical problem represented bythe various n-p article terms in the cluster expansion involves not only the st udyof the sequences of real, i.e., act ual, collisions taking place among n par ticles,but also tho se which involve "noninteract ing" collisions [71 . The lat ter incorporate the corrections for the incorrect dynamical t reatment of the full dynamicaln-par ticle problem by the lower order terms, which involve at most n - 1 part icles.

On the Sequences of Collisions Among Hard Spheres in Infinite Space 33

For the above reason, in addition to discussing actual collisions between hardspheres, we will also discuss some mathematical results pertaining to "noninteracting" collisions. This would not have come up naturally in a purely dynamicalcontext , but is motivated by the attempt to create a systemat ic physical th eoryof moder ately dense systems.

1.2 "Divergent" Terms. In the cluster expansion, certain sequences ofcollisions appear to lead to a divergent, that is, infinite contribution to various terms in the density expansion of the macroscopic transport coefficients.This is due to the fact that in the dynamical events described by the termsin the clust er expansions, long distances may be traveled by the n particles ina group between the initial and final collisions among them; if these distancesare large compared to the mean free path of a particle, the assumption thatthey travel in straight lines between collisions among members of the groupbecomes unrealistic . A similar problem arises in the equilibrium case when theinterparticle forces are of long range. For the Coulomb potential, for example,the contribution of n particles to the coefficients in the density expansion of thepressure diverges, since the range of the forces is large compared to the dist ancewhich characterizes the collective behavior of all N particles. Mayer [81 resolvedthis by a resummation of contributions from 2,3, . .. particles to produce finitecontributions ("Debye shielding"); a similar resummation of the cluster expansion in the kinetic theory case can likewise lead to finite terms in the transportcoefficients [91 .1

Even in these "divergent" cases, however, the calculations require knowledgeof the dynamics of the 2,3, . . . particle groups .

§2. The Dynamical Problem of n Hard Spheres

In the remainder of this paper we confine ourselves exclusively to the case ofisotropic2 Newtonian hard spheres moving in unbounded Euclidean space . Anadvantage of the choice of hard spheres for the illumination of the general collision dynamics problem is that for n particles the dynamics can be reduced toa sequence of binary collisions separated by free flight of the n particles. Thedynamics of each individual binary collision involve only two particles and areparticularly simple. As the number of particles increases, however, the overallproblem becomes increasingly complex due to the increasing number of possiblesequences of collisions. On the other hand , one general result can readily bederived:

1 It should be pointed out, however , that a resummed expans ion with only finiteterms cannot be obtained, due to collective hydrodynamic ("long time tail") effects .A discussion of this point goes well beyond th e scope of this pap er and we refer forthat to t he literature [9, 10].

2 i.e. with centrally symmetric mass distribution .


Lemma 2.1 For any combination of masses among n particles, initial condition s exist such that at least n(n - 1)/ 2 collisions occur.

A proof of this is given in Appendix A.We also know that for a finite numb er of spheres, the numbe r of collisions is

finite:

Theorem 2.2 (Vaserstein [11] ; Gal 'perin [12]) . In a system of spheres moving in R N (unbound ed Euclidean space of N dim ensions], the number of collisions is uniformly bounded over the whole tim e interval (- 00,+ 00) .

We will not attempt to describe the proof of thi s, but refer the reader to theoriginal papers .

2.1 Dynamics of Three Identical Hard Spheres. For the case considered by Boltzmann, that of two particles, t he "sequence" consists of at mostone single collision. It is somewhat surprising that for the next most simplecase, that of three par ticles, the set of possible collision sequences was not correctly enumerated until the 1960s, even for three hard spheres of equal massand diameter. The history of this enumerat ion is perhaps not without interest . As pointed out above, the problem arose in the context of the systema ticgeneralization of the Boltzmann equation to higher densities as, in particular ,proposed by Bogolubov , M. S. Green and one of us (EGDC) . At the tim e, 1963,it was generally known amongst the cognoscent i, such as Uhlenbeck and Wigner ,that in one dimension n identic al rods could not have more than a maximumof n(n - 1)/2 collisions. An element ary combin atorial proof of this is given inAppendix C. Then the argument ran that if in one dimension, where the particles could not "miss" each other , the maximum numb er of collisions amongthree particles is three, then a fortiori thi s should also be true in more thanone dimension, i.e., for hard disks (d = 2) or hard spheres (d = 3). A naggingquest ion about this argument was whether , in more dimensions, ways could notbe found for the three hard spheres to collide more than three tim es, exploit ingthe presence of a larger phase space. Stimulated by Uhlenbeck's hypothesis ofa maximum of three collisions, but driven by the uncert ainty of the possibilitiesfor d > 1, a graduate student at the Rockefeller University, J . D. Foch, wasasked by one of us (EGDC) to investigat e this, on the basis of a then existingtentative proof that three identical hard disks could indeed suffer a maximum ofthree collisions. To everyone 's consternat ion Mr. Foch found examples of init ialconditions which led to four collisions. Also stimulated by Uhlenbeck, Thurstonand Sandri at Princeton University announced this as well 1131. Subsequently,computer searches 114, 15] examined a tiny volume of phase space and indicatedthat it contained init ial condit ions which lead to four collisions; unfortunatel y, asfar as we know no one has yet published such a set of init ial condit ions, althoughthey were communicated in a letter to a skeptical Wigner. Thus the assert ionthat they exist cannot easily be verified by ot hers. We take the opportunity toremedy this by listin g Foch's original initi al condit ions in App endix B.


Figure 1: Four collision sequence between three hard disks at A, B ,0 , D (collisionsI,II,I1I,IV in the text) : (12)(23)(12)(13), respectively. After the (23) collision at B ,particle 3 has to "get around" particle 2 in order to collide with particle 1 in the fourthcollision. The distance that part icle 2 moves between B and C has been exaggeratedin the figure.

In orde r to discuss specific sequences of collisions, we will first establish a descriptive notation suitable to the problem and the elementary facts govern ingthe dynamics. For the tim e being we will confine ourselves to the case of "actual"as oppo sed to "noninteracting" collisions.

The part icles are numb ered 1,2,3. A collision, in which the particl es exchangethe momentum components along their line of centers at the time of the collision, is denoted e.g. (12). Without loss of genera lity we can denote the firsttwo collisions of any sequence by (12)(23) (rememb er that the particles travelin straight lines between collisions, so that e.g. (12)(12) cannot occur). Thenthere are just two possible sequences totaling three collisions: The "cyclic" sequence (12)(23)(13) and the "recollision" sequence (12)(23)(12). The sequencediscovered by Foch which totals four collisions is (12)(23)(12) (13), in which thefirst th ree collisions constitute a recollision sequence and the final three a cyclicsequence. The time-reversed version of this sequence, which contains no newinformation , is (12)(23)(1 3)(23), as can be seen by appropriate relabeling ofthe particles. We call thi s the "ant i-Foch" sequence. Computer calculations haveshown that the phase space of initial conditions which lead to the Foch sequenceis but a t iny fraction of that which leads only to a 3-collision sequence 114, 151 .One reason for this is a mutually contrastin g pair of geomet ric requirementsrestri ctin g the location of particle 1 at the tim e of the second collision, (23), inthe Foch sequence; at the tim e of the second collision one of these requires thatthe nonpar t icipating parti cle 1 be "near" the colliding particles but the otherrequires th at it be "not too near" to them.

36 T. J. Murphy and E. G. D. Cohen

>z

Figure 2: Posit ions of the particles at t n (Lemma 2.3) .

In order to explain clearly thi s cont ras t ing pair of requirements, we give herea lemma which is used repeatedly in the various geomet ric proofs whose out linesfollow, and which capt ures some of the spir it of the geomet ric approac h whichis taken in the proofs.

In all the following outlines, we denote collisions in the orde r that they occurby Roman numerals, denote the vector posit ion and velocity of the ith part icleas r , and V i respectively, and denote the time of the jth collision as tj with jin Roman numerals. Since the velocities of colliding part icles change instant aneously at the t ime of the collision, we add + or - superscripts if necessary todenote t imes immediately afte r or before a collision; e.g. tTl is the time immediately after the second collision. The dist ance scale is chosen so th at the particleshave unit diameter.

Lemma 2.3 (Murphy and Cohen [14, 16]). In the recollision sequence(12)(23)(12), let the z -axis run from its origin at r2(t Il ) to r3(tIl ), with v 2(tTl )= O. Let ' iz and Viz indicate the z -componen ts of r and v respectively, and ripand Vip the magnitude of their components perpendicular to this axis. Then

(i)rIp(tIl) < 1;

(ii) rI z(tIl) < 0; and

(iii) lIz (t Ill) < O.

Proof outline: Since 2 lies on the z-axis from I to III , (i) is required for Ito have occurred in th e past and III also to occur in the fut ure . (ii) is requiredso that , first , 1 is approaching 2 at tTl so that III can occur and, second, 1 wasmoving away from 2 at tIl so that I can have occurred. (iii) follows direct ly from(i) and (ii) since r 2 = 0 between tu and itn and 1 cannot "get past" 2.

We now present the two mutu ally contrasting conditions on rl (t Il) requir edfor the Foch sequence to occur:

Lemma 2.4 (Sandri and Kritz [15] ; Murphy and Cohen [14, 16]). For theFoch sequence (12)(23)(1 2)(1 3) to occur, it is necessary that at the tim e of thesecond collision, particle 1 must be within v'2 diam eters of particl e 2.

Proof outline: In the proof of thi s we use the same coordinates as in theproof of Lemma 2.3 (see Fig. 2). We show that for 1 to "catch up" with 3 soth at collision IV can occur we must have VI z(tIl) > v2z(tII ) = V3z(tjI ). Butif VIz (tII) > V2z (tII) then, looking now backwards in time, the z-component ofthe separation of 1 and 2 is increasing, so unless - ri z (tII) < 1 collision I willnot have occurred. But also, by Lemma 2.3, rIp(tIl ) < 1 and the result follows.

Lemma 2.5 (Murphy and Cohen [14, 16]). For the Foch sequence(12)(23)(1 2)(1 3) to occur, it is necessary that at the time of the second collision,particle 1 mu st be at least v'2 diameters away from parti cle 3.

(This is an immediate corollary of Lemma 2.3; see again Fig. 2.)Once it was known that a four-collision sequence was possible, the question

immediatel y arose as to wheth er a five-collision sequence might be possible.A moment 's thought shows that one has proved any five-collision sequence impossible once one has proved the following four sequences to be impossible:a) (12)(23)(12)(23)b) (12)(23)(1 3)(12)c) (12)(23)(12)(13)(12)d) (12)(23)(13)(23)(12)This is because a) and b) , together with the Foch and anti-Foch sequences,exhaust the list of four-particle sequences; and c) and d) are the only five-particlesequences such that both the first four and the final four collisions form eithera Foch sequence or an ant i-Foch sequence. Unfortunate ly, the geometric picturepresented by each one of these four sequences a)-d) is quit e different from thatpresent ed by the others; hence there appears to be no single uniform approachto the proofs of all cases and four separa te proofs are needed to eliminate thepossibility of five collisions.

Sandri, Sullivan and Norem stated that each of the four sequences a)-d) isimpossible, and th at "Detailed demonstrations of the foregoing assertions willbe published soon," [171 but, to the best of our knowledge, none of th eir fourproofs have app eared in the literature. (Sandri and Kritz [15J later stated "Sinceit has been demonstrated previously that the fifth collision among three hardspheres cannot exist, ..." but the only reference given for this assertion is to theSandri, Sullivan and Norem paper.) It was the present authors who for the firsttim e gave geomet ric proofs of the four assertions in a very concent ra ted formin the 1966 Lectures in Th eoretical Physics [14], and more recently in a moredetailed and discursive form [16J.

Let us now give the out lines of our four proofs. The general strat egic approach to these proofs is to choose a collision in the middle of the sequence andplace the origin of the coordin at e system at the location of one of the colliding

38 T . J. Murphy and E. G. D. Cohen

particles. The velocity of the coordinate system is chosen such that the particleat the origin is at rest either immediately before or immediately afte r the chosencollision. The axis of a cylindrical coordinate system runs from the particle toits partner in the collision, which it is touching. (See Fig. 2.) This takes advantage of the symmet ries of the problem. Looking forward then in tim e towardthe final collision, the properties of the paths of the particles are complicatedby at most one intervening collision. Looking backward in time toward s the firstcollision, the same holds. One then atte mpts to find some set of restrictions onthe axial and radial velocities of the two moving particles, as well as on the axialand radial distance from the origin of that particle which does not take part inthe chosen collision, which is requir ed for the final collision to take place; butwhich is incompat ible with a similar set of restr ictions that are required if thefirst collision (looking backwards in tim e) is to have taken place in the past .This general practice is well illustrat ed by the outline of the proof of Lemm a 2.3above.

Unfortunately, the geomet ries of the four cases are quit e different so th at thefour proofs have little else in common , as we shall now see. For details we referto Reference [161 .

T heorem 2.6 (Murphy and Cohen [14, 16]).The sequence (12)(23)(12)(23) cannot occur.

P roof outline: Here the choice of reference frame and z-axis is th e same asthat in Lemma 2.3. (See Fig. 2.) Tim es are assumed to be ttt unless otherwisespecified. The condition at tIl that 1 be moving away from 2 (so that , lookingbackward in t ime, collision I can have occurred) is

(3)

On the other hand, looking forward in tim e, a necessary condit ion for collisionIV to occur is v2z(t fII ) > V3 z(tfII) = v2z (tII ), so that 2 can "catch up" with 3.Thi s inequality, when subst ituted into Eq. (3) (using rl z < 0 as required bypar t ii of Lemma 2.3), yields

(4)

It is then shown trigonometrically that if Eq. (4) were to hold, then (lookingforward in time again) collision III would not occur . (Note that V2z (t fII ) is justthe z-velocity t ransferred from 1 to 2 at ttu)

Theorem 2.7 (Murphy and Cohen [14, 16]).The sequence (12)(23)(13)(12) cannot occur.

P roof outline: Thi s is the most complex of the four proofs. A referenceframe is chosen such that V2(tj) = 0 and the z-axis is chosen to run fromthe origin at rl (tIIr) to r3(tIIr) . This choice implies that immediat ely before

On the Sequences of Collisions Among Hard Spheres in Infi nite Space 39

z

Figure 3: The action sphere and positions of part icles used in the proof of Theorem 2.7.

tIII , V3z(tIII) < VIz (tIII) so that collision III occurs. An "act ion sphere" ofunit radius is const ructed around r2(tI) (see Fig. 3). It is then shown that forcollisions I and II to have occurred in the past and also collision IV to occur inthe future, the z-axis must pass through the "act ion sphere" and that rl (t I ) andr 3(tII) must lie on the + z hemisphere of the "act ion sphere". This is then shownto imply !r3z (tIII) - r3z(t II) ! > jrIz(t III ) - rIz(t II )1or v3z(tIII) > VIz(tIll)'contradicting the above requirement that III occur.

Theorem 2.8 (Murphy and Cohen [1 4, 161) .The sequence (12)(23)(12)(13)(12) cannot occur.

Proof outline: The approach in the proof of this theorem is quite differentfrom that in the ot her three. It centers on the magnitude e of the velocitytransferred at collision III. It is shown that if, for a given (, collisions I and IIoccurred in the past and collisions IV and V will occur in the future, then if (is decreased while holding all ot her components of the velocit ies and posit ionsconstant , collisions I and II will st ill have occurred in the past and collisions IVand V will st ill occur in the future. This is true down to ( = 0; however , this"grazing collision" is equivalent to a noncollision (no velocity is transferred) andtherefore is equivalent to the sequence (12)(23)(13)(12), which is ruled out byTheorem 2.7. Thus there cannot be an ( > 0 for which I and II occurred in thepast and also IV and V will occur in the future.

Theorem 2.9 (Murphy and Cohen [14, 161) .The sequence (12)(23)(1 3)(23)(1 2) cannot occur.

Proof outline: We choose a frame of reference in which v3(t fII ) = 0, andrun the z-axis from r3(tIII ) to rl (tII I ). (See Fig. 2 but with t = tttt andappropriate renumbering of the particl es.) Applying par t iii of Lemma 2.3 tothe recollision sequence II ,III ,IV it is shown that for 2 to "catch up" with 1 sothat collision V occurs in the future, we must have V2z (tIII ) > V3 z(t III )' (Thesituat ion is analogous to that in Lemma 2.4.) It can be shown, however, that ifthis inequality holds, I could not have occurred in the past.


At each stage of each of these four proofs, advant age has been taken ofcylindrical symmet ry so that the proofs are valid in 2, 3, ... dimensions.

2.1.1 Corresponding Results for "Noninteracting" Collisions. When"noninteract ing" as well as act ual collisions amon g identical spheres are considered, several result s are immediate ly apparent . First , in determi ning whethera given collision sequence can occur , it is not necessary to specify whether thefirst or last collision is actual or noninteracting. (To see this for the first collision,consider tim e - reversal , under which the first collision becomes the last .) Second , no three - collision "recollision" sequence is possible if the second collisionof the three is noninteracting.

Hoegy and Sengers [18] have shown that five collisions are impossible evenif one or more of the intermediat e collisions is noninteracting, and that the onlypossible four - collision sequences in which either the second or third collisionis noninteract ing are the Foch sequence with the second collision actual andthe third collision noninteracting, and, by t ime - reversal invar iance, the ant iFoch sequence in which the second collision is nonint eracting and the thi rdcollision actual. They also showed that for thi s case the collisions are separateand disti nct , that is, in the Foch case particle 3, which does not take part in th enoninteracting collision between I and 2, cannot touch either 1 or 2 while 1 and 2are "overlapping" ( i.e., at a distance of less than one diameter from each other) .

2.2 The Special Case of Motion of n Particles in One Dimension.The difficulty of generalizing the known results for three identi cal hard sphereslies largely in the innate geometrical complexity of the problem of th e analysisof each conceivable collision sequence, and th e lack of common features whichmight give rise to result s applicable to more than one sequence.

Some insight into the more general analysis might however be gained byexaminat ion of the problem of sequences of collisions among hard spheres constra ined to move in one dimension ("hard rods"). For this case the geometry iselementary and the problem is more an algebraic one, as can be seen in theargument given in Appendix C.

One crucial simplificat ion lies in the fact that (leaving aside the possibilityof "noninteract ing" collisions, which we will not examine) the particles are andremain ordered, so th at whatever the collision sequence may be, a given particlecan collide only with the particle to its left or with the particle to its right .

Another simplification lies in the fact that , unlike the case in more than onedimension where the geomet ry of the problem makes the relative radii of th e particles a crucial factor, in one dimension the lengths di of the rods are essent iallyirrelevant . To demonstrat e this, and for all future reference, we consider thez-ax is showing the positions of the particles to run from left to right , and number the n particles in order of increasing z. If then the location of the cent er ofa particle on the line is denoted by Zi, the transformation Y i = Zi - ~ - Z=;:i dj

maps the probl em onto tha t of a corresponding set of point particles with thesame masses (see Fig. 4).

On t he Sequences of Collisions Among Hard Spheres in Infinite Space 41

111 121 /31

Figur e 4: Positions of the centers of three hard rod s in the one-dimensional case . Ifth e rod s are "cut out" of th e line and t he line "rejoined", th en e.g. th e position Za of 3(now a point particle) goes from za to za - d1 - d2 - da/2 .

We will now examine the evolut ion of the one-dimensional system with tim e.After the last collision the velocities must be in increasing order: Vi+l > Vi.

The "job" of the collisions, t herefore, is to transfer momentum from left to rightamong the particles until thi s condit ion holds. Once the sequence of collisionshas been determined (the init ial positions Yi may be needed to determine thissequence) the dynamical laws are such that the evolut ion of this transfer ofmomentum among the n particles can be described complete ly in terms of n - 2dimensionless constants, each of which represents the mass of a particle (oth erthan the leftmost and the rightmost) relati ve to the masses of its immediat eneighbors, and n - 1 velocity variables, each of which represents the velocity ofa par ticle (oth er than the rightmost) relati ve to the velocity of the par t icle toits immediate right.

We first define the relative mass parameter Xi for any particle other thanthe leftmost particle 1 and the rightmost particle n :

2 VJ.l i - l .iJ.li .;+lXi =

m i

where m i is the mass of the i th par ticle and J.li .j is the reduced mass m imj / (mi +m j) of the pair i and j .

Xi characte rizes the "efficiency" of the transfer of momentum from left toright through the intermediate particle i. The dynamics of the system are suchth at the effect of the mass of par ticle i on the evolut ion of the system liesexclusively in its relationship to the masses of its immediat e neighbors . Theprecise definition of Xi is chosen so that the eventual result s will appear ina particularly simple form.

Note that Xi is dimensionless; it approaches zero as the mass of i becomeslarge relat ive to th e masses of its surrounding particles, and approaches a maximum value of 2 as the mass of i becomes small.

When all three masses are equal, Xi = 1. When Xi « 1 the magnitude of themomentum transferred to the (heavy) ith particle in a collision is a maximum;when 2 - X i « 1 the amount transferred becomes very small, so that forparticle i to transfer a given amount of momentum from th e particle on its leftto the particle on its right, the (light) ith par ticle must "rattle around" betweenthem many t imes.

We now also define the reduced relative velocity of each particle i except forthe rightmost particle n as ll i = VJ.li ,i+l (Vi - Vi+l ) , where Vi is the velocity of

42 T . J . Mur phy and E. G. D. Cohen

(5)

(6)

(7)

particle i ; the dynamics of the system are such that the effect of the velocityof part icle i on the evolut ion of the system lies exclusively in its relationshipto the velocity of the neighbor to its immediate right. Once again, the precisedefinit ion of U i is chosen so that the eventual result s will appea r in a particularlysimple form.

Note that immediat ely before a collision between particles i and i + 1, thevelocity of i must be greater than that of i + 1 so that Ui is posit ive.

In terms of these definitions, the laws of conservat ion of energy and momentum then determine the evolut ion of the system undergoing a known sequenceas follows: if, for a given sequence of collisions, u U) is a reduced relati ve velocityimmediately after the jth collision, and the j + 1st collision is between par ticlesi and i + 1, then all UU+ l ) = uU ) except for three (two if i is the first part icleor if i + 1 is the last par ticle):

llU+ 1) = u U) + x · u U), -1 , - 1 "

UU+l ) = -u(j ), ,UU+ l ) - u U) + z, uU)

i+ l - H I , + 1 i

Eq. (6) represents the direct result of the collision on the relative velocityof the colliding pair of par t icles. Eqs. (5) and (7) represent the change in thevelocities of the colliding pair relat ive to their immediate neighbors.

Iteration of Eqs. (5)-(7) for a given sequence gives the final reduced relativevelocit ies of the part icles as a linear combinat ion of the initial reduced relat ivevelocit ies u~O) , with coefficients involving sums of products of the Xi . Note that

since the j + 1st collision can be a collision between i and i+1 only if u~j) > 0, theiteration stops when all U i < °[i.e., all interpar ticle dist ances are increasing).

The above algorithm gives the final state after any given sequence of collisions. However, it does not determine the sequence of collisions which will occurfor a given set of init ial condit ions. While we know that for the j + 1st collisionin a sequence to be a collision between i and i + 1 we must have u~j ) > 0, thismay be true for several i and it will not be known which of the possible collisionswill occur first unless we know the complete trajectories of the particles; thesetrajectories depend on the initial particle positions as well as on their initialvelocities.

However , Gal 'perin has found an upp er bound for the number of collisionsamong n hard rods:

Theorem 2.10 (Gal'perin [1 2]). In the one-dim ensional case, n hard spheres( rcannot undergo more than 2 8nZ(n - 1) m max collisions, where m max andmml n

m m i n are the maximum and minim um masses.

Once again we will not attempt to describe the proof of th is, but refer the readerto the original paper.

On th e Sequences of Collisions Among liard Spheres in Infinite Space 43

Note that for m m ax = m m in this bound is rather large compared to theknown result that the maximu m number of collisions is n(n - 1)/2.

2.2.1 Collisions Among Three Hard Rods. There is one nontri vialspecial case in which only one sequence of collisions is possible: the case of threepar ticles. The only possible sequence is that of alte rnating collisions between theleft-h and pair and the right-h and pair, i.e. (12)(23)(12)(23) ... . (In this case,without loss of generality, the left-hand pair can be considered to collide first .)T hen the total numb er of collisions, and the final velocit ies of the particles,depend only on the initial velocit ies of the particles and not on their initialpositions. A further simplification is that in the three-par ticle case there is onlyone dimensionless mass param eter , X2, which we shall simply denote as x.

In all th at follows we assume u~o ) > 0 so that at least one collision occurs .The results [19] for the number of collisions and the final velocities involve

a set {Sk(X)} of polynomi als in x, called "the Chebyshev polynomials of the firstkind ," [20] which can be defined by the recursion relation Sk(X) = XSk- l (X) Sk- 2(X) with So = 1 and SI = x. The zeroes of Sk(X) lie at x = 2cos[m7l' /(k+l)]where m is a positive integer; the greatest zero lies at x = 2 cos [7l' /( k +1)]. SeeFig. 5. For the following we need a Lemma concerning these zeroes:

oa) I

SI =0

S3 =0

S5=0

3 3.5 4b) I I I

SI=O S2=0

S3=0 S.=O S5=0

S5=0

.6180I

1.25J.. 1.414 1.618 1.732 2

I I I 1"..

S2=0 S3 =0 S. =0 S5 =0S5=0

• (rJ9) + 1

Figure 5: a) Valu es of x ?: 0 such t hat 5k (x) = 0 for 1 :::; k :::; 5. Th e 5 k that equalzero for each value of x are list ed below th e line; if an 5k appears in t he mth row, th ezero is the mth greatest zero for th at value of k. The arrow corresponds to the valueof x obtained when th e masses of t he two outer particles are equal and t he mass oft he cent ra l par ticl e equa ls 3/ 5 that of an outer particle; for t his value of x a maximumof four collisions is possibl e. Note for comparison with Lemm a 2.1I t hat x is greatert han t he greatest zero of 52, is less t han the grea test zero of 53, and lies between th etwo greatest zeroes of 53 as well as t hose of 54. b ) Values of (-rr I O) + 1, 0 :::; -rr / 2,such t hat 5k(X) = 5k(2cosO ) = 0 for 1 :::; k :::; 5. The ar row corr esponds to t hevalue of x in par t a) of t he figur e. Par t b) of t he figur e is given to show more clearlythe relationship between t he masses and th e maximu m number of collisions; for eachm t he mth greatest zeroes of t he 5k are equa lly spaced a distance 11m apart. Themax imum numb er of collisions is t he integer value to t he left of t he arrow (four in thiscase) .

44 T. J . Murphy and E. G. D. Cohen

Lemma 2.11 (Murphy [19]). If x is greater than the greatest zero of Skbut less than the greatest zero of Sk+l , k > 0, then (i) Sk+l (x) < 0; (ii)Sk +2(X) < 0; and (iii) for j < k + 1, Sj(x) > O.

The results for three hard rods are then:

Theorem 2.12 (Murphy [1 9]). If at least j collisions occur, then after thej th collision the reduced relative velocities are given by

(8)

(9)

when j is odd (j th collisio n was (12)), except that u~1) = -So u~o) ; and by

(10)

(11)

when j is even (j th collision was (23)).

T his solut ion follows by recursion of Eqs. (5)-(7) and comparison with the recursion relation for the Chebyshev polynomials.

Theorem 2.1 3 (Zemljakov 121]; Murphy [1 9]). A necessary and suffi cientcondition that initial conditions exis t such that N > 3 collisions tak e place isthat x be greater than the largest zero of SN -2(X), that is,

or

ttx > 2cos -

N - 1

J f.l 12 f.l23m2 < r-r-ri-r-t-r-r-r-r-r-r-:

cos[7r/(N - 1)]

(12)

(13)

(Lemma 2.1 shows that init ial conditions always exist for which there will bethree collisions.) If the condition is satisfied , Lemm a 2.11(iii) gives Sj (x) > 0 forall j < N - 1 and, by recursion of Eqs. (9) and (10), the condition u (j ) > 0 for

all collisions up to and including N to occur is fulfilled for sufficient ly large u~o) .However unless x is also greater than th e largest zero of S N-l(X), Lemma 2.11(i)and 2.11(ii) give S N-l (X) < 0 and SN(X ) < 0, and then Eqs. (9) and (10) give

u(N) < 0, so that collision N + 1 does not occur. (Note that if u~o) < 0 theNth collision cannot occur; see Corollary 2.15 below.)

Note that if and only if x > 1 can there be more than three collisions.(x = 1 holds for the equal mass case.) As x ---t 2 the increase in N is seen asa rapid rat tlin g back and forth of the cente r particle transferring moment umfrom the particle on the left to the par ticle on the right .

Finally, we determine the initial condit ions which lead to a given numb er ofcollisions up to the maximum N determined by Theorem 2.13:

On t he Sequ ences of Collisions Among Hard Sph eres in Infin ite Space 45

Theorem 2.14 (Murphy [19]) . If according to Theorem 2.13 N collisionsare possible but N + 1 are not , k collisions will occur , 1 < k s:; N , if and onlyif

(14)

This follows from Eqs. (9) and (10) and the fact th at th e next collision willoccur if and only if the appropriate u (j) > 0, in much the same way as in theproof of Theorem 2.13.

Corollary 2.15 If u~O) = 0, then the number of collisions will be one lessthan the maximum number perm itted by Th eorem 2.13.

This also follows from Eqs. (9) and (10) and Lemma 2.11.

Corollary 2.16 Wh en Theorem 2.13 allows at most N collisions to occur,the phase space volume of initial conditions which lead to exactly k collisi ons,2 < k s:; N , is proportional to SSk - l - ---"'-!L.

SS .

k -2 k -l

T his follows from T heorem 2.14. T his corollary is relevant to the problem of dilute gas mixtures, since the effect of any given collision sequence on the nonequ ilibrium properties of a gas is dependent on the volume of phase space occupiedby the initial condit ions which lead to that sequence. (See e.g. Sengers et al [7] .)

§3. Open Questions

a) Extension of the results in more than one dimension to cases involving four ormore particles is likely to prove quit e difficult , as the complexity of the geomet ricproblem increases enormously with each addit ional collision. One possible approach would be a computer search for examples of initi al condit ions th at mightlead to a targeted sequence. This however will run into the difficulty that theoccurrence of the targeted sequence is likely to be exquisite ly sensitive to initi alcondit ions, so that lit tle confidence can be given to conclusions th at a sequenceis impossible drawn from failur e of a comput er search to demons tr ate initialcondit ions for which it occurs. Th is is already apparent from the t iny fract ionof that phase space leading to a recollision, which gives rise to a Foch sequence.

b) Of greate r interest would be genera l theorems covering all sequences. Forexampl e, theorems 2 .6~2.9 make use of cylindrical symmetry which disapp earsonce more than three particles are involved; therefore a theorem concern ingfour par ticles in two dimensions may not hold in three dimensions. We conjecture [16] that the maximum number of collisions among n ident ical hard spheresis indep endent of th e dimensionality d, provided d 2: n - 1.

c) Prospects for extension of the one-dimensional results from three to fouror more hard rods are more promising. Each additional rod introduces only oneadditional constant , the mass parameter x, and one more variable, the reducedrelative velocity u. Especially in the case of four particles, it shou ld not be toodifficult to find ranges of the two mass parameters which permit a given numberof collisions to occur. Also, more general results might be obtained. For example,we conject ure that if the masses of the particles form a concave-downward sequence, that is, if no interior particle has mass less than the arithmetic mean ofthe masses of its immediate neighbors, then the maximum number of collisionsamong n particles will prove to be n(n - 1)/ 2.

Append ices

A. Proof That for Any n Hard Spheres,n(n - 1)/ 2 Collisions Can Occur

It suffices to prove this for the case where the particles are constrained to moveon a line; we therefore use the notat ion of Subsection 2.2 . The proof is byinduction: it is evident ly true for two particles. If it is true for n - 1 particles,and an additional particle (numbered 1) is added to the left , uiO) can be chosento be large enough so that, once the initial distance between 1 and 2 is chosenlarge enough so that for i > 1 all Ui < 0 (all interparticle distances amongparticles 2,... ,n are increas ing) before the first (12) collision occurs, there will beat least n -1 additional collisions. (At the (12) collision U2 becomes positive, andlarge enough so that when 2 collides with 3, U3 becomes positive, etc .) Whenthese n - 1 collisions are added to the previous (n - l)(n - 2)/2 collisions thetotal is n(n - 1)/2 collisions.

B. Initial Conditions Leading to Four Collisions Among ThreeIdentical Particles

t Xl Yl X2 Y2 X3 Y3

0 -2.450000 + 0.242893 -0.542893 + 0.250000 + 3.490000 +2.4903840.100000 0 -0.707107 + 0.707107 0 + 1.990000 + 0.2146050.109430 + 0.207460 -0.820267 + 0.848554 0 + 1.848554 00.113243 + 0.291350 -0.866026 + 0.791350 0 +1.905755 -0.0867980.321799 + 4.034199 -4.833150 -1.491579 + 1.464404 +5 .034155 -4.8331600.421799 + 5.534199 -6.735300 -2.586194 +2 .166554 +7 .008770 -7.108939

Table 1: Positions 0.1 t ime units before t he first collision, at the time of each collision,and 0.1 time units after the fourth collision.

t V l x V l y V2x V2y V 3 x V3y

0.100000 + 24.500000 -9.500000 + 12.500000 -2.500000 -15.000000 -22.757785+22.000000 -12.000000 + 15.000000 0 -15.000000 -22.757785

0.109430 +22.000000 -12.000000 + 15.000000 0 -15.000000 -22.757785+ 22.000000 -12.000000 -15.000000 0 + 15.000000 -22.757785

0.113243 + 22.000000 -12.000000 -15.000000 0 + 15.000000 -22.757785+ 17.946153 -19.021503 -10.946153 + 7.021503 + 15.000000 -22.757785

0.321799 + 17.946153 -19.021503 -10.946153 + 7.021503 + 15.000000 -22.757785+ 15.000000 -19.021503 -10.946153 + 7.021503 +17.946153 -22.757785

Table 2: Velocities immediately before and immediately after each collision.

C. Proof Th at for n Hard Rods of Equal Mass,No More Than n(n - 1)/2 Collisions

Can Occur

The dynam ics of hard rod s of equal mass are par ticularly simple: in any collisionth e two par t icles simply exchange their velocitie s. Therefore the total numb erof collisions can be determined from the initial velocit ies as follows:

The initial velocities viol compr ise an ordered list. The collisions then sortt he list by the binary sort algorithm until th e list is in increasing order , afte rwhich there can be no more collisions. Specifically, choose a neighboring pairi , i + 1 at random. If Vi > Vi +! , exchange the positi ons of th e two velocit ieson the list ; otherw ise, do nothing. Repeat until t he list is ordered. T hen th etotal numb er of collisions is the to tal number of particle pairs i < j such th at

viol > viO) , since these two velocit ies will be exchanged with one anot her in

a collision exactly once before th e list is ordered ; if viol ::; viOl the two velocit ieswill never be excha nged. The ord er in which the exchanges take place does notaffect t he total numb er of exchanges; furth ermore, t he numb er of collisions isindepend ent of the initial posit ions of the par ticles. The maximum numb er ofcollisions occurs when th e init ial velociti es are in decreasing order from left toright ; t hen all part icle pairs i < j have viol > viO) , and the total numb er ofcollisions equa ls the tot al numb er of pairs, or n( n - 1)/ 2 collisions among npar ticles.

48 T. J . Murphy and E. G. D. Cohen

References

III J . R. Dorfm an and H. van Beijeren , "The Kinetic Theory of Gases", in: Stat ist icalMechani cs, Par t B, B. J . Berne, ed ., Plenum (New York ) 65-179 (1977).

[2] E. G. D. Cohen, "Fifty Years of Kinetic Th eory", Physica A 194, 229-257 (1993);"Twenty-five Year s of Non-equilibrium Statist ical Mechani cs", in: Lecture Notesin Physics 445 , Springer, 21-50 (1995).

131 G. E. Uhlenbeck and G. W. Ford , "T he T heory of Linear Graphs With Applications to the Theory of t he Virial Developm ent of the Properties of Gases", in:St udies in Stati st ical Mechanics I Part B, J . de Boer and G. E. Uhlenbeck, eds.,Nort h Holland , 119-211 (1962).

14] Ref. 2; E. G. D. Cohen, "Kinet ic Theory: Underst anding Nature T hrough Collisions", Am. J. Phys. 61 , 524-533 (1993).

15] D. Ruelle, "Correlat ion functions of classical gases", Ann . Ph ys. 25 , 109-120(1963) ;J. L. Lebowitz and O. Penro se, "Convergence of Virial Expans ions", J . Math .Phys. 5, 841-847 (1964); D. Ruelle, in Statistical Mechani cs: Rigorous Resu lts ,Addison-Wesley, Reading, Mass, pp . 85,99 (1989).

16] E. G. D. Cohen, "On th e Kinet ic Theory of Dense Gases", J. Math. Phys. 4 ,183-189 (1963).

17] J . V. Sengers, M. H. Ern st and D. T . Gillespie , "T hree-Part icle Collision Int egral sfor a Gas of Hard Spheres", J . Chern. Phys. 56 , 5583-5601 (1972); J . V. Sengers,D. T. Gillespie and J . J . Perez-Esand er , "T hree-Part icle Collision Effects in th eTransport Properties of a Gas of Hard Spheres", Physica A 90 , 365-409 (1978).

18] J . E. Mayer , "Theory of Ionic Solutio ns", J . Chern. Phys. 18 , 1426-1436 (1950) .19] E. G. D. Cohen, "T he Kinetic Theory of Dense Gases", in: "Fundamental Problems

in Statistic al Mechanics II", E. G. D. Cohen, ed ., Nort h Holland , Amsterd am228-275, (1968); J. R. Dorfman , "Kinet ic and Hydrodynamic Theory of TimeCorrelation Functions", ibid. III , 227-330 (1975).

110] E. G. D. Cohen, "Bogolubov and Kineti c Theory: T he Bogolub ov Equ ations",~I3AS: Math emati cal Models and Methods in Appli ed Sciences, 7(7), 909-933(1997).

II1] L. N. Vaserstein, "On Systems of Par ticles with Finite-Range and/or Repu lsiveInteractions", Commun. Math. Phys. 69, 31-56 (1979).(For th e one-dimensional case this had previously been proven in G. A. Gal' perin,"Elast ic Collisions of Par ticles on a Line", Russian Math . Surveys 33 :1, 199-200(1978) and Ya. G. Sinai , "Billiard Trajectories in a Polyhedr al Angle", RussianMath. Surveys 33:1,219-220 (1978) .)

[12] G. A. Ga l'perin , "On Systems of Locally Interacting and Repelling Particles Moving in Space", Trans. Moscow Math. Soc. Issue 1, 159-215 (1983) .

113] W. Thurston and G. Sandri, "Classical Hard Sphere Three-Body Problem", Bull.Am. Phys, Soc. 9 , 386 (1964).

[141 E. G. D. Cohen, "On th e Statist ica l Mechani cs of Moderately Dense Gases", in:Lectures in Theoreti cal Physics Vol. 8A (Univ. of Colo. Press, Boulder , Colo.) ,167-178 (1966).

[151 G. Sandri and A. H. Kri tz , "Approach to the N-Body Problem with Hard-Sph ereInteraction Applied to t he Collision Domains of Three Bodies", Phys. Rev. 150 ,92-100 (1966) .

On t he Sequences of Collisions Among Hard Spheres in Infinite Space 49

[161 T . J . Murphy and E. G. D. Cohen , "Max imum Number of Collisions amongIdentical Hard Spheres", J . Stat . Phys. 71 , 1063-1080 (1993).

[17] G. Sandri , R. D. Sullivan , and P. Norem, "Collisions of Three Hard Spheres",Phys. Rev. Lett . 13,743-745 (1964) .

[18] W . R. Hoegy and J. V. Sengers, "T hree-Part icle Collisions in a Gas of HardSpheres", Phys. Rev. A 2, 2461-2471 (1970) ; J. V. Sengers, D. T . Gillespie andW . R. Hoegy, "Dyna mical Th eorems for Th ree Hard Spheres", Phys. Lett. A 32 ,387-388 (1970) .

[191 T. ,J. Murphy, "Dynamics of Hard Rods in One Dimens ion", J . Stat . Phys. 74 ,889-901 (1994).

[20] Cf. e.g. T . J. Rivlin, The Chebyshev Polynomials, (Wiley-In terscience, New York)(1974) .

[211 A. N. Zemljakov, "Arit hmetic and Geometry of Collisions", Kvan t No. 4, p. 14(1978) .

Hard Ball Systems and Semi-DispersiveBilliards: Hyperbolicity and Ergodicity

N. Simanyi!

Contents

§1. Th e Models . . . . . . . .1.1 Hard Ball Systems1.2 Box Billiards . ...1.3 The Lorentz-Pro cess and Lorentz-G as .1.4 Stadia .1.5 Falling Balls . . . . . . . . . . . . . . .1.6 Cylindric Billiards - A Common Generalization

§2. Some Technical Aspects of the Results .2.1 Full Hyperbolicity - Invariant Cone Fields2.2 Local Ergodicity .2.3 Global Ergodicity .2.4 Irration al Mass Rat io .

References . . . . . . . . . . . . .

52525960626466696972788385

Abstract. The purpose of this survey art icle is two-fold: First , we intend tointroduce the reader into the world of several types of semi-dispersive billiard s,such as Sinai's hard sphere systems in tori or rectangular boxes, the Lorentzgas, stadia (including Bunimovich's celebrated one), and Wojtkowski's systemsof I-D falling balls. Th e second part of the survey deals with some crucia ltechnical aspects of proving full hyperbolicity (nonzero Lyapunov exponentsalmost everywhere) and ergodicity for such models of statist ical mechanics.

1 Research supported by the Hungarian National Found ation for Scientific Resear ch,grants OTKA-26176 and OTKA-29849.

52 N. Simanyi

§1. The Models

1.1 Hard Ball Systems. Hard ball systems or, a bit more generally, mathematical billiards consti tute an important and quite interesting family of dynamical systems being intensivel y studied by dynamicists and researchers ofmathematical physics, as well. These dynamical systems pose many challengingmathemati cal questions, most of them concern ing the ergodic (mixing) prop erties of such systems. The introduction of hard ball systems and the first majorsteps in their investigations date back to the 40's and 60's, see Krylov's paper[K(1942)] and Sinai 's groundbreaking works [Sin(1963)], [Sin(1970)]. In the art icles [Sin(1970)] and [B-S(1973)1 Bun imovich and Sinai prove the ergodicityof two hard disks in the two-dimensional uni t torus 11'2. T he genera lizat ion ofthis result to higher dimensions v > 2 took fourt een years, and was done byChernov and Sinai in [S-Ch(1987)]. Although the model of two hard balls inTV is already rather involved technically, it is still a so called strict ly dispersivebilliard system, i. e. such that the smooth components of the boundary 8Q ofthe configuration space are strictly convex from inside Q. (They are bend ingaway from Q.) The billiard systems of more than two hard spheres in TV areno longer strict ly dispersive, but just semi-dispersive (strict convexity of thesmooth component s of 8Q is lost , merely convexity persist s!), and this circumstance causes a lot of addit ional technical troubles in their st udy. In the seriesof my joint papers with A. Kramli and D. Szasz [K-S-Sz(1989)], [K-S-Sz(1990)],[K-S-Sz(1991)], and [K-S-Sz(1992)] we developed several new methods, andproved the ergodicity of more and more complicated semi-dispersive billiardsculminat ing in the proof of the ergodicity of four billiard balls in the torusTV (v 2:: 3), [K-S-Sz(1992)]. T hen, in 1992, Bunimovich, Liveran i, Pellegrinottiand Sukhov [B-L-P-S(1992)] were able to prove th e ergodicity for some systems with an arbirarily large numb er of hard balls. The shortcoming of theirmodel, however , is that , on one hand , they restrict the types of all feasibleball-to-ball collisions, on the other hand , they int roduce some ext ra scatteringeffect with the collisions at the st rict ly convex wall of the container. The onlyresult with an arbitrarily large numb er of spheres in a flat unit torus TV wasachieved in [Sim(1992-A-B)], where I managed to prove th e ergodicity (actually,the K-mixing prop erty) of N hard balls in TV, provided that N ::; u, The annoying shortcoming of that result is that the larger the number of balls N is, largerand larger dimension v of the ambient container is requir ed by the method ofthe proof.

On the other hand , if someone considers a hard sphere system in an elongated torus which is long in one direction but narrow in the others , so that thespheres must keep their cyclic order in the "long direction" (Sinai' s "pencase"model), then the technical difficulties can be handled, thanks to the fact that thecollisions of spheres are now restricted to neighbouring pairs. The hyperboli cityof such models in three dimensions and the ergodicity in dimension four havebeen proved in [S-Sz(1995)].

Hard Ball Syste ms and Semi-Disp ersive Billiar ds : Hyperbolicity an d Ergodi city 53

The positivity of the metric entropy for several systems of hard spheres canbe proven relat ively easy, as was shown in the pap er [W(1988)]. The art icles[L-W(1995)1and [W(1990-C)] are nice surveys describing a general setup leading to the technical problems treated in a series of research papers. For a comprehensive survey of the result s and open problems in this field, see [Sz(1996)].

Fin ally, in our latest joint venture with D. SZ{lSZ [S-Sz(1999)] we prevailedover the difficult y caused by the low value of the dimension v by developinga brand new algebraic approach for the study of hard ball systems. T hat result ,however, only establishes complete hyperboli city (nonzero Lyapunov exponentsalmost everywhere) for N balls in 1[''' . The ergodicity appears to be a muchharder task.

Consider the v-dimensional (v 2:: 2), standard, flat , unit torus 1[''' = jR" I'll"as th e vessel containing N (2:: 2) hard balls (spheres) B l , .. . , BN with positi vemasses m l , .. . , m N and (just for simplicity) common radius r > O. We alwaysassume that the radius r > 0 is so small that even the interior of the arisingconfiguration space Q is connected. Denote the center of the ball B, by qi E 1[''' ,

and let Vi = iIi be the velocity of the i-t h par ticle. We investigate the uniformmotion of the balls B l , ... , BN inside the contai ner 1[''' with half a unit of totalkinetic energy: E = ~ L~l miIlvi 11 2 = ~. We assume that the collisions betweenballs are perfectly elast ic. Since - besides the kinet ic energy E - the totalmoment um I = L~l m iVi E jR" is also a tri vial first integral of the mot ion, wemake the stand ard reduction I = O. Due to the apparent tr anslation invarianceof the arising dynamical system, we factori ze out the configurat ion space withrespect to uniform spatial translat ions as follows: (ql , .. . , qN) rv (ql+a, . . . ,qN +a) for all translat ion vector s a E jR" . The configuration space Q of the arising

flow is then the factor torus ((1[''' t I rv ) ~ 1[',,(N- 1) minus the cylinders

C· . - {( ) 1[',,(N-l ). di ( . .) 2}' ,J - ql , .. ·, qN E . ist q"qJ < r

(1 ~ i < j ~ N) corresponding to the forbidden overlap between the i-t h andj -t h cylinders. Then it is easy to see that the compound configuration point

( ) Q 1[',,(N- l ) \q = ql , · ·· , qN E = U c.,l ~ i<j ~N

moves in Q uniformly with unit speed and boun ces back from the boundaries8Ci ,j of the cylinders Ci,j according to the classical law of geometric optics: theangle of reflect ion equals the angle of incidence. More precisely: the post-collisionvelocity v+ can be obtained from the pre-collision velocity o: by the orthogonalreflection across the tangent hyperplane of the bound ary 8Q at the point ofcollision. Here we must emphasize that the phr ase "ort ogonal" should be understood with respect to the natural Riemannian metric IIdql12 = L~l m il ldqiWin the configuration space Q. For the normalized Liouville measure fJ of thearising flow {st} we obviously have dfJ = const dq-dv , where dq is the Riemannian volume in Q induced by the above metric and dv is the surface measure

54 N. Simanyi

(determined by the restriction of the Riemannian metric above) on the sphereof compound velociti es

T he phase space M of the flow {st} is the unit tangent bundle Q x §d-l ofthe configurat ion space Q. (We will always use the shorthand d = v(N - 1) forthe dimension of the billiard table Q.) We must, however, note here that at theboundary 8Q of Q one has to glue together the pre-collision and post-collisionvelocities in order to form the phase space M , so M is equal to the unit tangentbundle Q x §d-l modulo this identification.

A bit more detailed definition of hard ball systems with arbitrary masses , aswell as their role in the family of cylindric billiards, can be found in section 4of [S-Sz(1998)] and in section 1 of [S-Sz(1999) 1. We denote the arising flow by(M, {sthEIR ,Jl).

The first major result came from Ya. G. Sinai [Sin(1970)1 in 1970: He provedthere the ergodicity of (M , {st hEIR ,Jl) for the case N = v = 2, technicallyspeaking under the assumption ml = m 2.

The joint work by Ya. G. Sinai and N. I. Chernov [S-Ch(1987)] paved theway for further fundamental results concerning the ergodicity of (M , {sthEIR, Jl) .They proved there a strong result on local ergodicity: An open neighbourhoodU c M of every phase point with a hyperbolic trajectory (and with at most onesingularity on its tr ajectory) belongs to a single ergodic component of the billiardflow (M , {st h EIR ,Jl) , of course, modulo the zero measure sets . An immediateconsequence of this result is the (hyperbolic) ergodicity of the hard ball systemswith N = 2 and v 2': 2.

In the series of articles [K-S-Sz(1989)], [K-S-Sz(1991)j, [K-S-Sz(1992)], and[Sim(1992-A-B)] the authors developed a powerful, three-step st rategy for proving the (hyperbolic) ergodicity of hard ball systems. First of all, all these proofsare inductions on the number N of balls involved in the problem. Secondly, theinduction step itself consists of the following t hree major steps:

Step I. To prove that every non-singu lar (i. e. smooth) trajectory segments [a,b]xQ with a "combinatorially rich" (in a well defined sense) symboli c collisionsequence is automat ically sufficient (or, in other words, "geomet rically hyperbolic", see below), provided that the phase point XQ does not belong to a countableunion E of smooth submanifolds with codimension at least two. (Containing theexceptional phase points.)

1.1.1 D efinitions of t he neutral space and sufficiency. Consider a non singular trajectory segment s [a,b]x . Suppose that a and b are not moments ofcollision . Before defining the neutral linear space of this trajectory segment, wenote that the tangent space of the configurat ion space Q at interior points can

Hard Ball Systems and Semi-Dispersive Billiards: Hyp erboli city and Ergodicity 55

be ident ified with the common linear space

Defini t ion 1.1 The neutral space No(s[a ,bJ x) of the trajectory segments[a,b]x at time zero (a < 0 < b) is defined by the following formula:

No(s [a ,bJx) = {w E Z : 3(<5 > 0) such that Va E (-<5,<5)

v (sa (q(x) + aw ,v(x))) = v(sax) &

V (Sb (q(x) +aw, v(x ))) = v(Sbx )}.

It is known (see (3) in section 3 of [S-Ch (1987)]) that No(s [a ,blx) is a linearsubspace of Z indeed, and v(x) E No(s [a ,bJx). The neutral space M(s [a,bJx) ofthe segment s[a,bl x at tim e t E [a,b] is defined as follows:

M(s [a ,bJx) = No (s[a-t ,b-tJ(st x)) .

It is clear that the neutral space M(s [a ,blx) can be canonically identified withNo(s[a ,bJx) by the usual identification of the tangent spaces of Q along thetrajectory S( - oo,oo) x (see, for instance, section 2 of [K-S-Sz(1990)]).

Definition 1.2

• The non- singular trajectory segment s [a,bJx is said to be sufficient if andonly if the dimension of M(s [a ,b]x) (t E [a,b]) is minimal, i.e.dirn.N;; (s [a ,b]x) = 1.

• The trajectory segment s [a ,bJx containing exactly one singularity is saidto be sufficient if and only if both branches of this trajectory segment aresufficient.

For the notion of t raj ectory branches see, for example, the end of section 2in [Sim(1992)-A].

Definition 1.3 The phase point x E M with at most one singularity is saidto be sufficient if and only if its whole trajector'y S( -00 ,00)x is sufficient, whichmeans, by definition, that some of its bounded segments s [a ,blx is sufficient.

In the case of an orbit S(- 00,00)x with exactly one singularity, sufficiencyrequir es that bot h branches of S(- 00,00)x be sufficient .

The advantage of the except ional set E c M (being a countable union ofsmooth submanifolds of M with codimension at least two) is that its complementM \ E contains an arcwise connected set of full measure, see the subsection "Basicfacts from general topology" from section 2 of [K-S-Sz(1991)1.

The concept of a "slim" subset S c M is the non-smooth vari ant of the abovenot ion of a negligible set E:

56 N. Simanyi

Definition 1.4 We say that the subset S of M is slim if and only if the setS can be covered by a countable union U:=l Fn of closed zero-measure subsetsFn c M whose topological dim ension (say, the sm all inductive dim ension) is atmost dimM - 2.

Similarly to the smooth case, the key advantage of a slim set is that its complem ent M \ S necessarily contains an arcwise connected set with full measure,see the reference above.

Step II. Assume the induction hypothesis, i. e. th at all hard ball systemswith n balls (n < N) are (hyperbolic and) ergodic. Prove tha t th ere existsth en a slim set S cM with th e following property: For every phase pointXo E M \ S t he entire trajectory SlR xo contains at most one singularity and itssymboli c collision sequence is combinatorially rich, just as required by the resultof Step I.

Step III. By using again the induction hypothesis, prove that almost everysingular traj ectory is sufficient in the time int erval (to ,+ 00), where to is thetime moment of the singular reflection. (Here th e phrase "almost every" refersto the volume defined by th e induced Riemannian metric on the singularitymanifolds.)

We not e here that the almost sure sufficiency of the singular trajectories(featuring Step III) is an essential condit ion for the pro of of the celebratedTheorem on Local Ergodicity for semi-dispersive billiards proved by Sinai andChernov, see Theorem 5 in [S-Ch(1987)] . Under this assumpt ion that th eoremasserts that a suitable, open neighbourhood Uo of any sufficient phase pointXo E M (with at most one singularity on its t rajectory) belongs to a singleergodi c component of the billiard flow (M, {sthEIR ,Jl).

Steps I and II together ensure that there exists an arcwise connected setC C M with full measur e, such that every phase point Xo E C is sufficientwith at most one singularity on its trajectory. Then the cited Theorem on LocalErgodicity (now takin g advantage of the result of Step III) states that for everyphase point Xo E C an open neighbourhood Uo of Xo belongs to one ergodiccomponent of th e flow. Finally, the connectedness of th e set C and Jl(M \ C) = 0easily imply th at t he flow (M, {st hEIR ,Jl) (now with N balls) is indeed ergodic,and actually fully hyperbolic, as well.

In the series of art icles [K-S-Sz(1991)], IK-S-Sz(1992)] the authors followedth e stra tegy outlined above and obtained the (hyp erbolic) ergodicity of threeand four hard balls, respectively. Technically speaking, in those pap ers we alwaysassumed tacit ly that th e masses of balls are equal.

The twin papers [Sim(1992-A-I3)] of mine brought new topological and geometric to ols to attack th e problem of ergodicity. Namely, in [Sim(1992-A)]a brand new topological method was develop ed, and that resulted in settlingStep II of th e induction, once forever .

In th e subsequent pap er [Sim(1992-B)] a new combinatorial approach forhandling Step I was developed in the case when the dimension l/ of th e toroidal

Hard Ball Systems and Semi- Dispersive Billiards: Hyperbolicity and Ergodicity 57

contai ner is not less than the numb er of balls N . This proves the ergodicity ofevery hard ball system wit h v ~ N .

1.1.2 Restricted Interaction. There are some results establishing theergodicity of certain hard ball systems, provided that some additional (naturalor artificial) mechanism restri cts the possibilit ies of interactions (elast ic collisions between the balls) or , in other words, makes the collision graphs of orbi tssimpler. By the collision graph of an orbit (or, of an orbit segment) we meanthe set of unordered pairs of balls (i ,j) (the edges of the collision graph) whichhave at leas t one non-tangent ial collision in the considered orbit. Such modelsare essent ially one-dimensional in the following sense: Th e balls can be arrangedin a cyclic order for which it is t rue that the interactions (collisions) can onlyoccur if the balls are neighbours in this cyclic order. In other words, the graphof allowed collisions is a circle of length N.

In th e article [B-L-P-S(1992)j the authors introduced a sophist icated roomsand-passages struct ure for th e container tha t only allowed ball-to-ball collisionsbetween the k-th and k +1-st balls (k = 1, .. . , N ) in accordance with th e cyclicorder. Th e size of the container was comparable with the common size of theballs in v -1 dimensions, and it was elongated in the first dimension up to a sizebeing proportional to the number of balls N . Furthermore, the boundary BCof the v-dimensional container C, by being concave from the out side (bendingaway from th e configuration space) , provided some addit iona l dispersing effect.

At th is point we have arrived at Sinai 's celebrated "pencase model" of hardballs. Here the container to rus is elongated in one direct ion, i. e. the balls withthe common rad ius r > 0 move in a torus

In order to have a given cyclic order (1,2 , . . . , N , 1) preserved , one must assumethat~/4 < r < 1/2. (Which, in turn , requir es that v :S 4.) Th en thephase space of the pencase model consists of N ! connected, mutually isomorphi ccomponents, and the ergodicity (or , stronger ergodic properties) can only holdon each of th em, separately. In the early 80's Ya. G. Sinai suggested thi s modelfor st udy ing its ergodic properties, and also as a good caricature model forunderst anding its - essent ially one-dimensiona l - hydrodyn amics.

By genera lizing th e concept of cyclic interactions, we can consider a widerclass of models with restricted interactions. Namely, assume th at th e graph 9 :=

(V , t:) of interactions is given, where the set of edges £ forms a non-orient ed tree(i. e. a connected graph with out loop) on th e set of vertices V := {I,2, . . . , N} .(In the case of th e mechanical model of N hard balls in TV t his graph is, ofcourse, the complete one.) Fur th er , for every {i , j}: 1 :S i, j :S N , i f:. j , thepotential funct ions Ui ,j : lR+ -t n, U {oo}are given as follows

U. ' (r ) = {oot ,] 0

o:S r < d i ,j

r > d ·- t ,)

58 N. Simanyi

where the numb ers di ,j = dj ,i are nonnegative. Th ey determine the set of edgesE: {i,j} E £, if and only if di,j > O. Moreover, the interaction between the pair{i, j} of particles with configuration coordinates qi , qj E TV is

where 11 .11 denotes the euclidean distance in the torus TV.The first result of [8-8z(1995)] is the following

Theorem 1.5 Ass1lme that the edge-degree of the tree of interactions is D ,N 2: 2, 1/ 2: D + 2, and intQ (or , equivalently intM) is conn ected. Then thestandard billiard ball flow with a V -interaction is a K-flow.

Remark 1.6 In general, if intM decomposes into a finit e number of connected components, then the methods of {S-Sz(1995)j provide that the staruiardbilliard ball flow with a D-inieractioti is a K- system on each of these connectedcomponents.

Remark 1.7 As it has been proved by Chernov and C. Hask ell {C-H(1996)jand by Ornstein and Weiss , {O- W(1998)J, the K-mixing property of a semidispersive billiard flow actually implies it s Bernoulli property.

The second result of [8-8z(1995)1 concerns the so-called cyclic in teraction,where £'C:= {{I , 2}, {2, 3}, .. . , {N - 1, N} , {N, I}} . Let D; be a correspondingmatrix of interaction ranges.

Theorem 1.8 Th e standard billiard ball system with a cyclic interaction isa K-system if 1/ ::::: 4 and intQ is connec ted.

As mentioned before, an interesting mechanical realization of the cyclic inter action is Sinai's pencase model. Now, as a particular case of th e theoremabove, we obt ain the following

Corollary 1.9 Sinai 's pencase model is a K-system if 1/ = 4 and V3/4 <r < 1/2.

By strengthening the inductive arguments of the proofs of the theoremsabove, the authors of [8-8z(1995)I also obtained the following results:

Theorem 1.10 Assume that the edge-degree of the graph of interaction isD . If N 2: 2, 1/ = D+1, then, almost everywhere, none of the relevant Ly apunovexponents of the standard billiard ball flow with a Ti-isu eract ion vanishes . Furthermore, the ergodic components of such a system are open (and thus of positi vem easure) , and on each of them the flow has the K-property.

Hard Ball Syst ems and Semi-Di spersive Billiards: Hyperbolicity and Ergodicity 59

Theorem 1.11 If the dimension v is equal to three for the hard ball systemwith a cyclic in teracti on, then all relevant Lyapunov expone nts of thi s flow arenonzero almost everywhere . Furth ermore, the ergodic components of the systemare open (an d thus of positive m easure) , and on each of them the flow has theK -property.

1.1.3 Hard Ball Systems With Different Masses. In the art icle [88z(1999)1 we introduced a new, algebraic approach to prove the almost sureexistence of geometrically hyperbolic (sufficient) orbits, provided that their symbolic collision sequences are combinatorially rich, which in this case means thatthese tr ajectories are already proved to possess infinitely many, consecut ive, connected collision graphs. In other words, by complexifying the system and usingsome basic algebraic geomet ry, we were able to carry out Step I of the previouslyoutlined stra tegy with the codirnension-two submanifold replaced by prop er (i.e. of codimension at least one) submanifolds. T his project was done th rough theuse of the Connected Path Formula (CPF) , which first app eared in [8im(1992B)]. The Connected Path Formula paves the way for proving the main result of[8-8z(1999)], namely th at for almost every (N + Ll-tuple (ml , ... , m N ;r ) of theouter geometric parameters the hard ball system with masses ml , . . . ,m N andcommon ball radius r is fully hyperbol ic, i. e. all of its relevant characterist icexponents are nonzero almost everywhere.

1.2 Box Billiards. None of the above result s took up the problem of handling hard balls in physically more realistic containers, e. g. rectangular boxes.The extra technical hardship in their investigation is caused by the loss of thetotal momentum and center of mass. This amounts to th e increase of the dimension of the configura t ion (phase) space without any addit ional scattering effectas a compensation . The problem of proving ergodicity for N hard spheres in av-dimensional rectangular box is so difficult that the only result so far is the(hyperboli c) ergodicity of such systems in the case N = 2, v 2 2.

Let us consider the billiard system of two hard balls with unit mass (just forsimplicity) and radius r (0 < t: < 1/4) moving uniformly in the v-dimensional(v 2 2) Euclidean container

C = [-r, 1 + r]k x ll'v-k = [- r, 1 + rlk x (lRv- k/zv-k )

(0 S; k S; v) and bouncing back elastically at each other and at the boundary BCof C. Denot e the cente r of the i-th ball (i = 1,2) by qi , and its t ime derivativeby Vi = qi . Also denote by A = {I , 2, .. . ,k} the set of the first k (i. e. thenon-p eriodic) coordin ate axes of C , and by 7l"2( .) the projec tion (of a positionor a velocity) onto the second , periodic factor ll'v- k of the container C . We usethe usual reductions 2E = IIvIW + IIv2 W= 1, 7l"2(ql + q2) = 7l"2(VI + V2 ) = o.Plainly, the configurat ion space is the set

Q = {(ql , q2) E ([0, Ilk x ll'v-k) x ([0, Ilk x ll'v- k) Idist(ql ' q2) 2 2r and 7l"2(ql + q2) = o}

60 N. Simanyi

with a connected interior. The phase space M of the arising semi-dispersivebilliard flow is almost the unit tangent bundle of Q. The only modification tothe unit tangent bundle is that the incoming and outgoing velocities at 8Q areglued together , just as it is prescribed by the law of elastic collisions: the postcollision (out going) velocity is the reflected image of the pre-collision (incoming)velocity across the tangent hyperplane of 8Q at the point of collision. We call thearising semi-dispersive billiard flow the standard (v , k , 1' ) model, or the standard(v ,k,r) flo w.

Theorem 1.12 For every triple (v ,k,r) (v ~ 2,0 S; k S; u , 0 < r < 1/4)the standard (v , k , r) model is ergodic, hence it is actually a Bernoulli flow.

Remark 1.13 It becam e clear from the proof of this theorem that the assumption s

• on the equality of the side lengths of the container C ,

• on the equality of the ma sses of balls, and

• on the equality of the radii of balls

are not essential, but m erely notation al simplification s. Th e proo]easily carri esover to the general case when these equalities do not hold .

1.3 The Lorentz-Process and Lorentz-Gas. Let Gl , G2 , • • • be an infinit e sequence of mutually disjoint , compact , strictly convex subsets of ]Rd (d ~ 2)with smooth (G2) boundaries 8Gi . The convex sets C, will be called "scatterers". We assume that th e configuration of the scatterers {Gd~l is d-periodic,i. e. there exist linearly independent vectors t l , . . . , td in ]Rd such that the unionC = U:l Ci is invariant under the translations by th e vectors t j , j = 1,...,d.We also assume that the configuration {Cd~l has "finite horizon", i. e. the closure Q of the set ]Rd \ G does not contain an ent ire line. The dyn amical system(M, {5 t h EIR , p,) describing the uniform motion of a point in the curvature freeconfigurat ion space Q (with elastic reflections at th e boundary 8G = U:l 8Gi )

is called the Lorentz-process with periodic configuration of scatterers. Here thenatural invariant (Liouville) measure p" being invar iant under the translationsby the vectors tj , is obviously infinite. We have tha t dp, = dq x dv , where dqis the Riemannian (Lebesgue) volume in Q and dv is the natural Riemann ianvolume in the unit sphere of velocit ies § d- l .

1.3.1 The Lorentz-gas. Besides th e Lorent z-process outlined above, thereis another approach to a gas model. Namely, instead of considering the (uniform)motion of a single point-particle in Q, we can as well consider the equilibriumtime-evoluti on of a syste m where infinite ly many point-particles are placed on Qaccord ing to a Poisson process. Thi s Poisson measure for configurat ions, along

Hard Ball Systems and Semi-Dispers ive Bill iards: Hyperbolicity and Ergodicity 61

with the independent Gauss distribut ions for th e velocities of the points definean equilibrium probability mesure v on the phase space M . The arising flow(M , {st} ,v) is often called th e Lorentz-gas.

In his paper [Sin(1979)] Sinai proved that the Lorentz-gas (M ,{St},v) hasthe K-mixing prop erty, by effectively const ructing a K-partition for the flow.Then, in the art icle IB-S(1981)1 Bunimovich and Sinai proved the Central LimitTheorem (CLT) for the two-dimensional Lorentz-process (M,{sthEIR ,fl) , i. e.they proved tha t the distribution of the phase point s tx / Vi converges to a nondegenerate Gaussian law (t ---+ (0) if the initi al phase point x E M is selectedfrom a given, bound ed subdomain of M according to the condit ional measureof fl . Moreover , they proved Donsker's "Invariance Principle", i. e. that the processes

{satx / va :0 ::::; t::::; I}

weakly converge (as a ---+ (0) to a Wiener-pro cess with a non-degenerat e covariance matri x. In these result s th e authors heavily used the Markov par tition forthe associated dispersive billiard syste m const ructed by them in their precedingpaper [13-S(1980)1 . By the associated dispersive billiard system one means theLorent z-process (M , { st }tEIR , fl) factored out by the linearly independent spati altranslation vectors tl , t2 with respect to which the configuration of scatterers isinvar iant . We have to point out here, however, that these results heavily usedthe assumpt ion d = 2.

In the paper IK-Sz(1983)] Kramli and Szasz reproved the CLT for th e Lorent zprocess by using perturbat ion theory. However, the problem of recurrence forthe 2D Lorent z-pro cess remained wide open at that time. Two years later , in[K-Sz(1985)] the same authors proved a relaxed form of recurrence for the 2DLorentz-process, namely, th ey showed that the phase point stnx returns to theball B (0, 10gPn) (the ball with radius 10gPn cente red at the origin) infinitelyoften for fl-almost every phase point x . Here p > 3/2 is an arbitrary, fixedconstant, and tn denotes the t ime moment of the n-t h collision of the positivetrajectory S (O,oo)x. The authors used (and proved) a logarithmic, quasi-localversion of the Central Limit Th eorem for the Lorentz-process.

In the arti cle [Sim(1989)] I proved that, even if the 2D Lorentz-process istransient , the invariant measure of a wandering set W c OM must be finite,and an explicit upper bound for the measure of the set W was given there inte rms of the limiting covariance mat rix of the process

featuring in the Invariance Principle proved earlier by Bunimovich and Sinai in[13-S(1981 )1 · By a wandering set We OM we mean the following: We considerthe Poincare sect ion (OM,T ,v) of the Lorent z-process (M , {sth EIR , fl) , so thatthe tra jectories are only observed at the moments of collisions. Th en a measura ble set W c OM is T-wandering if and only if v(Tnw n Trnw) = 0 form i= n (m ,n E Z) and v (OM \ U n EZ T" W) = O. It is an elementary tas k to

62 N. Sirna nyi

see that the measur e I/(W) of th e wand ering sets is uniquely determined by th edynamical system (8M ,T , v). The relat ion v(W ) < 00 means, in some sense,being close to be recurrent.

Fin ally, in the articles [C(1999)] and [Sch(1999)] J .-P. Conze and K. Schmidtproved (independently, and by using different approaches) the recurrence of th e2D Lorentz-process, by successfully answering a long standing open question.

1.4 Stadia. The first convex billiard with full hyperbolicity and ergodicitywas discovered by L. A. Bunimovich [B(1979)], and it is now called the "Bunimovich stadium". Let the configuration space (the "billiard table") be the convexhull Q of two disks D 1 and Dz of the same radius r in the plane ]Rz . (We assume, of course, th at th ese disks are not concent ric.) The Bunimovich stadiumis th e flow (M,{SthEIR,fL) describing th e uniform motion (with unit speed) ofa point par ticle inside Q, so that thi s point always boun ces back at the boundary8Q totally elastically. The Liuoville measure fL is aga in the norm alized Lebesguemeasur e in the unit tangent bundle Qx § l . It is more convenient to st udy th e flow{st } at the moments of reflections, switching this way to the natural Poincaresect ion map (8M, T ,v). Here the phase points x = (q,v) E 8Q X § l = 8M areunderstood to be supplied with the outgoing (post-collision) velocity v = v+.Thus, 8M is naturally isomorphic to the product 8Q x (- 7[/2, 7[/2) , where th eopen interval (-7[/ 2, 7[ / 2) contains the oriented angle ¢ subtended between theunit (inside) norm al vector n(q) of 8Q at th e point q E 8Q and th e outgoingvelocity vector v of th e phase point x = (q,v) . The boundary 8Q , on th e otherhand, is naturally parametrized by th e arc length s , 0 :::; S :::; C, where C isthe circumference of the stadium Q. The arc length parameter s ought to beconsidered periodi cally with period C .

In his insemin al paper [B(1979)] Bunimovich proves that th e billiard map(8M ,T , 1/) is fully hyp erb olic and ergodic. The govern ing mechanism behindthese phenomena was then fully explored and clarified by Wojtkowski in theseries of papers [W(1985)j, [W(1986)], and then by Liverani and Woj tkowskiin [L-W(1995)]. Thi s mechanism is nothing else but th e shea r existence of a socalled "eventually strictly invariant , measurable cone field" C(x), (x E 8M).(For some technical det ails, please see sect ion 2.1. in thi s survey.) As a matterof fact , for proving ergodicity one needs to use Sinai 's celebrated method ofproving local ergodicity, laid down in the groundbreaking pap er [Sin(1970)]. Forsome technical details of that proof, cf. sect ion 2.2. of the present survey.

In the paper [W(1985)] Wojtkowski developed the necessary linear- algebr aicbackground for the theory of invariant cone fields in two dimensions. Th enthe achievements of this paper had fru itful appli cations in [W(1986)], whereWojtkowski was able to define a robu st class of 20 billiard ta bles - billiards withthe so called "convex scattering property" - with apparent full hyperbolicityand ergodicity. Here we are outlining the definition of a 2-D billiard with theconvex scattering property.

Hard Ball Systems and Sem i-Dispersive Billiard s: Hyp erbolicity and Ergodici ty 63

Consider a bounded, connected domain Q C IR2 (the billiard table) witha piecewise smooth bound ary 8Q = U~=l Gi , where GI , .. . , Gn are the smoothcomponents of 8Q. We assume that if C, n C, i- 0 (i. e. when O, and Gj sharejust a common endpoint {q}) then the tangent lines of C, and Gj at q aredifferent. We will distin guish between three types of boundary components Gi :

T he ones that are convex (with st rict ly positive curvat ure and bending towardsthe region Q) , the ones that are flat (i. e. line segments ), and the ones th at areconcave (with strictly positive curvat ure and bending away from the region Q) .If C, is convex, q E Gi , denote by D2(q) the disk tha t is tangent to C, at q andhas the radius half of the curvat ure radius of C, at q. (D 2(q) is on the same sideof C, as Q .) For an arbitrary phase point x = (q,v) E 8M with the propertyth at q lies on a convex arc Gi , denote by d(x) the length of the intersect ionof the disk D 2 (q) with the line {q + tv : t E IR} , in other words , the relationq+ tv E D2(q) holds if and only if 0 ~ t ~ d(x). Let

{x(t) = (q(t),v(t )) = s t (x(O)) : 0 ~ t ~ b}

be an arbit rary, non-singular orbit segment of the billiard flow (M, {st h EIR , J.L)with the following prop ert ies:

• q(O) E 8Q is on a convex arc Gi ;

• q(b) E 8Q is on a non-flat arc Gj ;

• for 0 < t < b the point q(t) is either in intQ or it lies on a flat arc Gk .

The convex scatte ring prop erty of {x(t) : 0 ~ t ~ b} means, by definition, that(i) d( x(O)) + d( x(b)) ~ b if q(b) belongs to a convex component of the

bound ary;(ii) d (x(O)) ~ b if q(b) belongs to a concave component of the boundary.We say that th e billiard flow (M, {st h EIR , J.L) enjoys the convex scattering

prop erty if every orb it segment listed in (1)-(3) above enjoys that property.If, in addit ion, almost every orb it contains a finite segment with the convexscattering prop erty so that in (i) or (ii) above stri ct inequality holds, then wesay that the billiard has the event ually strict convex scattering prop erty. Th eresults of the paper [W(1986)] are two-fold: Wojtkowski shows, on one hand ,that every 2-D billiard with the event ually stric t convex scattering prop erty isfully hyperboli c and ergodic, and he presents to the reader, on the other hand ,a straightforward geomet ric method for designing a robust family of billiardtables with th e eventually strict convex scattering prop erty.

We need to point out , however , that such a const ruction of a robust (stableunder small G2 perturbations of the billiard table) family of convex scattering billiards is essent ially two-dimensional, despite the claims in the papers[B-R(1997)] and [B-R(1998)]. Indeed, the higher dimensional construct ions ofconvex scat tering stadia in those papers merely provide very specific billiardtables; the const ruct ion far from being generic.

64 N. Simanyi

Finally, in the art icle [L-W(1995)1 Liverani and Wojtkowski generalized tohigher dimensions the linear-algebraic background of invariant cone fields,paving the way to use the result s of [W(1985)] for higher dimensionsal symplectomorphisms with singularities. An appropriate generalization to symplectomorphisms with singularities of the Chernov-Sinai Theorem on Local Ergodicity is also given there. For more technical detailes related to thi s topi c, seesections 2.1 and 2.2 of this survey.

1.5 Falling Balls. In his paper [W(1990-A)] M. Wojtkowski int roduced thefollowing Hamiltonian dynam ical system with discontinuities: There is a verticalhalf line {ql q 2 O} given and n (2 2) point particles with masses ml 2 m 2 2.. . 2 m n > 0 and positions 0 ~ ql ~ q2 ~ .. . ~ qn are moving on this half lineso that they are subj ected to a constant gravitational acceleration a = - 1 (theyfall down) , they collide elastically with each other, and the first (lowest) par t iclealso collides elastically with the hard floor q = O. We fix the total energy

by taking H = 1. The arising Hamiltonian flow with collisions (M, {Sth EIR ,J1.)(J1. is the Liouville measure) is called a "system of one-dimensional balls withgravity" or a "system of one-dimensional falling balls".

Before formulat ing some results on one-dimensional systems of falling balls,however , it is worth mentioning here three important facts:

(1) Since the phase space M is compact, the Liouville measure J1. is finite.(2) The phase points x E M for which the orbit s[a,bJxo hit s at least one

singularity (i. e. a multiple collision) are contai ned in a countable union ofprop er , smooth submanifolds of M and, therefore, such points form a set of J1.measure zero.

(3) For J1.-almost every phase point x E M the collision moments of theorbi t s [a ,bJxo do not have any finite accumulat ion point, see the Appendix of[Sim(1996)] .

In the pap er [W(1990-A)] Wojtkowski formulated his main conjecture pertaining to the dynamical system (M ,{sthEIR ,J1.) :

Wojtkowski's Conjecture. If ml 2 m2 2 . . . 2 m n > 0 and ml #- m n ,

then all but one characteristic (Lyapunov) exponents of the flow (M , {SthE~ ' J1.)are nonzero.

Remark 1.14 1. The only exceptional zero exponent must correspond to theflow direction .

2. The condition of non increasing masses (as above) is essential for establishing the invariance of the symplectic cone field - an important conditionfor obtaining nonzero characteristic exponents. As Wojtkowsk i point ed out in

Hard Ball Systems and Semi-Dispersive Billiards: Hyp erbolicity and Ergodicity 65

Proposit ion 4 of {W{1990-A)J, if n = 2 and ml < m2, then there exists a linearly stable periodic orbit , thus dimming the chances of proving ergodicity.

In pursuing the goal of proving this conjec ture, Wojtkowski obtain ed the following results in {W{1990-A)J:

Proposition 1.15 For every E > 0 there is a 8 > 0 such that if ml >m2 > .. . > m n > 0 and ml~;nn < 8, then (M , {st h EIR , J-t) has exactly one zerocharacteristic exponent except possibly on a se t of J-t measure < E.

Proposition 1.16 If there are exactly l groups of particles with equal masses,l ~ 2, conta ining k1, • . • , k1 particles respectively, the greatest common divisorof ki , . . . , k1 is one and ml ~ m2 ~ ... ~ m n > 0, then {'Ij/} has exactly onezero characteristic exponent on a set of posit ive Liou ville measure.

Proposition 1.17 If n = 3 and ml > m2 > m3, then {1ft} has exactly onezero charact eristic exponent u-olmosi everywhere.

In the subsequent art icle IW(1990-B)] Wojtkowski replaced the linear potenti al U(q) = q of constant gravit ation by a varyin g gravitat ional force withpotenti al U(q) for which U'(q) > 0 and UI/(q) < O. (The usual gravitat ional potenti al U(q) = q~~O belongs to this category!) He proved there that in the fallingball system with such a potential U(q) all relevant char acterist ic exponents arenonzero almost everywhere.

The result of the paper ISim(1996)] is a proof for a slightly relaxed versionof Wojtkowski's original conjecture :

Theorem 1.18 If ml > m 2 ~ m 3 ~ . .. ~ m n > 0, then u-olmosi everywhere all but one charact erist ic exponents of the flow (M, {st h EIR , It) arenon zero.

We are closing thi s section by mentioning that in his work [Ch(1993)1 Chernov significant ly relaxed a condit ion of the Liverani-Wojtkowski local ergodicitytheorem for symplectomorphisms, [L-W(1995)]. (This theorem is a generalization of the celebrated Theorem on Local Ergodicity for semi-dispersing billiardsby Chernov and Sinai, [S-Ch(1987)].) The ominous condit ion is the "proper alignment" of the singulari ty manifolds, Condit ion D in Section 7 of [L-W(1995)].Thi s condition is easily seen to be violated by the system of falling balls (seeSection 14.C of [L-W(1995)]), but the relaxed condit ion 5' of Chernov's paper[Ch (1993)1 is very likely to hold for thi s system. However , checking the condition5' (the transversality of the st able and unstable foliati ons) for the falling ballmodel seems very difficult , if not hopeless. Therefore, proving the ergodicityfor systems of falling balls seems much more difficult than just showing thatsuch systems are fully hyperbolic (i. e. that all of their relevant characterist icexponents are nonzero almost everywhere) .

66 N. Simanyi

For a more detailed introduct ion to this subject, and for a thoroughly assembled collection of references and historical remarks, the reader is kindly referredto the introduction of the paper [W(1990-A)I.

It is easy to see that the st udying of the falling point par ticles on the vert icalhalf line {qjq 2: O} is not a rest rict ion of genera lity as compared to the systemsof I-D balls (hard rods) of length 2r . Namely, the simple change in the kineticdata q i t-t qi - (2i - l )r , Vi t-t Vi , Ho t-t Ho - r L~l (2i - l )m i (the changeof the fixed level of energy) establishes an isomorphism between the hard rodsystem and the point par ticle model.

1.6 Cylindric Billiards - A Common Generaliza tion. Non-uniformlyhyperbolic systems (possibly, with singularit ies) playa pivot al role in the ergodictheory of dynamical systems. Their systemat ic st udy started severa l decades ago,and it is not our goal here to provide the reader with a comprehensive review ofthe histor y of these invest igat ions but, inst ead, we opt for presenting in nutshella cross sect ion of a few selected results.

In 1939 G. A. Hedlund and E. Hopf [He(1939)], [Ho(1939)], proved the hyperbolic ergodicity of geodesic flows on closed, compact surfaces with constantnegative curvature by inventing the famous method of "Hopf chains" constitutedby local stable and unst able invari ant manifolds.

In 1963 Ya, G. Sinai [Sin(1963)] formul ated a modern version of Boltzmann 'sergodic hypothesis, what we call now the "Boltzmann-Sinai ergodic hypothesis":the billiard syste m of N (2: 2) hard balls of unit mass moving in the flat torus']['11 = ]RII / ZII (v 2: 2) is ergodic after we make the standard reductions by fixingthe values of the trivi al invar ian t quantit ies. It took seven years unti l he provedthis conject ure for the case N = 2, v = 2 in [Sin(1970)]. Another 17 years laterN. 1. Chernov and Ya. G. Sinai [S-Ch (1987)] proved th e hypothesis for the caseN = 2, v 2: 2 by also proving a powerful and very useful theorem on localergodicity.

In the meantim e, in 1977, Ya. Pesin [P (1977)] laid down the foundations ofhis theory on the ergodic properties of smoot h, hyperbolic dynamical systems.Lat er on this theory (nowadays called Pesin theory) was significant ly extendedby A. Katok and J-M . Str elcyn [K-S(1986)] to hyperboli c systems with singular ities. That theory is already applicable for billiard systems, too.

Unti l the end of the sevent ies the phenomenon of hyperbolicity (expon entialinst ability of the tra jectories) was almost exclusively at t ributed to some directgeometric scat tering effect, like negative curvature of space, or st rict convexityof the scatterers. Thi s explains the profound shock that was caused by thediscovery of L. A. Bunimovich [B(1979)]: certain focusing billiard tables (likethe celebrated stadium) can also produce complete hyperbolicity and, in th atway, ergodicity. It was par tly this result that led to Wojtkowski's theory ofinvariant cone fields, [W(1985)], [W(1986)].

T he big difference between the system of two balls in ']['11 (v 2: 2, [S-Ch (1987)])and the system of N (2: 3) balls in ']['11 is that the latter one is merely a so called

Hard Ball Systems and Semi-Di spersive Billiards: Hyp erboli city and Ergodicity 67

semi-dispersive billiard syst em (the scat terers are convex but not st rictl y convexsets , namely cylinders), while the former one is st rict ly dispersive (the scatterers are st rict ly convex sets). Thi s fact makes the proof of ergodicity (mixingprop ert ies) much more complicated. In our series of papers jointly writ ten withA. Kramli and D. Szasz [K-S-Sz(1990)], [K-S-Sz(1991)], and [K-S-Sz(1992)] wemanaged to prove the (hyperbolic) ergodicity of three and four billiard balls ina toroidal container 1[''' . By inventing new topological methods and the Connecting Path Formula (CPF) , in the two-volume pap er [Sim(1992-A-B)] I proved the(hyperbolic) ergodicity of N hard balls in 1[''', provided that N :::; t/ ,

Th e common feature of hard ball systems is - as D. SZ&'lZ pointed thi s outfirst in [Sz(1993)] and [Sz(1994)] - that all of them belong to the family ofthe so called "cylindric billiards", the definition of which can be found later inthis paragraph. However , the first appearance of a special, 3-D cylindr ic billiardsyst em took place in [K-S-Sz(1989)], where we proved the ergodicity of a 3-Dbilliard flow with two orthogonal cylindric scatterers. Later D. Szasz [Sz(1994)]presented a complete picture (as far as ergodicity is concerned) of cylindricbilliard s with cylinders whose generator subspaces are spanned by mutuallyorthogonal coordinate axes. The tas k of proving ergodicity for th e first nontrivial, non-orthogonal cylindric billiard system was taken up in [S-Sz(1994)].

In our joint article with D. Szasz [S-Sz(1999)] we managed to prove thecomplete hyperboli city of typical hard ball systems.

1.6.1 Cylindric billiards. Consider the d-dimensional (d ~ 2) flat torus'j['d = IRd / E supplied with the usual Riemannian inner product ( . , .) inheritedfrom the standard inner produ ct of the universal covering space IRd • Here £ C IRd

is supposed to be a lattice, i. e. a discrete subgroup of the additive group IRd

with rank(£) = d. T he reason why we want to allow general lat tices, other thanjust the integer lat tice Zd, is th at otherwise th e hard ball systems would notbe covered. The geomet ry of the st ructure lat tice E in the case of a hard ballsystem is significant ly different from the geomet ry of the standard rectangularlat tice Zd in the standard Euclidean space IRd.

The configuration space of a cylindric billiard is Q = 1['d \ (C1 u· · ·u Ck),where the cylindric scatterers C, (i = 1, .. . , k) are defined as follows:

Let Aj C IRd be a so called lattice subspace ofIRd, which means that rank(Ajn£) = dimzl.. In th is case the factor Ad(A j n £) is a sub torus in 1['d = IRd/ Lwhich will be taken as the generator of the cylinder C; C 1['d, i = 1, . . . ,k. Denoteby L, = At the orthocomplement of Aj in IRd. Throughout this survey we willalways assume that dim.L, ~ 2. Let , furthermore, the numb ers r j > 0 (the radiiof the spherical cylinders Cj) and some translation vectors t j E 1['d = IRd/ L begiven. The translation vectors tj play a role in positioning the cylinders C, inthe ambient torus 1['d. Set

C j = { x E 1['d : dist (x - t j, Ad(A j n £)) < rd .

68 N. Sima nyi

In order to avoid further unnecessary complicat ions, we always assume that theinterior of the configurati on space Q = ']['d \ (C1 u ·· ·u Ck) is connected. Thephase space M of our cylindric billiard flow will be the unit tangent bundle ofQ (modulo some natural glueings at its boundary) , i. e. M = Q X § d- l . (Here§ d- l denotes the unit sphere of ]Rd.)

The dynamical system (M, {SthEIR ,I.L) , where st (t E ]R) is the dynamicsdefined by uniform motion inside the domain Q and specular reflections at itsboundary (at the scat terers), and J.L is the Liouville measure , is called a cylindricbilliard flow. (As to notion s and not ations in connect ion with semi-dispersivebilliard s, the reader is recommended to consult the work [K-S-Sz(1990)].)

1.6.2 Transitive cylindric billiards. T he main conjecture concerning the(hyperboli c) ergodicity of cylindric billiards first appea red as Conjecture 1 inSection 3 of [S-Sz(1998)]:

Main Conjecture. A cylindric billiard flow is ergodic if and only if it istransitive. (As for the definition and basic features of t ransit ivity, see later inthis sect ion.) In th at case the cylindric billiard system is actually a complete lyhyperboli c Bernoulli flow , see [C-H(1996)] and [O-W(1998)].

The th eorem of [Sim(1999-B)] proves a slight ly relaxed version of this conjecture (only full hyperboli city without ergodicity) for a wide class of cylindricbilliard systems, namely the so called "transverse systems" (see later in thissection) which include every hard ball system.

Theorem [Sim(1999-B)j. Assume that the cylindric billiard system istransverse. Then this billiard flow is complete ly hyperboli c, i. e. all relevantLyapunov exponents are nonzero almost everywher e. Consequentl y, such dynamical systems have (at most count ably many) ergodic components of positivemeasure, and the rest rict ion of the flow to the ergodic components has theBernoulli prop ert y, see [C-H(1996)] and [O-W(1998)].

Corollary of the theorem. Every hard ball system - necessarily beinga transverse cylindric billiard system - is completely hyperbolic.

Thus, the theorem of [Sim(1999-B)] generali zes the main result of [S-Sz(1999)],where the complete hyperboli city of almo st every hard ball system was proven.

1.6.3 Transitivity. Let L 1 , . . . , Lk C ]Rd be subspaces, dimL, ;:;:: 2, A; =

Lr , i = 1, . . . ,k. Set

9; = {U E SO(d) : UIA; = IdA.} ,

and let 9 = (91 , .. . ,9k) c SO(d) be th e algebraic generat e of th e compact ,connected Lie subgroups 9; in SO(d) . The following notions appea red in Section3 of [S-Sz(1998)].

Definition 1.19 We say that the system of base spaces {L 1, . . . , L k} (or,equivalently, the cylindric billiard system defin ed by them) is transitive if andonly if the group 9 acts tran sitively on the unit sphere § d- l of ]Rd.

Hard Ball Systems and Semi-Dispersive Billiards: Hyperbolicity and Erg odicity 69

Definition 1.20 We say that the system of subspaces {L 1 , . . . , Lk } has theOrthogonal Non-Splitting Property (ONSP) if there is no non-trivial orthogonalsplitti ng IRd = B 1 EB B 2 of IRd such that for every index i (1 :S i :S k) L , C B 1

or i; C B 2 .

The next result can be found in Section 3 of [S-Sz(1998)] (see 3.1-3.6 thereof) .

Proposition. For the system of subspaces {L 1 , .•• , Lk } the following threeprop erties are equivalent :

(1 ) {L1 , oo . ,Ld is transitive;(2) {L 1 , 00 • , L d has the ONSP ;(3) the natural represent at ion of 9 in IRd is irreducible.

1.6.4 Transverseness.

Definition 1.21 We say that the system of subspaces {L 1, . 00 , L d of IRd

is transverse if the follow ing property holds: For every non-transi tive subsystem{L i : i E I} (1 C {I , . 00 ,k}) there exists an index jo E {I , 00 . ,k} such thatPE+(Aj a ) = E+, where A j a = L~ , E+ = span{L i : i E I}, and PE+( . )denotes the orthogonal projection onto the subspace E+. We not e that in thiscase, necessarily, jo 'f. 1, otherwise PE+(A j a ) would be orth ogonal to the subspaceL j a C E+ . Therefore, ever y transverse system is automatically transiti ve.

By using a qui te different approach (actu ally, by carefully scrut inizing thedynamical and geometric properties of all, combinatorially different symboliccollision sequences) P. Balint [B(1999)] was able to obt ain the following results:

Theorem [B(1999)]. If the t ransitive cylindric billiard system consists ofat most three cylinders, then it is fully hyperbolic.

Theorem [B(1999)]. If the cylindric billiard system is transitive and theaxes of cylinders A j (j = 1, . . . , k) have the property that Ail n Ah = {O} fori, -1-12 , then the billiard flow is fully hyperbolic and ergodic.

§2. Some Technical Aspects of the Results

2.1 Full Hyperbolicity - Invariant Cone Fields. In our outline of thegeneral theory of invariant cone fields we will closely follow the set-up of §7 ofthe excellent survey [L-W(1995)].

Let (M 2d, w) be a smooth, connected , compact , simplect ic manifold , pos

sibly with boundary aM. Suppose tha t - besides the simplect ic form w - Malso possesses a smooth Riemannian metric 9 in such a way that wand 9 areproperly coupled, i. e. for every point x E M there exists a g-orthonormal ba-

70 N. Simany i

sis {el ' .. . ,e2d } in the tangent space TxM which is also w-orthonormal in thefollowing sense:

{

I if j = i +d,

w(ei , ej) = -1 ifj = i- d,

ootherwise.

In thi s case the Riemannian volume element du is automat ically equal to thesimplect ic volume element w 1\ w 1\ . . . 1\ w (d copies). Assume, moreover , thattwo compact subsets S + and S - are given with th e following properties:

(Sl ) S + and S - are finite unions of (2d - 1)-dimensional, smooth, compactsubmanifolds {S: : 1 :::; i :::; I+} and {Si- : 1 :::; i :::; I -} (possibly with boundaries 8Si±);

(S2) ueu (S ±) = 0;(S3) S: nS! = 8S: n 8S! , s; n S; = 8Si- n 8Sj- for i i- j .Here ueu denotes the surface measure (indu ced by the restriction of g) on

8M. The last component of our dynamical syste m (M,T , /-l) is the mapping T .Suppose that a simplectomorphism T : M \ S + -+ M \ S - is given with thefollowing, addit ional properties:

(T1) For every smooth component S f of S ± th e mapping T has two continuous exte nsions on Si±' one from each side of the submanifold Si± in M ;

(T2) The derivatives DT, DT- I are well behaved near sf ,i. e. they satisfythe Katok-Strelcyn condit ions [K-S(1986)], so th at we can apply th eir result s[K-S(1986)] on the existence of stable and unstable foliations and their absolutecontinuity.

We will say that T is a (discontinuous) simplectomorphism T : M -+ M .Formally, T is not well defined on the set S + (and T- I is not well defined on theset S - , respectively) . The mapping T has several values (actually, two valuesin the interior of each component S: , and more on the intersections S: n S! 'i i- j ) on its singular set S +, and the inverse map T - I also has several valueson its own singularity set S - . We adopt the convent ion that the image of a setA c M under T (under T - I ) contains all such values.

We will use the not at ions

S+ = S + U T -IS+ U· · · U T -n+lS+(n) ,

S (;.) = S - U TS - U · · · UTn-IS- .

The compact set S~) is the singular ity set of the iteration T" ; while S(-;,) is the

singularity set of 'I": " ,

2.1.1 The cone field. (F) For every point x E M \ S = M \ (S +US- ) thereis a pair of transversal Lagrangian subspaces VI (x) , V2 (x) in TxM given in sucha way th at VI (x) and V2 (x ) depend on x in a measurable manner. (A Lagrangiansubspace of TxM is a d-dimensional, linear subspace L of TxM on which therestricted form wlL vanishes.)

Hard Ball Systems and Semi-Dispersive Billiards : I-Iyper bolicity and Ergodicity 71

Using the above axiom (F), we can now define the quadratic form Qx (sometimes called "infinitesimal Lyapunov exponent") in TxM for every x E M \ Sas follows: Qx(v) = W(VI ,V2 ) for v E TxM , v = VI + V2, Vi E V; (x) . The (measura ble) cone field C(x) (x E M \ S) associated with the pair of transversalLagrangian fields (VI (x), V2 (x)) is defined as

C(x) = {v E TxM : Qx(v) 2': O} .

2.1.2 The monotonicity of T . The key property of the simplectomorphism T is post ulated in the axiom of monotonicity

(M) DxT (C(x)) c C(Tx) for all x E (M \ S) n T -I(M \ S).We say that a point

has an eventually strictly monotonic orbit (or, shortly, the point x itself is eventually strictly monotone) if one can find two integers m < n so that

We postulate the axiom of strict monotonicity as (SM) tL-almost every point

is eventually strictly monotonic.The main result of the paper [W(1985)1and [L-W(1995)] is (as far as only the

complete hyperbolicity is concerned) that every dynam ical system (M2d , T , tL)fulfilling the above listed axioms has only nonzero characteristic (Lyapunov)exponents IL-almost everywhere or, in other words, it is completely hyperbolicor chaotic.

We should note here that the main result of [L-W(1995)] is not the st at ementof complete hyperbolicity, but a strong result on local ergodicity, being thegenera lizat ion (to simplectomorphisms with singularities) of the main theoremon local ergodicity for semi-dispersive billiards proven by Sinai and Chernov in[S-Ch (1987)]. We opted, however, not to cite that result here for at least tworeasons. First is that the main theorem of [L-W( 1995)] requires two additional,very technica lly sounding assumpt ions. Secondly, we will survey the problem oflocal ergodicity in the upcoming sect ion 2.2 of this article .

2.1.3 Examples. 1. The first major fami ly of examples is constituted bythe celebrated Bunimovich stadium [B(1979)] and 2 - D billiards with the socalled "convex scattering property" from [W(1986)], see also sect ion 1.4 above.

72 N. Simanyi

2. The second group of models fulfilling the above axioms is formed by thesystems of one-dimensional balls falling freely under the action of gravitationwith either a linear or a concave potential.The case of linear pot enti al (constantgravitational field) with non-increasing masses ml 2: m2 2: ... 2: ttu. (as we goupwards from below) was thoroughly treated in [W(1990-A)] and [Sim(1986)],see also section 1.5 in this survey. The system with ml > m2 2: m3 2: ... 2: ttu;

proved to be completely hyperbo lic due to t he validity of the above axioms[Sim(1996)J. The paper [W(1990-B)1 proves the complete hyperbolicity in thecase of a concave potential U(q) (q 2: 0 is the vertical posit ion), U'(q) > 0,UI/(q) < O. In [Ch(1991)] Chernov discusses in detail the case of two onedimensional falling balls subjected to the effect of such a potential.

3. Finally, the third major fam ily of examples with an invariant cone fieldis constituted by hard ball systems, see also section 1.1 in this survey. In thiscase the appropriate infinitesimal Lyapunov function Q is the quadratic formQ = I:[:l bqibvi , where bql," " bqN;bVl, . .. , bVN are the components of a tangent vector in the phase space of an N-ball system. In the pap er [S-Sz(1999)] D.Szasz and I proved that for almost every selection of the outer geometric parameters (ml , ' .. , m N ; L) (ml , " " mN are the masses , L is the size of the toroidalcontainer) the hard ball system fulfills all the axioms listed in this section (notably, the axiom of event ual strict monotonicity (SM), too) and , therefore, thesemodels are typically completely hyperbolic.

2.2 Local Ergodicity. By "local ergodicity at the phase point x E M"we mean that a suitable, open neighbourhood of the phase point x belongs toa single ergodic compo nent of the system. (Modulo the zero sets , of course .)

We purs ue the goal of proving this property for any hyperbolic phase point(with at most one singularity on its whole orbit) in a broad class of dynamicalsystems.

T he celebrated Theorem on Local Ergo dicity for semi-dispersive billiardswas proven by Chernov and Sinai, T heorem 5 of [S-Ch(1987)J. Here we wouldlike to formulate a slight generalization of it [K-S-Sz(1990)], which proved tobe rath er useful in th e subsequent proofs of ergodicity for 3 and 4 hard balls,[K-S-Sz(1991)J, [K-S-Sz(1992)J.

2.2 .1 Semi-dispersive billiards a n d invariant m anifolds.

Billiards. A billiard is a dynamical system describi ng the motion of a pointparticle in a connected, compact domain Q C jRd or Q C ']['d = jRd j Zd, d 2: 2with a piecewise C2-smoot h boundary. Inside Q the motion is uniform while thereflect ion at the boundary 8Q is elastic (the angle of reflection equals the angleof incidence) . Since the speed is a first integral of the motion, the phase spaceof our system can be identified with the unit tangent bundle over Q . Namely,the configuration space is Q while the phase space is M = Q X §d-l where§d-l is the unit sphere in jRd. In ot her words, every phase point x is of the form

Hard Ball Syst ems and Semi-Dispersive Billiards: Hyp erb olicity and Ergo dicity 73

(q,v) where q E Q and v E § d-l . The natural project ions 1r : M -t Q andp : M -t § d- l are defined by 1r(q, v ) = q and by p(q, v ) = v, respectively.

Suppose th at 8Q = U~8Qi , where 8Qi are the smooth components of theboundary. Denot e 8M = 8Q x § d-l , and let n(q) be the unit normal vector ofthe boundary component 8Qi at q E 8Qi directed inwards Q .

The flow ({s t} : t E IR} is det ermined for the subset M' c M of phase pointswhose trajec tories never cross the intersections of the smooth pieces of 8Q anddo not contain an infinite number of reflections in a finite tim e interval. If /-ldenotes the Liouville (probabil ity) measure on M , i.e. duiq , v ) = const · dq -dv ,where dq and dv are the Lebesgue measures on Q and on § d- l, respectively,then under certain condit ions /-l(M') = 1 and /-l is invariant , cr. [K-S-F (1980)1.The interior of the phase space M can be endowed with the natural Riemanni anmetric. For our present purpose it is sufficient to pose the following assumpt ion.

Condition 2.1 (Slimness of the set of orbits with an accumulation ofcollisions in finite time). The set of phase points whose trajector y contai nsan accumulat ion of collisions in finite t ime is slim, i. e. it can be covered bya countable family of closed zero-measure sets with topological dimension atmost dimM - 2 = 2d - 3.

We note that a st rong form of Condition 2.1 holds for billiard syste ms isomorphic to systems of elastic hard balls in the torus TV, 1/ 2': 2. As a matter offact , for these syste ms the above mentioned set is act ually empty, cr. [G(1981)]or [V(19791 .

The dynamical system (M, {st} , /-l) is said to be a billiard. Notice, that(M, {st} , /-l) is neither everywhere defined nor smooth.

The main object of the present paragraph is a particularly inte resting classof billiards: that of semi-dispersive ones where , for every q E 8Q, the secondfundament al form K(q) of the boundary is non-negative (if, moreover, for everyq E 8Q , K(q) is positive, th en the billiard is called a dispersive one).

It will be convenient to denote (q, - v ) by -x if x = (q,v ); then, of course,for y = S'« we have -x = st(-y) .

2.2.2 Invariant manifolds. We recall that a CI-smooth , connected submanifold "(S C M without boundary is called a local stable (invariant) manifoldfor {st} at x E M if

(i) x E "( S.

(ii) :lCi = Ci("(S) > 0 (i = 1,2 ) such that, for any YI,Y2 E "(S, t > 0

A local stable manifold for {S - t} is called a local unstable manifold for {st} .The billiard flow {st} has singularit ies of different types. Since we shall

always work with the Poincare section map T it is sufficient to give a more

74 N. Simanyi

det ailed description of the set R C 8M of the singularities of T. R = R' UR" isa 2d - 3 dimensional cell complex which consists of two types of cell complexes:

a) R' := /T-l{the union of the boundaries of smooth pieces of 8Q} ("multiple reflection");

b) R" is the set

{x = (q ,v) E 8M : (v,n(q)) = O}

For n E 12 let

"tangency".

Denot e for arbitrary n E N by ~n the set of double singularities of maximalorder :s:; n. ~n consists of points x E 8M for which there exist two differentintegers k1 and k2 (I k11 < n , Ik21< n) such that Tk, x and T k

2 x belong tothe set of singular reflections ("multiple" or "tangent ial" ones) . Int roduce thefollowing not ations:

8MO := 8M \ Uu:n Eil

00

8M* := 8M \ U~nn =l

2.2.3 Formulation of the m ain theorem. The main aim of th is section isto formulate the Theorem on Local Ergodicity in its most general and applic ab leform. We note that there are two dual forms of thi s theorem: The first oneproviding an ample set of not too short local stable invariant manifolds and theother one stating the same prop erty for the local unst able manifolds . Now weare going to draw up the first (stable) version of the Fundamental Theorem; thedualiz ation is left to the reader.

In order to phrase the theorem we need three important preliminary conditions.

Condition 2.2 (Chernov-Sinai Ansatz) . For fLR.-almost every point x E

R (R being the singularity set) we have x E 8M* and the positive semitrajectory of the point st(x )x is sufficient, where fLR. denot es the measure on thecodimension-one cell complex R of 8M induced by th e Riemannian met ric pand t (x) is the moment of the singular reflection in the trajectory of x.

The other important regularity condition needed for the proof of the Theorem on Local Ergodicity is:

Condition 2.3 (Regularity of the se t of degenerate t angencies) . Theset

{x = (q ,v) E 8Q X §d-l : (v,n(q)) = 0 and K(q)v = O}

Hard Ba ll Syste ms and Semi-Di spersive Billiards: Hyp erbolicity and Ergodici ty 75

is a finite union of compact smoot h submanifolds of 8M (usually with boundary), i.e, thi s set is a compact cell subcomplex in R . Here K(q) denot es th esecond fundament al form of 8 Q at th e point q E 8Q.

Our last regular ity condit ion concerns the sets b.n of double singularit ies:

Condition 2.4 (Regularity of double singularities). For every n E Nthe set b.n is a finite union of compact smoot h submanifolds of 8M (i. e. it isa compact cell complex).

Definition 2.5 Let x E 8M ' and let U(x ) be an open neighborhood of xin 8M diffeomorphic to IR2d- 2 and U(x ) = U a EB d- l f a a smooth foliation ofU(x) with (d - 1)-dimensional smooth submanifolds f a which are uniformlytronsuersal to all possible local stable invariant manifolds in U(x) (Bd- 1 is thestandard (d - I )-dimensional open ball, i. e. the factor of U(x) by the foliation) .

The parametrized family

go= {Gf : i = I ,2 , .. . ,1(J) (0 < J < Jo)}

of finite, open coverings of U(x ) is called a family of regular coverings if th efollowing five requirements are fulfilled:

(a) all the sets Gf are open parallelepip eds of dimension 2d - 2, i. e. t hey areimages of the standard unit cube [0, I]2d-2 C IR2d- 2 under inhomogeneouslinear mappings IR2d- 2 -7 U(x ) where linearity is defined in terms ofa fixed coordina te system in U(x ), say the exponent ial coordinates usingthe mapp ing expx;

(b) th e cente rs wt E 8Mo of Gt (according to this coordinate system) haveth eir own (d- 1)-dimensional stable invariant manifolds "(S (wt) and, moreover , th e tangent spaces

are parallel (according to th e coord inat e system) to some (d - 1)-dimensional faces of Gt.The faces of Gt parallel to Twn S(wt) are called s - faces

(there are 2d- 1 of th em), while those faces of Gf parallel to Twor( wf) are

called Ivfaces (th ere are also 2d-

1 of them) and they are supposed to becubes with edge-length J;

(c) if Gt n GJ=1= 0, then

1/ (Go n GO) > c . J2d- 2,..1 , J _ 1 ,

where Cl > 0 is a fixed number (not depending on J);

76 N. Siman yi

(d) for every 15 < 150 there are at most 22d - 2 different indices 1 ::; i 1 < i 2 <.. . < i k ::; 1(15) such that

knG1; -I- 0;j=l

(e) the system of the centers

{wf : i = 1,2 , . . . ,I(J)}

const itutes asymptot ically (as 15 -+ 0) a (2d - 2)-dimensional linear lattice with edge-length (1 - 0.01)15 such that the stable- and Iv faces of theelementary parallelepipeds of this lattice are cubes. (In the not ion of thislinear lattice again the fixed coordinate syste m in U(x) is used.)

The following lemma, stat ing the existence of regular coverings, can be obtainedby using simple geomet ric argu ments, see [S-Ch(1987)] :

Lemma 2.6. Let x E 8M' be a sufficient point and let Uo be an openneighborhood of x in 8M with the smooth foliation U = U UEBd- l I'U as above(recall that the manifolds I'U are uniformly transversal to all possible localstable invariant manifolds) . Then there are arbit rarily small neighborhoods Uof x in 8M having families of regular coverings with respect to the foliat ion

U = U UEBd-l r u ·

Now we are in the position of formulating th e "Tra nsversal" Fund amentalTheorem for semi-dispersing billiards generalizing Lemma 3 of [S-Ch (1987)]. Assaid before, the present version is stronger because our form of the Ansatz issimpler, our condition of sufficiency is weaker , and any t ransversal foliation U =U UE B d-l I'U can be used inst ead of the partition into local unstable invariantmanifolds. These improvements are important in applicat ions, e.g. in the caseof three and four billiard balls in tori , [K-S-Sz(1991)], [K-S-Sz(1992)].

We int roduce the following notation: 8r (Gf) is the union of tho se (2d - 3)dimensional faces of Gf which contain at least one f-face of Gf . We call 8 r (Gf )the f -jacket of Gf. The notion of the s-jacket 8 S (Gf ) of Gf is quite similar: Itis the union of the remaining (2d - 3)-dimensional faces of Gt . (They are justthose (2d - 3)-dimensional faces which contain at least one s-face of Gn It isclear that 8(Gt) = 8r (Gt ) U8S(Gt). If the condition

8(Gf n-"s(y)) c 8r(G f)

is fulfilled for a point y th en we say that the invariant manifold -ys (y) inte rsectsthe parallelepiped Gt correctly.

Theorem 2.7 "Transversal" Fundamental Theorem, theorem 3.6 of[K-S-Sz(1990)]. Suppose tha t

(i) the cond tions 2.1-4 above are fulfilled for the semi-dispersing billiard flow(M ,{St} ,{L) ;

Hard Ball Systems and Semi-Dispersive Billiards : Hyperboli city and Ergodicity 77

(ii) the point x E 8M' is sufficient and has at most one singularity on itsent ire trajectory;

(iii) Cl is a fixed consta nt between zero and one.

Then th ere exists a small neighborhood UC 1 (x) of x in 8M such tha t forevery open neighbourhood U (x ) C UC 1 (x ) of x and for every family

go = {G~ : i = 1,2 , . . . ,1(8) (0 < 8 < 8o)}

of regular coverings of U( x) t he covering go can be divided into two disjointsubsets g~ and gg such that (th e subscript "g" stands for "good", while "b"indicates "bad")

(I) for every parallelepiped G~ E g~ and for every s-face ES of G~ t he set

{y E G~ : g(y , E S) < c18 and :J (d - 1) - dimensional '-yS (y )

such th at 8(G~ n"l (y)) C 8r (G~)}

has posit ive Ill-measure;

(II)

Remark 2.8 Assume that x E 8M' n 'R" , (n ~ 0) . Th e statement of theFundame ntal Th eorem also rem ain s true in this case. We only need the foll owing,relaxed version of (II) :

(8---+0).

Remark 2.9 The grid condition (e) in the definition of regular covering s(Defin ition 2.5) is not necessary in the Fundamental Theorem , none theless inall applications it is enough to kno w that the theorem is tru e for every family of regular coverings satisf ying the condition (e) . The other reason for retain ing the condition (e) is of didactics: The best way fo r cons tructing fa milies of regular coverings (Lemma 2.6) is to begin with an almost lin ear lattice{wt : i = 1,2 , . . . ,1(8)} of centers of the parallelepipeds to be cons tructed. In thisway the geom etric and com binatorial st ructure of the covering go = {G~ : i =1, 2, .. . , I (8)} will be mo re tmnsparent than it would have been without condition(e).

In the applications one often uses two corollaries of the fundamental th eoremwhose formulation is less technical. The first one, is called the Zig-zag Theorem in [K-S-Sz(1989)], and is also derived th ere from the fundamental th eorem

78 N. Sima nyi

phrased for the case when the foliation r is I'" = b U} . As a matter of fact , I'"

is not a smooth foliat ion as r was supposed to be in Definition 2.5, but it isclear from all our proofs tha t about r its tran sversality to I" and the cont inuityof 1y'Y were only exploited.

Corollary 2.10 (Zig-zag Theorem). Assume the conditions 2.1- 4 for th eflow (M, {st} , f/,) , and let the base-point x E 8M* be sufficient . Then there existarbitrarily small neighborhoods U(x) of x in 8M such that for every null-setN C U(x) there exists a set A = A(N) c U(x) of full measure such th at wehave:

(i) An N = 0,

(ii) for every pair of points y , z E A there exist two finite sequences 'YL 'Y~ , . . . 'Yk 'and 'Yf , 'Y~ , .. . ,'Yk of local stable and unstable invariant manifolds in U(x)s.t.

(a) y E 'YL Z E 'Yk;

(b) 0~ 'Yi n 'Yi c A (i = 1,2 , , k),

o~ 'Yi n 'Yi+1 C A (i = 1,2 , , k - 1).

We note that , because of t ransversality, the non-empty sets 'Yi n 'Yi and'Yi n 'Yi+1 must contain exactly one point.

Corollary 2.11. Assume the condit ions of Theorem 2.7 for the flow(M, {s t} , J.l) and for the base point x E 8M n M* = 8M*. Then there existsa neighborhood U(x) of x in 8M contained in a single ergodic component of thesystem (8M ,T, J.l1) '

Proof. Using Hopf' s classical method and the Zig-zag Theorem we get th estatement of th is corollary in a st raight forward way.

2.3 Global Ergodicity. Pesin's theory [P(1977)] on the ergodic propertiesof non-uniformly hyperbolic, smooth dynamical systems has been genera lizedsubstantiall y to dynamical systems with singularities (and with a relat ively mildbehaviour near the singularit ies) by A. Kat ok and J-M . Strelcyn [K-S(1986)].Since then th e so called Katok-S trelcyn theory has become part of folklore inthe theory of dynamical systems. It claims t hat , under some mild regularitycondit ions, especially near the singularit ies, every non-uniformly hyperboli c andergodic flow enjoys the Kolmogorov-mixing prop erty, short ly the K-mixing property.

Lat er on it was discovered and proven in [C-H(1996)] and [O-W(1998)] thatthe above metioned fully hyperboli c and ergodic flows with singularit ies turnout to be automatically having the Bernoulli-property (B-property). It is worthnotin g here th at almost every semi-dispersive billiard system, esp. every hardball system enjoys those mild regular ity prop erties imposed on the systems

Hard Ball Systems and Semi-Dispersive Billiards: Hyperb olicity and Ergodicity 79

(as axioms) by [K-S(1986)], [C-H(1996)], and [O-W(1998)]. In other words, fora hard ball flow (M, {st},J-t) the (global) ergodicity of the systems actuallyimplies its full hyperbolicity and the B-property, as well.

As said in section 1.1 above, in the series of pap ers [K-S-Sz(1989)1, [K-SSz(1991)1, [Sim( 1992-A)], [Sim(1992-B)], and [K-S-Sz(1992)] the authors developed a fruitful , three-step stra tegy for proving the (global) ergodicity of hardball systems. Those three steps I-III (see in 1.1) consti tu te the whole inductionstep (N - 1) ===} N of proving the wanted ergodicity by an induction with respect to the number of balls N k 2). Without wanting to repeat the contentsof section 1.1 above, here we only point out two interesting features of steps Iand II.

2.3.1 The Connecting Path Formula (CPF). The Connecting PathFormula, abbreviated as CPF, was discovered for particles with identi cal massesin [Sim(1992-B]. Its goal was to give an explicit description (by introducing a useful syste m of linear coordinates ) of the neutral linear space No(SI-T,O]xO) in th elanguage of the "advances" of the occurin g collisions by using, as coefficients ,linear expressions of the (pre-collision and post-collision) velocity differences ofthe colliding particles. Since it relied upon the conservat ion of the momentum,it has been natural to expect th at the CP F can be generalized for par ticles withdifferent masses as well. The case is, indeed, thi s, and next we give this generalization for particl es with different masses. Since its st ructure is the same as th atof the CP F for identi cal masses, our exposit ion follows closely th e st ructure of[Sim(1992-B].

Consider a phase point Xo E M whose trajectory segment S [- T,O] xo is notsingular , T > O. In the forthcoming discussion the phase point Xo and thepositive number T will be fixed. All the velocit ies Vi (t ) E lRv i E {I , 2, . . . ,N } ,- T :::; t :::; 0 appearing in the considerations are velocities of cert ain balls atspecified moments t and always with the starting phase point Xo. (Vi(t) is thevelocity of the i-th ball at time t .) We suppose that the moment s 0 and -T arenot moments of collision. We label the balls by the natural numbers 1,2, .. . ,N(so the set {I , 2, , N } is always the verte x set of the collision graph) and wedenote by el, e2, ,en the collisions of the t rajec tory segment SI-T,O]xo (i.e.th e edges of the collision graph) so that the time order of these collisions isjust th e opposite of the order given by the indices. A few more definitions andnotations:

1. t, = t(ei) denot es the time of the collision e., so 0 > t l > t2 > .. . > tn >-T.

2. If t E lR is not a moment of collision (- T :::; t :::; 0), th en

is a linear mapping assigning to every element W E NO(S[-T,O] xO) the displacement of th e i-th ball at tim e t, provided that the configuration displacementat t ime zero is given by W . Originally, thi s linear mapping is only defined for

80 N. Simanyi

vectors W E No(S I- T,O]xo) close enough to the origin, bu t it can be uniquelyexten ded to th e whole spac e No(SI-T,Ol xo) by preserving lineari ty.

3. a(ei) denotes the advance of the collision e., thus

is a linear mapping (i = 1,2, .. . , n).4. The integers 1 = k(l) < k(2) < .. . < k(lo) ::::; n are defined by the

requirement th at for every j (1 ::::; j ::::; lo) the gra ph {el ' ez, . . . ,e k(j) } consists ofN - j connected components (on the vertex set {I , 2, . . . , N }, as always!) whilethe graph {el , ez, . .. ,ek(j) - d consists of N - j + 1 connected components and,moreover , we require th at th e number of connected components of the wholegraph {el ' ez, . . . ,en } be equal to N - lo. It is clear from this definition that th egraph

T = {ek(l )' ek(Z), . . . ,ek(lo) }

does not contain any loop , especially lo ::::; N - 1.Here we make two remarks comment ing the above notions.

Remark 2.12 We often do not indicate the variable W E No(Sl- T,O]xo) ofthe linear mappings tlqi(t) and a(e. ), for we will not be dealingwith specific neutral tangent vectors W but, instead, we think ofW as a typical (running) elementof No(S l- T,O]xo) and tlqi(t) , a(ei) as linear mappings defined on No(Sl-T,O]xo)in order to obtain an appropriate description of the neutral spaceNo(SI - T,O]XO) '

Remark 2.13 If W E No(S l- T,O] xO) has the property tlqj(O)[W] = ).Vi(O)for some). E lit and for all i E {I , 2, .. . ,N } (here Vi (O) is the velocity of thei-th ball at time zero), then a(ek)[W ]= ). for all k = 1,2, ... , n . This particularW corresponds to the direction of the flow. In the sequel we shall often refer tothis remark.

Let us fix two distinct balls a , w E {I , 2, . . . ,N } th at are in the same connected component of the collision graph On = {el ' ez, ... , en}. The CP F expresses th e relative displacement tlqn(O) - tlqw(O) in te rms of the advances c( e.)and the relative velocities occuring at these collisions e. . In order to be able toformulate th e CPF we need to define some gra ph-t heoretic notions concerningthe pair of ver tices (a, w).

Definition 2.14 Since the graph T = {ek(l ), ek(Z), . .. , ek(lo)} contains noloop and the vertices a , w belong to the same connected component of T , thereis a unique path II(a ,w) = {iI , fz ,..., f,,} in the graphT connecting the verticesa and w. The edges f i E T (i = 1,2 , . . . , h) are listed up successively along thispath II(a ,w) starting from a and ending at w. The vertices of the path II(a ,w)are denoted by a = Bo, Bl , B2 , . . . , B h = w indexed along this path going froma to w, so the edge f i connects the vertices B j- l and B, (i = 1,2 , .. . , h).

Hard Ball Systems and Semi-Dispersive Bill iards: I-Iyp erbolicit y and Ergodicity 81

When trying to compute ~qQ(O) - ~qw(O) by using the advances a(ei) andth e relative velocities at th ese collisions, it turns out that not only the collisionsfi (i = 1,2 , ... , h) make an impact on ~qQ(O)-~qw(O) , but some other adjacentedges too. This motivates the following definition:

Definition 2.15 Let i E {I , 2, . .. , h - I} be an integer. We define the setA i of adjacent edges at the vertex B, as follows:

Ai = {ej : j E {1,2 , .. . ,n } & (t(ej) - t(f;)) · (t(ej) - t(fHd) < 0 &

s, is a vertex of ej }.

We adopt a similar definition to the sets Ao, Ah of adjacent edges at th everti ces Bo and Bh ' respectively:

Definition 2.16

Ao = {ej : 1:::; j :::; n & t(ej ) > t(f]) & Bo is a vertex of ej };

We note that the sets Au,A] , ... ,A h are not necessarily mutually disjoint.Finally, we need to define the "cont ribut ion" of th e collision ej to ~qQ (0)

~qw(O) which is composed from the relative velocities just before and after themoment t(ej ) of th e collision ej '

Definition 2.17 For i E {I , 2, . . . , h} the "contribution" f(fi) of the edgefi E II( Q , w) is given by the formula

V Bi_1(t(fi)) - VB, (t(f;)) ,

if t (fi - d < t(fi) & t(fHd < t(f;);

V~i _ l (t(f;)) - vl(t(fi)) ,

if t(f;-d > t(fi) & t(fHd > t(fi);

f(f;) = mBi_1\mBi [mBi_1(vBi_1(t(fi)) - vBi(t (fi )))

+mBi (V~i _l (t(fi)) - V~i (t(f;)))] ,

ift (fH d < t(fi) < t(fi-d

mB \ mB [mBi_l (vl _1 (t(fi)) - v~i(t(f;)))t - l t

+m Bi (VBi_1(t(fi)) - VBi(t(Ii)))] ,

ift(fi-d < t(1i) < t(fHd ·

82 N. Simanyi

Here vB,(t(fi)) denot es the velocity of the Bi-th particle just before thecollision f i (occuring at time t(fi)) and, similarly, V~(t(fi)) is the velocityof the same particle just after the ment ioned collision. 'We also note tha t, byconvention , t(fo) = 0 > t(fd and t(fh+d = 0 > t(fh) ' Apparently, the timeorder plays an important role in this definition .

Definition 2.18 For i E {O,1 ,2 , . .. , h} the "contribution" fi (ej ) of an edgeej E Ai is defined as follows:

where 0 is the vertex of ej different from Bi ,

Here again we adopt the convent ion of t (fo) = 0 > t(ej) (ej E Ao) andt(j,,+d = 0 > t(ej) (ej E Ah). We note that , by the definition of the set A ,exact ly one of the two possibilities t(fi+d < t(ej) < t(fi) and t(fi) < t (ej) <t(fi+1) occurs. The subscript i of I' is only needed because an edge e j E Ail nAi2

(i 1 < i 2 ) has two contri butions at the vertices Bi; and Bi2 which are just theendpoints of ej '

We are now in the position of formulating the Connecting Path Formula:

Connecting Path Formula (CPF). Using all definitions and notationsabove, the following sum is an expression for ~qQ(O) - ~qw(O) in te rms of theadvances and relati ve velocit ies of collisions:

h h

~qQ(O) - ~qw(O) = L a(f;)f(fi) +L L a(ej)fi(ej) .i = l i = O ejE A,

The proof of the proposition follows the proof of Sirnanyi's CP F (Lemma2.9 of [Sim(1992-B)]) with the only difference that Lemma 2.8 of [Sim(1992-B)]is replaced here by the following

Lemma 2.19 If e is a collision at time t between the particles Band 0 ,then

v~ ( t ) - vc (t ) = me (v~(t) - v; (t )) + m B (vB(t) - vc (t ))mB+me m B +me

and

v~ (t ) - vB(t ) = me [(v~(t) - v; (t )) - (vB(t) - vc (t ))] .mB+me

Hard Ball Systems an d Semi-Disp ersive Billiards: Hyp erbolicit y and Ergodici ty 83

By writing down the CPF's

h h

a(e*) · [v~ (t(e*)) - v;;; (t(e*))] = La(fi)r(fi) + L L a(ej)ri(ej) (*)i = l i = O e j EA i

for all collisions e* = (Q ,w) of the orbit segment S[- T,O]xO (for which th e twocolliding balls Q and ware in the same connected component of the collisiongraph of Sl-T,t(e' ))xo , where t(e*) denot es the t ime moment of the collision e* =(Q,w)), we obt ain a system of homogeneous linear equat ions, and this facilitatesthe computing of the dimension of the neutral linear space No (Sl - T,O]XO)' Thi sdimension is equal to the dimension of the solut ion set of the system of linearequations (*) described above. An inequality dinINo (S [- T,OJxO) 2: t5 will then beequivalent to the simultaneous vanishing of certain velocity polynomials, namelysome minors of the coefficient matrix of the system (*). This characterizat ion ofdinINo (S I- T,O]xO) played a fundament al role in proving the "codimension-oneversion" of 8tep I of our st rat egy in 18-8z(1999)J.

2.3.2 A strong ball avoiding theorem. 8t ep II in the indu ct ive proof ofergodicity is sett led by

Theorem 5.1 of [Sim(1992-A)].Let (M, {st} ,tt) be the standard billiard flow of N (2: 3) particles on the unittorus 1['V (v 2: 2). Assume that for all n < N the n-billiard flow on 1['v is aK-flow. Let , moreover, P = (P1,PZ ) be a given, two-cl ass partition of the Nparticles. Then the set

F+ = {x E M : SIO,oo)x is par ti tioned by P}

is a closed, zero set with codimension at least two (i.e. a closed slim set).We note th at th e phrase "a trajectory segment is part itioned by P" means

that on this traj ectory segment there is no proper (non-t angentional) collisionbetween particles being in different classes of the parti tion P .

This theorem belongs to the cat egory of "strong" ball avoiding theorems forthe following reasons: The event "the orbi t segment SIO,oo)x is parti tioned byP" means th at this orbit segment avoids an open set (and , thus, it also avoidsa ball) , and the theorem claims the smallness of this event. The adject ive "st rong"refers to the circums tance that th e smallness of the ball avoiding set is not onlyunderstood in measure-theoretic, but also in topological terms . (More precisely,in te rms of topolo gical dimension theory founded by Menger and Urysohn.)

For an extended review of ball avoiding theorems, the reader is kindly directe d to the nice survey [8z(1999)J.

2.4 Irrational Mass Ratio. Due to the natural reduction I:~1 mivi = 0(which we always assume), in section 1.1 we had to factorize out the configuration space with respect to spat ial tran slations: (q1 " .. , qN) rv (q1 +a, .. . ,qN+a)

84 N. Simanyi

for all a E jRv . It is a remarkable fact , however, that (despit e the reduction2::[:1 m i v i = 0) even without thi s translation factori zation the system still retains the Bernoulli mixing prop erty, provided that the masses ml , . .. , m N arerationally independen t and the factori zed syste m enjoys the Bernoulli prop erty!For the case N = 2 (i. e. two balls) thi s was proven in IS-W(1989)1 by successfully applying D. Rudolph's theorem on the B-property of isometric groupextensions of Bernoulli shifts IR(1978)].

Suppose that we are given a dynamical system (M, T ,p,) with a probabilitymeasure p, and an automorphism T . Assume that a compact metric group Gis also given with the normalized Haar measure v and left invari ant metri c p.

Fin ally, let t.p: M -+ G be a measurable map . Consider the skew productdynamical system (M x G,S,p, x ,\) with S( x ,g) = (T x , t.p(x) · g), x E M ,9 E G . We call the system (M x G , S , p,x ,\) an isometric group exte nsion of thebase (or factor) (M, T , p,) . (The phrase "isomet ric" comes from the fact that th eleft t ranslations t.p (x ) . 9 are isometries of the group G.) Rudolph 's ment ionedtheorem claims th at th e isometri c group extension (M x G , S , p, x ,\) enjoysthe B-property as long as it is at least weakly mixing and the factor system(M,T ,p,) is a B-mixing syste m.

But how do we apply this theorem to show th at th e system of N hard ballsin TV with 2::[:1 m i v i = 0 is a Bernoulli flow, even if we do not make thefactoriz ation (of the configuration space) with respect to spat ial t ranslat ions?It is very simple. The base system (M,T , p,) of the isometric group extension(M x G, S,p, x ,\) will be the time-one map of the factoriz ed (with respect tospatial translations) hard ball syst em. The group G will just be the containertorus TV with its standard Euclidean metri c p and normalized Haar measure'\ . Th e second component 9 of a phase point y = (x ,g) E M x G will be justthe position of the center of the (say) first ball in TV. Fin ally, the govern ingtranslation t.p (x ) E TV is quit e naturally the total displacement

of the first particle while unity of time elapses. We assume that the B-property(ergodicity) of the factor map (M,T ,p,) has been proven successfully. Thenthe key step in proving the B-property of the isometri c group extension (M xG, S, p, x ,\) is to show that th e latter system is weakly mixing . This is justthe essent ial contents of the art icle IS-W(1989)], and it takes advantage of theassumption of rational independence of the masses. Here we are only presentingto the reader th e outl ine of that proof in nutshell. As a matter of fact, we notonly proved the weak mixing prop ert y of the extension (M x G, S,p, x ,\) , butwe showed that this syste m has in fact the K-mixing prop erty by proving thatthe Pinsker par tition 7r of (M x G, S,p, x ,\) is trivi al. (The Pinsker partitionis, by definition, the finest measurable partition of the dynamical system withrespect to which the factor system has zero metric entropy. A dynamical system

Hard Ball Systems and Semi-Dispersive Billiards: Hyperbolicity and Ergodicity 85

is K-mixing if and only if its Pinsker partition is trivial , i. e. it consists on ly ofsets with measure zero and one , see [K-S-F (1980)1. ) In order to show that thePinsker partition is trivial, in [S-W (1989)] we constructed a pair of measurablepartitions (C, ~U) for (M x G,S,1l x >') made up by open submanifolds ofthe local stable and unstable manifolds, respectively. It followed by standardmethods (see [Sin(1968)]) that the part ition 7r is coarser than each of C and ~u .

Du e to the S-invariance of tt ; we then have that 7r is coarser than

1\ sne 1\ 1\ sn~u .

nE Z nE Z

(**)

In the fina l step , by using now the rational independence of the masses , weshowed that the partition in (**) is, indeed , trivial.

Acknowledgement. The author would like to express his most sincere gratitude to Domokos Szasz (Technical University of I3udap est) for the numeroussuggestions and remarks he made during the preparation of this paper .

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Decay of Correlationsfor Lorentz Gases and Hard Balls

N. Chernov' and 1. S. Young/

Contents

§1. Definitio ns .§2. Historical Overview .§3. Statistical Properties of Chaotic Dynamical Systems .

3.1 Reference Set and Distribution of Return Tim es(for the Cat Map) . . . . . . . .

3.2 T he Abstract Model F : ~ -t ~ .. ..

3.3 Discussion . . .. . . .. .. . . . . . .§4. Correlation Decay for Planar Lorentz Gases

4.1 Geometric Properties of the Map lP : M -t M4.2 Comparison with the Cat Map .4.3 Growth of Unstable Curves .

§5. Correlation Decay in Related Billiard Models .5.1 Sinai Billiard Tables .5.2 Lorent z Gases with Infinite Horizon .5.3 Lorent z Gases under External Forces5.4 Mult i-Dimensional Lorentz Gases5.5 Multiple Correlations5.6 Real Time Dynamics5.7 Gases of Hard Balls .

References . . . . . . . . . . . .

1 N. Chernov is partially supported by NSF grant DMS-9732728.2 L. S. Young is part ially supported by NSF grant DMS-9803150.

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90 N. Chernov and L. S. Youn g

Abstract . We discuss rigorous results and open problems on the decay of correlations for dynamical syste ms characterized by elastic collisions: hard balls,Lorentz gases, Sinai billiards and relat ed models. Recently developed techniquesfor general dynamical systems with some hyperbolic behavior are discussed.These techniques give exponent ial decay of corre lations for many classes of billiards and open the door to further investigat ions.

Decay of Correlations for Lorentz Gases and Hard Balls

§1. Definitions

91

This is a survey on the statistical properties of Lorentz gases and gases of hardballs. The properties of interest to us include rates of correlation decay, centrallimit theorems, and invariance principles. We begin with some precise definitions.

Let t : M ---+ M be a dynamical system preserving a probability measureu. The time t here is either discrete, i.e. tEll, or continuous, i.e. t E JR. Letf : M ---+ lR be a real-valu ed measurable function which we think of as anobservable. Then the family

~t = f 0 t, t Ell or lR,

defines a stationary stochastic process with (M ,J-l) as the underlying prob abilityspace, st ationarity following from the invariance of the measure u.

For t > 0, let S, : M ---+ lR be the accumulation function of ~t . That is tosay,

St = f + f 0 1 + f 0 2 + ... + f 0 t-l

in the case of discrete time and

in the case of cont inuous time . (We assume in the latter that as a function of t,f 0 t(x) is continuous or at least piecewise continuous for a.e. x E M, so thatthe integral is well defined) . The function St!t is the time average of the process~t . We denote by (-) the expected value of a function with respect to J-l.

The Birkhoff Ergodic Theorem asserts th at if (M, t, J-l) is ergodic and f isintegrable, then St/t converges almost surely to (J) as t ---+ 00 . In probabilitytheory, this is also called the strong law of large numbers.

As we shall see, under suitable assumpt ions on the system t : M ---+ M andthe observable I , many other results in probability theory can be carried overto the stochastic process ~t and its accumulation function St .

An important object of study is the time correlation function

(1.1)

This function measures the dependence between the values of f at tim e 0 andtime t. It is also common to study the asymptotics of more general correlationfunctions than (1.1), nam ely

(1.2)

where 9 : M ---+ lR is another measurable function . The function Cf(t) in (1.1)is called an autocorrelation function.

92 N. Chernov and L. S. Young

(1.3)

In sta tist ical physics, autocorrelation functions are involved in transporttheories. Transport coefficients (such as diffusion coefficient , heat or elect ricalconductivity, shear viscosity) can be expressed through integrals of certain autocorrelation functions. See the survey IBu].

Next , we say that ~t satisfies th e central limit theorem (CLT) if

lim J1 {St - t(f ) < z } = _1_jZe-~ dsH 00 Vi V2ira - 00

for all - 00 < z < 00 . Here a = af 2 0 is a constant. (In the case af = 0, t heright side of the equat ion is to be read as 0 for z < 0 and 1 for z > 0.) Equation(1.3) is equivalent to the convergence of (St - t(f ))/ Vi in distribution to thenormal random variable N(O,aJ) . We remark that the central limit theorem isconsiderably more refined than the Birkhoff Ergodic Theorem ; it tells us thatthe distribution of the deviations of the tim e average St/t from its limit value(f ), when scaled by 1/ Vi, is asymptot ically Gaussian.

The variance aJ in the CLT is relat ed to the corre lat ion function (1.1) by00

aJ = Cf(O) + 2.L Cf(n)n= !

(1.4)

in the case of discrete time and

aJ = I:Cf(t) dt (1.5)

in the case of continuous tim e. It follows that a prerequisit e for the centrallimit theorem is the integrability of the correlation function Cf(t) . Most existing proofs of the CLT for dynamical systems follow essent ially (though notimmediatel y) from slightly stronger estimates on the speed of correlat ion decay,i.e. the speed with which Cf(t) tends to 0 as t ~ 00 .

We remark also that under very mild assumptions, aJ = 0 if and only ifthe function f is cohomologous to a const ant. Thi s means, in the discrete timecase, that f = K +9 - 9 0 ell for some 9 E L2(M) and a const ant K , and in thecontinuous time case, that f = K + -1t It=o(g 0 ell t).

A good survey on issues related to the central limit theorem for dynamicalsystems may be found in [Del.

An invariance prin ciple often accompanies the central limit theorem in thest udy of random processes and dynamical syste ms. For large T > 0, x E M and0 :::; s :::; 1, consider the function WT(s ;x) defined by

TI T ( • ) _ St( x) - t(f ) h TYY T S , X - 1m were t = S .

avT

In the discrete tim e case, we interpol at e linearly between integer values of t ,letting

WT(S;x) = (k + 1 - sT) WT(k /T;x ) + (sT - k) WT((k + 1)/T;x )

for kiT < s < (k + 1)/T.

Decay of Correlatio ns for Lorentz Gases and Hard Balls 93

For fixed T , the family {WT ( s;x ), x E M} induces a probability measure onthe space of piecewise continuous function s on [0,1] . We say that ~t sati sfies th eweak invariance principle (WIP) if this measure converges, as T -t 00, to th eWiener measure. We say that ~t satisfies an almost sure invariance prin ciple(ASIP) if there is a standard Brownian motion B( s;x ) on M with respect tothe measure /l so that for some A > 0,

IWT(S;X) - B(s;x)1 = O(s-.\)

for /l-almost all x E M : The invariance pr inciple assert s, t herefore, that theaccumulation funct ion Bt , afte r a prop er rescaling of space and tim e, convergesto the Wiener process (or Brownian motion) . The weak invariance prin ciple issomet imes called the functional central limit theorem. More detailed discussionsmay be found in [C3, DP , PS].

Other refinements of the cent ral limit theorem and related probabilist ic limitlaws also have their corresponding versions for dynamical systems. For example,one can prove a local central limit theorem, the law of iterated logarithms,renewal theorems, Borel-Cantelli lemmas, Poisson distribution for return t imesetc. These extensions would ta ke us too far from the main topic and will not bediscussed here.

Finally, let us relat e the discussion above to some of the st andard notions inergodic theory. Recall that a dynamical syst em (M, <p t , /l) is said to be mixingif for any two measur able sets A, B c M , we have /l(A n <p - t B) -t /l(A)/l(B)as t -t 00 . The following fact is standard in ergodic theory:

Fact 1.1 Cj,g(t ) -t 0 as t -t 00 for all [ . 9 E L2(M) if and only if (M, <p t , /l)is mixing.

One might surmise that stronger versions of mixing (for example, multiplemixing, K-mixing, or Bernoull i) imply fast decay of correlations. Th is is nottrue. Even the Bernoulli propert y cannot guarantee any speed of convergenceof Cj,g (t ) to zero for arbitrary funct ions f ,9 E L2(M) , not even for bound edor cont inuous funct ions! Moreover , let <p t be the "most chaot ic" of all knowndynamical systems, such as an expanding interval map or a hyperboli c toralautomorphism. Even then, for typical integrabl e or cont inuous functions f ,9 onM , the convergence of Cj ,g(t) to zero is arbitrarily slow, and the central limittheorem (1.3) fails (d . [C3, CC, JR, VD. In order to have a reasonable speedof corre lation decay common to a family of observables, or to guarantee theCLT, it is necessary to restrict ourselves to obscrvables with some regularity. Ingeneral, Holder cont inuity is sufficient. Fortunat ely, all interesting functions inphysics (such as temp erature, energy, velocity) are smooth or at least piecewisesmooth. Thi s makes it possible to obtain st rong statist ical prop erties for manyphysically relevant obscrvables.

§2. Historical Overview

Among the dynamical systems first rigorously st udied are geodesic flows onmanifolds of constant negative curvature IHa, He, Ho]. E. Hopf [Ho] provedergodicity for these flows as early as 1940. His argument relies on the existenceof a pair of transversal foliat ions whose leaves are made up of stable and unstablemanifolds (called horocycles in the case of geodesic flows). Hopf's argument issimple yet far reaching. While it alone is not adequ at e in more complicatedsituations, it lies at the heart of many proofs of ergodicity, including those forthe dynamical systems considered in thi s survey.

Following Hopf' s work , the dynamics of geodesic flows on manifolds of constant negative curvature were invest igat ed by Ya, Sinai . In the lat e fifties, Sinainoticed a st riking resemblance between these flows and stationary random processes. He proved the cent ral limit theorem in 1960 [Si1] and the K-mixingprop erty in 1961 [Si2] . In 1967, D. Anosov [AnI completed a study of geodesicflows on manifolds of variable negative curvature, proving ergodicity, in particular , along the lines of Hopf. The Bernoulli prop erty was lat er proved byD. Ornstein and B. Weiss [OW] and by Ratner IRa2].

Generalizing a key prop ert y of geodesic flows on manifolds of negative curvat ure, Anosov [An] introduced a class of flows with the property that t leavesinvar iant a pair of foliati ons transversal to th e vector field, uniformly expanding distances in one of them and uniformly cont ract ing distances in the other.He called these flows and their discret e-time versions C-systems; they are nowknown as Anosov diffeornorphism s and flows, and the dynamical prop erty described above is called unifo rm hyperbolicity. At roughl y the same t ime, S. SmaleISm] introduced his famous horseshoe model and the notion of A xiom A . TheAxiom A condit ion requires only th at the system be uniformly hyperbolic oncertain recur rent sets and not necessarily on the ent ire manifold . Axiom A systems, therefore, are more general than Anosov syste ms. Unlike geodesic flows,however, Anosov and Axiom A systems do not always admit invariant probability measures that are compatible with volume . Thi s perhaps motivated thestudy of more genera l invariant measures.

An important class of invariant measures for dynamical systems has its origins in st ati sti cal physics. This came about in a cur ious way. In 1967 Adler andWeiss [AW] const ructed Markov partit ions for linear toral automorphisms. In1968 Sinai [Si3, Si4] const ructed Markov par ti tions for all Anosov diffeomorph isms. Via a Markov partition, the t rajectory of each point is coded by aninfinite sequence of symbols from a finite alphabet , and the dynamics of themap is represent ed by a topological Markov chain. Spaces of symbol sequencesare naturally reminiscent of one-dimensional lat t ice models in statistical mechanics. For lat tice models, Gibbs measures were const ructed in 1968-1969 byR. Dobrushin [Db1, Db2, Db3] and by O. Lanford and D. Ruelle [LR], who alsoshowed th at translationally invariant Gibb s measures are equilibrium states, i.e.they are characterized by a variat ional principle. Using Markov partitions, Sinai

Decay of Correlat ions for Lorentz Gases and Hard Balls 95

developed a theory of Gibb s measures for Anosov diffeomorphisms in 1972 [Si6]in analogy with that for lat tice models. In the meantim e, Bowen constructedMarkov partit ions for all Axiom A diffeomorphisms [B1], making it possibleto exte nd the theory of Gibbs measures to thi s larger context. See [B3] for anexposit ion.

For dynamical systems, certain invariant measures are more important thanot hers from the st andpoint of physics. Here we take the view that observableevents correspond to sets of positive Lebesgue measure in the phase space. Itfollows tha t the invariant measures of interest in physics are those that reflectthe distribut ions of orb its start ing from sets of posit ive Lebesgue measure. Th isis all very natura l for conservative systems, i.e. for syste ms that preserve smoothmeasures. In the presence of dissipati on, the situat ion is more subtle: there isno reason a pri ori why any invariant measure with the desired property shouldexist . For Anosov diffeomorphisms and Axiom A attractors, it turns out thesemeasures can be found among Gibbs measures [Si6]. Equivalent characterizations emphasizing their connection to Lebesgue-almost every init ial condit ionare given in IRu2, BR]; see also [B3] . Today these special invariant measures areknown as Sinai-Ru elLe-Bowen measures or SRB measures.

In the discrete t ime case, stro ng statistical prop erties for Gibbs measureswere obtained (most ly) by Sinai [Si6] and Ruelle [Rul , Ru2]; a version of it isgiven in [B3]. In these papers, exponent ial decay of correlat ions for Holder continuous funct ions is proved for Anosov and Axiom A diffeomorphisms. From thisthe centra l limit theorem is easily deduced. For Axiom A flows, the Bernoulliproperty and cent ral limit theorem were proved by Ratner [Ra2], and the ASIPwas obtained by M. Denker and W. Ph ilipp [DP]. Asymp totic bound s on correlat ions in cont inuous time have turned out to be considerably more delicate.First , Ruelle [Ru4] and M. Pollicott [Po] obtained negat ive results: they foundAxiom A flows with arbit rarily slow rates of correlat ion decay. Recently D. Dolgopyat [Dol] proved the exponent ial decay of corre lat ions for Anosov flows undercertain addit ional assumpt ions and for all geodesic flows on surfaces of negativecurvature. He also proved [Do2] that there is an open and dense set of Axiom A flows that enjoy rap id mixing in the sense of Schwarz. Wheth er "most"Anosov flows have exponential decay of correlat ions is unknown at this t ime.

Today, the theory of Anosov and Axiom A systems can be regarded as fairlycomplete . In statistical physics, these syste ms have become a reference model,a paradigm for many heuristic studies of chaotic multiparticle systems - gasesand fluids (both in and out of equilibrium). G. Gallavotti and E. Cohen [GC]spelled th is out in their Axiom C in 1995. They stated that chaotic muit iparticlesystems could be regarded, for the purpose of averaging of phase observables,as Anosov systems.

Since the sevent ies, mathemat icians have tri ed to exte nd the theory of Axiom A syste ms to dynamical systems having some hyperbolic behavior but sat isfying less stringent conditions. Particularly important to physicists are billiardmodels, including hard-ball gases and Lorentz gases. Analogies between colli-


sions of hard balls and geodesic flows on manifolds of negative curvature hadbeen noticed by N. Krylov IK] decades earl ier. Krylov pointed out that the convex surface of hard balls produces the same scat te ring effect on phase trajectoriesas the negative curvat ure of the manifold on geodesic curves.

In 1970 Sinai [Si5] undertook a syste matic st udy of plan ar periodic Lorentzgases and, more generally, of plan ar billiards in tables with concave bound aries(now called dispersing billiards or Sinai billiards) . He investigat ed the hyperboli cprop erti es of these billiards and proved ergodicity and the Kvmixing propert y.A major difference between billiards and Anosov systems lies in the fact th atbilliard flows are not cont inuous. Singularity sets break up stable and unst ablecurves into arbitrarily small pieces, making the proof of ergodicity considerablymore involved and ruling out the exist ence of finite Markov partitions, which,as we recall , were the main tool in unde rst anding the ergodic theory of Anosovsystems. Sinai 's seminal work paved the way for many subsequent developmentsin this direction.

In 1980, L. Bunimovich and Sinai considered billiard maps associated withplanar periodic Lorentz gases and constructed for them countable Markov part itions [BS]J; see also [BSC1J. (Billiard maps are return maps on the Poincaresect ions of billiard flows corresponding to collisions.) In 1981, Bunimovich andSinai IBS2] established the CLT and WIP with the help of these Markov partitions. Note that from the point of view of physics, th e WIP for Lorentz gaseshas the following important inte rpretation: it says that typical particle trajectories (Xt ,Yt) on the covering plane converge, afte r a suitable rescaling of t imeand space, to Brownian motion. See also IBSC2]. Bunimovich and Sinai IBS21obtained, in fact , an upper bound on the t ime corre lat ion function. They showedthat

ICf ,g(t )1::::: const · exp (- at' ) (2.1)

for some a > 0 and "f E (0,1). (As usual, f and 9 are Holder cont inuous functionsand cpt here is the billiard map ). This mode of decay, lat er termed stretchedexponential decay of correlat ions, is slower th an exponent ial but fast enough toallow them to derive their results on the CLT and WIP.

The true asymptot ics of the tim e correlation funct ion Cf ,g(t) for dispersingbilliards were not known for some time. Numerical result s, including est imateson the constant "f, were produ ced by var ious people (e.g. [BID, CCG)). See also[FM1, MR, FM2, GG]. Th ere was disagreement in the mathematical physicscommunity on whether this decay rate is in fact exponential or if it is substant ially slowed down by t he cut t ing and folding action of the singulari ty set.Analyt ic evidence in favor of exponent ial esti mates came in the early 1990s:Chernov [C1] and lat er C. Liverani [L] proved exponential decay of corre lat ionsfor certain piecewise hyperboli c maps (with singularit ies) in 2-dimensions.

For Lorent z gases, thi s quest ion was resolved in the lat e nineties. In 1998,L.-S. Young [Y1] developed a general method for determining if a map has exponenti al decay of correlat ions with respect to its SRB measure. She [Y1] appli ed

Decay of Correlatio ns for Lorentz Gases and Hard Balls 97

her method to planar Lorentz gases with finite horizon and obt ained exponent ialdecay of corre lations. Shortly thereafter , Chernov applied the scheme in [Yl1 toother classes of planar dispersing billiards [C6] and to Lorent z gases under externa l forces and their SRB measures [C7], obtaining in both cases exponent ialdecay rates.

Young [Y2] has since expanded her results to deal with arbitrary decay rates.With tools for establishing polynom ial decay now available, there is hope thatrigorous correlat ion decay results for certain billiard models with very nonuniform hyperbolic behavior, such as the stadium, may be forthcoming.

The main ingredient of Young's approach is a tower construct ion that captures the renewal prop erties of a dynamical system. She focuses on return t imesto a reference set rather than the partitioning up of the phase space. In particular , Markov partitions are not used. It is becoming increasingly clear thatthis approach is qui te generic, in the sense that it has given a unified way ofunder st anding corre lat ion decay and relat ed statistical properties for many (different) dynamical syste ms th at have some degree of expansion or hyperbolicity.In Section 3 we describe in a fairly general context her tower const ruct ion andthe statistical informati on it car ries. In Section 4 we explain what this te lls uswhen applied to Lorent z gases. Further applicat ions of thi s method are discussedin Sect ion 5.

Finally, we turn to gases of hard balls, whose ergodic and statist ical prop erties are among the less tractable problems in dynamical systems. In the generalsett ing consisting of an arbit rary number of balls on a torus , full hyperbolicity(i.e. the absence of zero Lyapunov exponents) has been proved only recent ly byN. Simanyi and D. Szasz [SSII. A proof of ergodicity is not yet available exceptin certain special cases (see the survey [SS2]) . Nothing is known for gases ofn 2: 3 balls in a rectangular box (see [Sim] for the case n = 2). What is clear isthat hyperbolicity is very nonuniform in hard ball systems: there are "t raps" ofvarious kind in the phase space . Trajector ies temporarily lose hyperbolicity asthey get caught in these t raps, and they may remain there for arbit rarily longperiods of time. A careful quantitat ive analysis of these traps is necessary forrigorous result s on correlation bound s and the CLT. This appears to be hopelessly out of reach at the present t ime. Numerical and heuristic studies have,however, been car ried out by physicists. Some of their results and conjecturesare discussed in Sect ion 5.

§3. Stati stical Properties of Chaotic Dynamical Systems

In th is sect ion we discuss a general method for capturing stat ist ical informationfor chaot ic dynamical systems. This approach is introdu ced in [Yl], expandedin [Y2], and has been used successfully to obtain result s for a number of thesystems considered in this survey.

Some of the material in this section is valid only for discrete time systems,i.e. for t with t E 7l or 7l+ . It can be appli ed , in principle, to all discrete timesystems that are predominantly hyperbolic , that is to say, t does not haveto be uniformly hyperbolic or Anosov; singularities and other nonhyperboli cbehaviors are allowed as long as there is "enough" hyperbolicity.

The key idea of this approach is to ext ract statistical information from certain distributions of return times. This is motivated by similar considerationsin probability th eory, in, for example, th e theory of countable state Markovchains, the stat ist ical properties of which are known to be closely related to therecurrence properties of the "tail st ates" to a fixed block of st at es.

Leaving more precise discussions for later, we give an indication here ofwhat the proposed scheme ent ails: Pick a suitable reference set A in the phasespace, and regard a subset of A as having "renewed" itself when it makes a"full" or "Markov" return to A, meaning its image covers all of A or at leaststretches across all of A in the unstable direction. When successfully carr iedout , this construction gives rise to a representation, or a model, of the syst em inquestion, described in terms of a reference set and return tim es. As a dyn amicalsystem, this model is often much simpler than the original one. Consequently,its st ati stical properties are more easily accessible, and, as we will see, they canbe expressed explicit ly in terms of the tail distribution of t he return t imes.

This, in summary, is th e approach proposed by Young: Set aside the individual characteristics of the original dynamical system, focus only on returntimes to a reference set , study the statistical propert ies of the resulting (abstra ct)mod el, and pass the findings back to the original dynamical system.

In the rest of thi s section, we will limit our selves to discrete time systems.Writing = 1

, we let : M -+ M denote the given dynamical system (whichwe do not need to assume a priori to have an invariant density or to admitan SRB measure) . All test function s f : M -+ lR are assumed to be at leastHolder cont inuous. In Sect . :l.1 we discuss the construction of reference sets andMarkov return maps , using a very simple example, namely the "cat map" or "21-1-1 map" of the 2-torus, to illustrate how precisely this is done. The abstractmodels that result from th ese constructions will be denoted by F : D. -+ D. . InSect . 3.2 we describe F : D. -+ D., and discuss its statistical properties and theirimplications for . Sect. 3.3 contains a general discussion.

3.1 Reference Set and Distribution of Return Times (for the CatMap). We begin with a discussion of th e "cat map". This example will be usedto illustrate (1) how to pick a reference set A, (2) how to define legitimate returnmaps, and (3) how to estimate the tail of the distribution of return times. Oncethis example is understood, we will comment on how it differs from the generalsituat ion.

Decay of Correlations for Lorentz Gases and Hard Balls 99

Let <fI : T 2 -+ T 2 be the "cat map" or any invertible map indu ced froma hyperboli c linear map of lR2

. We require that A be a rectangle, with two ofits sides aligned with the st able direction and two with the unstable direction;other th an that , its choice is ent irely arbitrary. Next we describe a procedurefor defining a return map to A. T his map will be denoted by <fiR : A -+ A, whereR here is to be thought of as an integer-valued function or random variable andnot a fixed integer. That is to say, R : A -+ 7l+ is a function , and <fiR evaluatedat x E A is equal to <fIn(x) with n = R(x).

To describe the Markov prop erty of the return map , we introduce the following language: r cA is called an s-subrectangle if it spans A in the st abledirection, a u -subrectangle if it spans A in the unstable direction . As <fI is itera ted, A is transformed into a long and thin ribbon running parall el to theunstable direction. Let nl > 0 be the first time when part of <fin!A contai ns au-subrectangle of A as shown, and let A l , A2 , · · · ,Ak ! be the s-subrectangles ofA th at are mapped under <fin! onto u-subrectangles of A. We declare that theseAi have "returned " with return time R = nl and stop considering them. Focusing on the part that has not returned, we cont inue to itera te until its imagecontains a u-subrectangle of A, say at t ime n2 > nl ' Label the s-subrectanglesth at return at this tim e Ak ! +l , .. . ,Ak 2 and set their return tim e to be R = n 2.The process is continued ad infini tum . We will show momentarily that almostall points in A eventually return to A under th is procedure, defining, moduloa set of Lebesgue measure zero, a return map <fiR on A.

Since we are interested in the process of dynamical renewal and not jus trecurrence alone, parti al crossings are not counted as returns in the procedureabove. The fact that the <fiR-image of each Ai is a u-subrectangle of A meansth at <fIR(Ai ) contains a sample of all possible fut ure trajectories start ing fromA; it is as though afte r <fiR steps, the syst em is starting anew. Thi s is what wemean by "Markov returns".

•

Figure 1: Markov returns

100 N. Chern ov and L. S. Young

Note that we do not insist that R(x) be the first time the orbi t st artingat x returns to A. T he choice of R is, in general , quite flexible in the sensethat many reasonable choices will not significantly affect the outcome of thisdiscussion. To facilitate the estimation of the distribution of R, we will, in fact ,use the following rule: fix a number J > 0, comparab le, say, to the size of A,and count an s-subrectangle I' C (A n {R > n - I}) as returning at tim e n ifand only if the component of n (A n {R > n - I}) containing n (f) not onlycrosses A but extends beyond A on both sides by lengt hs at least J .

Next we show how to est imate the tai l distribution of R. Let J1. denoteLebesgue measure on the torus. We claim that

J1.{R> n } < con for some 0 < 1.

We will see in the next subsection that this tail est imate implies exponentialdecay of corre lat ions.

To prove th e claim, observe that at step n, the set n-l (A n {R > n - I})is the union of a finite numb er of very thin "ribbons", disconneted due to thefact that the parts that have returned have been removed . Our rule of whatconst itutes a return in the last paragraph ensures that each component of thi s"ribbon" has length at least J. We divide n-l (An {R > n -I}) into segments oflength ", J , consider them one at a tim e and argue as follows: By the topologicaltransitivity of , there exists N = N(J) such that if B 1 and B2 are any twoballs of radius *, we have kB1 n B2 -I- 0 for some k :'::: N . Now let B1 becentered at the midpoint of one of the J-segments I' of n- l (An {R > n - I} ),and let B2 be centered at the midpoint of A. Suppose kB1n B2 -I- 0. Then bythe geometry of hyperbolic maps, k(r) crosses A with two pieces sticking outon both sides as required (see Figure 2). We have th us shown that for every n ,J1.{R :'::: n + N IR > n - I } is greater than some c > 0, proving the claim.

We discuss next some of the similarities and differences between the exampl eabove and what one may expect to find in general.

With regard to the choice of A and R , the construction above is quiteindicat ive of what is done in general. To make sense of the "Markov" property,we need to be able to talk about s- and u-subrectangles, and A should be chosenso that it has a product structure of stable and unstable man ifolds. When isnot Anosov, there may not be "solid" rectangles mad e up of stable and unstableleaves; when that is the case, take A to be the product of two Cantor sets . Also,

B (1/

I

B

I::

"...... " I~ ,< _ _ , .2

I I II , ,

, , ' --,

Figure 2: The geometry of hyp erbolic maps

Decay of Correlations for Lorent z Gases and Hard Balls 101

since our main focus is on smooth invariant measures or SRB measures, A mustbe chosen so that each unstable leaf meets A in a set of positive (J-dimensicnal)Lebesgue measure. These differences aside , the general construction is quit esimilar to that for th e cat map.

As to the estimation of the measures of {R > n} , the argument abovesuggests that pure and uniform hyperbolicity leads to return time distributionswith exponent ially decaying tails . Indeed, essenti ally the same argument givesan alternate proof of the exponential decay of correlat ions with respect to SRBmeasures for all Anosov diffeormophisms, a result proved earlier in [Rul , Ru2 ,Si6].

For syste ms that are not Anosov, other aspects of the map , such as discontinuities or other forms of nonhyperbolic behavior, may affect the distributionof return times, the nature of which depends entirely on the dynamical systemin question.

3.2 The Abstract Model F : ~ ---+ ~. Let <P : M ---+ M be the dynamicalsystem of interest . As indicated at the beginning of th is sect ion, our strategy isto construct from <P another dynamical system F : ~ ---+ ~ which we think ofas an abstract model of <P . The reason for passing from <P to F is that F willbe a much simpler object , with the relevant information tha t determines thestatistical prop erties of <P conveniently displayed. The goal of thi s subsect ion isto take a closer look at F and the stati stical information it contains.

Th e maps F : ~ ---+ ~ that may pot ent ially arise as abstract models forsome <P have the structure of a tower or skyscraper , in which every point movesupward until it reaches its highest level, i.e. there is nothing above it , before itreturns to th e base of the tower which we denote by ~o (see Figure 3). For thecat map , F is relat ed to the construction in Sect . 3.1 as follows. The set ~o isobt ained from A by collapsing stable manifolds into points, and the return mapof F from ~o to itself is the quotient map of <pR : A ---+ A.

We now give a more precise description of F : ~ ---+ ~. The bottom levelof this tower, denot ed by ~o , is partitioned into a countable numb er of sets~O, i , i = 1,2,· · · . Th e first level of thi s tower is denoted by AI, the next levelup A2 etc. Under the action of F , each ~O, i moves upward one level per iterate ,to ~l , i , then ~2, i , etc. Think of this upward movement as rigid translat ions.Nothing interesting happens until a point reaches the highest level above ~O, i ,

which we call ~Ri , i ' At the next iteration, F maps ~Ri , i bijectively onto ~o . Weassume that the partition {~£,;} separates points, meaning for every x ,y E ~ ,

the re exists n 2: 0 such tha t Fnx and Fny lie in different elements of thepar tition. Thi s completes the description of the coarse structure of F : ~ ---+ ~.

We are interested in returns from ~o to ~o . The return time function R :~o ---+ 7l+ is given by RI~o , i = Ri, and the return map F R : ~o ---+ ~o isdefined by FRI~o, i = FRi.

Moving on to the finer structures of F : ~ ---+ ~, we assume th at thereis a reference measure m on ~, not necessarily finite and not necessarily F-

~O,i

-~~~2 '

~I----------;--';--'i------r------,

~o---------'-------'-------------'----

Figure 3: The tower map F : ~ --+ ~

invariant. The map F as well as its local inverses are assumed to be measurableand nonsingular with respect to m , so tha t measures with densities with respectto m are transformed under the act ion of F to measures with densities .

Fin ally, we introduce a notion of Holderness or Lipschitzness for functionson b.. A metric on b. that measures symbolic distances between points can bedefined as follows: Let the separation time between x, y E b.o be given by

s(x, y) = the smallest n such that (FR)n x and (FR)n y belong in different b.O,i'

For x, y E b.o, we define d(x , y) = (3"(x, y) where /3 < 1 is a numb er arbitrarilychosen but fixed. For x , y E b., we define d(x , y) = 1 if they do not belong in thesame b.e,i' and if they do we let d(x ,y) = d(F - ix ,F- iy) . Letting JF denotethe Jacobian of F with respect to m , we assume that J F = 1 on b. \ F - 1b.o,and impose the following regularity condit ion on JFIF- 1b.o or , equivalently,on J FRJb.o: we require th at log J FRIb.o,i be uniformly Lipschitz with respectto the metric defined, i.e. th ere exists C > 0 such that for all i and for allx , y E b.O,i'

IJFR(x) - 11 < C /3s (x ,y)JFR(y) - .

Furthermore, all the observables considered will be assumed to be uniformlyLipschitz with respect to the same /3.

Thi s completes our description of the dynamical system F : b. --+ b.. Weremark that the structure of F is chosen so that it displays clearly the information of great est import ance to us, namely the sequence of numbers m{R > n}as n --+ 00 .

Decay of Cor relat ions for Loren tz Gases and lI ard Balls 103

Before stating result s on the statistical prop erti es of F , we explain moreprecisely how F : t. ---+ t. is derived from the const ruction in Sect. 3.1. Fir st let be the cat map , and let R : A ---+ A be as in Sect. 3.1. Identifying points inA that belong to the same stable leaves, we obt ain a quotient set It, a quotientmeasure jl on It, and an induced map <p R : It ---+ It. Clearly, It can be thought ofas an interval and jl the I-dimensional measure on It. Moreover , It is part itionedinto a countable numb er of intervals each one of which is mapp ed affinely by is the tower with t. o = It, F R = n} < c()n for some () < 1.

In general , we proceed as with the cat map , but when nonlineari ties arepresent , the situation is a lit tle more complicated: Topologically, the map <p R :

It -+ It is defined, but it is a priori not clear that J FR makes sense. Sometechnical work is needed. We refer the reader to [VII for details.

Results on the statistical properties of F : t. ---+ t. and their implications for : M -+ M, the dynamical system from which F is derived!Yl, Y2]:

• If JRdm < 00, then

(i) F has an invariant probabili ty measure v equivalent to m;

(ii) has a smooth invariant probability measure or an SRB measuref-L .

Remarks. 1. (F ,v) is automat ically ergodic. In the result s below, let f-L be theSRB measure with f-L(Un nA) = 1. Then ( , f-L) is also ergodic. 2. We do notclaim th at has no other SRB measures: th rough information on R : A ---+ A,one cannot possibly know about the existence of SRB measures whose supportsdo not intersect UnnA.

• If JRdm < 00 and the greatest common divisor of {R} is 1, then

(i) (F,v) is mixing

(ii) ( ,f-L) is mixing.

We assume from here on that gcd{R} = 1. In the next th ree bullets, all testfunctions are understood to be uniformly Lipschit z as stipulated earlier.

• If m{R > n} = O(n- n) , 0: > 1, then

(i) the ra te of corre lation decay of (F,v) is O(n-n+l) ;

(ii) the rate of correlation decay of ( , f-L) is O(n - n +1 ) .

• Ifm{R > n} = Ct)" ; () < 1, then

(i) the rate of corre lation decay of (F,v) is < c,(),n for some ()' < 1;

(ii) the rate of corre lation decay of ( , f-L) is < C'()1n for some ()'.

• If m{R > n} = O(n- a ) , a> 2, then

(i) the cent ral limit theorem holds for (F, v) ;

(ii) the central limit theorem holds for ( ,JL) .

3.3 Discussion. 1. To what kinds of dynamical systems would the m ethodof this section apply, and for what kinds of statis tical propert ies ? The successof this approach depends on(1) the const ruct ion of R : A ---+ A, and (2) success in passing informat ion for Fback to the original system. (1) generally works when has "enough hyperbolicity", although it is hard to axiomat ically formulate what exactly that means.As mentioned at the beginning of this sect ion, this construction has been successfully carr ied out for a numb er of th e systems considered in this survey. Inaddition to that, it has been shown to work for various examples of interest indynamical systems, including logist ic interval maps fYI, BLS], expand ing mapswith neutral fixed points [Y2], piecewise hyperbolic maps fYI, C5], Henon at tr act ors [BY] and their generalizat ions [WY], and certain partially hyperbolicsystems. As for (2), for almost-sure prop erties and properties expressed in termsof the expectation of a random variable (such as correlat ion decay) , this passage is largely (though not completely) formal ; for other prop ert ies such as thePoisson law for large returns, this passage is less tra nsparent .

2. What aspects of a dynamical system determin e its statis tical properties? Wehave seen from the cat map that uniformly hyperboli c systems have exponent ialdecay or correlations. For syste ms that are predominantly but not purely hyperboli c, the following phenomenon has been observed: Initially, the hyperboli cpart of the system gives rise to an exponent ial drop-off in m{R > n}. However,as n increases, m{R > n} is determined more and more by the least hyperbolicpart of the syste m. To illustrat e how this works , imagine that the phase spaceof has certain (localized) regions of nonhyperboli city. These regions behavelike "traps" or "eddies", in which orb its may linger for arbit rarily long t imes.While most orb its st arti ng from A ret urn relat ively quickly, a fract ion of initial conditions will, by ergodicity, get into these "traps" and remain there fora long time . The ultimat e decay rate of m{R > n} , therefore, is determined byhow fast these orbits are able to break free. Indeed, the following message isclear : if a dynamical syste m has identifiable sources of nonhyperboli city, thenthe t ime it takes to overcome these nonhyperboli c parts determine m{R > n }and consequent ly th e type of statist ical properties in Sect. 3.2.

3. Wh at are the standard m ethods for obtaining rates of correlation decay indyna mical systems? In dynamical systems as in statist ical mechanics and probability, the existence of a spect ral gap is one of the most commonly used methodsof proof of exponent ial decay. The operator in question for dynamical syst ems isthe Perron-Frob enius or t ransfer operat or (see e.g. [Ru3, HK, YI]) . Equivalent

to the existence of a spectral gap but technically more flexible is the existenceof strict ly invariant cones with projective met rics ([FS, LJ). A method used bySinai et. al. ([BS2, BSC2J) to st udy correlation decay of billiards is approximation by Markov chains. Yet anot her standard method first used by Doeblinand well known in probabili ty but not exploited seriously in dynamical systemsuntil recently is th e coupling method, which is first used in [Y2] to prove theresults sta ted in Sect . 3.2. Thi s method goes as follows: Consider two processesconsisting of iterating F : ~ --+ ~ with different initi al distributions>' and N.We run th ese processes independently, all the while trying to "match" F:: >. withF::N. The rate at which the LI-norms of the densities of F:: >. - F::N tend tozero is a measure of the speed of convergence to equilibr ium, which in turn givesa bound for the speed of correlat ion decay.

§4. Correlation Decay for Planar Lorentz Gases

T he purpose of this section is to explain the proof of exponent ial decay ofcorrelat ions for a class of billiard maps using th e met hod discussed in the lastsect ion.

The class we will focus on is the 2-dimensional periodi c Lorentz gas , whichis a model for elect ron gases in metals. Mathematically it is represented by themotion of a point mass in lR2 bouncing off a (fixed) periodic configurat ion ofconvex scatterers. T his model was first studied by Sinai around 1970 [Si5]; itis somet imes called th e Sinai billiard. Pu t tin g the dynamics on the torus, weassume that the billiard flow takes place on n = T 2 \ U~= l n i where the n i 'sare disjoint convex regions with C3 bound aries (see Figure 4). The section map is defined on M = an x [- ~ , H We denote point s in M by p = (x, B) wherex E an is the footpoint of th e arrow indicating the direction of the flow and Bisthe angle this arrow makes with the normal pointin g into n . It is straight forwardto check that leaves invar iant the probabili ty measure f.l = ccos Bdx dB wherec is the normalizing constant. For simplicity, we will assume in th is sect ionth e finite horizon cond ition, which requires that the time between collisions beuniformly bounded.

Thi s ent ire sect ion is devoted to explaining the ideas behind the followingresult. Let Co. denote th e class of Holder functions on M with Holder exponenta .

Theorem 4 .1 ([Y1]) . Let ( , f.l) be as above. Then correlation decays exponentially fast for observables in C", More precisely, there exists (3 = (3(a) > 0such that for every f ,9 E c o. ,

ICj ,g(n)1 ~ C exp(- (3n)

for some C = C(J,g).

Figure 4: Billiard on T 2 with convex scatterers

As mentioned in the introduction, a weaker version of this result, namelythat of "stretched exponent ial decay", is first proved for this class of billiardsin [BSC2] along with other statistical prop erties. The finite horizon conditionin the theorem above is dropped in [C6] . This and other results on correlationdecay for billiards based on a similar approach are discussed in Section 5.

4.1 Geometric Properties of the M ap : M -+ M. Th e followinggeometric propert ies of play important roles in determining its st ati stic alproperties. For more background informat ion, we refer the reader to [Si5] and[BSCl , BSC2!.

(1) The discontinuity set S It is easy to see that is discontinuous at a point(x,B) E M if and only if the straight line segment st art ing at x and goingin the direction determined by Bmeets an tangentially at the first point ofintersection . We claim that the geometry of S as a subset of (x, B)-space isas shown in Figure 5: (i) S is the union of a finite number of smooth curvesegments (this number is infinite in the infinite horizon case) . (ii) Fixingan orientation for an, the slopes of these segments all have the same sign;let us assume they are negatively sloped. (iii) Some of the segments inS run from the top edge to the bottom edge of M, whereas others endabruptly as they join one of the "main" branches.

To understand these assert ions better, let us imagine starting from a fixedcomponent of an, aiming in a certain general direct ion and having anobstacle in front of us. (See Figure 6.) It is easy to see that the set ofdirections that give rise to a tangency forms a smooth curve . Moreover , ifthe obstacle in question is the "nearest " obstacle, meaning there is no other

Decay of Corr elations for Lorent z Gases and Hard Balls 107

r---•• x

Figure 5: The singularity set S in phase space

obst acle in front of it , then this curve extends from B= - ~ to B= ~. If,on the other hand, the obstacle in question is the "second" obstacle, thenthe curve of tangencies cannot be extended beyond the point of "doubletangency" . Points of mult iple tangency, therefore, are points in S at whichtwo or more smooth segments of S meet. They are called multiple points.We will return to them later on in the discussion.

(2) Hyperbolicity Hyperbolicity of <Jl is first proved in [Si5j. Intuitively, it iseasy to see from the strict convexity of the scatterers th at D<Jl has strictlyinvariant cones ([WD, the existence of which is a standard way or provinghyperbolici ty:

-_ ... -----::::--;: ...- --- ... ---_ ... --_ ....... -----.. ------

Figure 6: Curves of singularity. The top set of arrows represents a branch of S t hatextends from top to bot tom of M. Th e bottom set of arrows represent s a branch of St ha t ends abruptly at a doubl e tangency

Tangent vectors at (x ,O) E M are represent ed by curves in M passingthrough (x ,O), and curves in M are parametriz ed fam ilies of "arrows" inD. Consider the cone corresponding to all l -parameter fami lies of arrowsthat are divergent. Since divergent families upon reflection at a convexcurve become even more divergent , we see that the cones in question arest rict ly invariant . Thi s proves that on the projective level, at least , Dip

is uniformly hyperb olic. A lit tle more work shows that EU, the unstabledirections of D ip , are positively sloped and transversal to the curves in S.

(3) Unbounded derivati ves In addit ion to having discontinuit ies, anot heraspect of ip that complicates its analysis is that Dip is unbo unde d as(x ,O) tends to S. The technical difficulties that arise from this have beentaken care of in [BSC2] and will not be discussed in this article.

4.2 Comparison with the Cat Map. According to the scheme discussedin Section 3, it suffices to pick a reference set A, construct an acceptable returnmap ipR : A ---7 A, and show that J.L{R > n } < con for some 0 < 1. Theexponential decay of corre lat ions and Central Limit Th eorem for ip will thenfollow automatically.

Instead of repeat ing the const ruction from scra tch, let us remember what isdone for the cat map (Sect . 3.1) and focus on the differences. Fir st , ip here isnonlinear and Dip is unbounded, but as we have said earlier, we will not concernourselves in this art icle with these technical aspects of the problem. There aretwo concept ual differences between the cat map and the situation here, bothcaused by the presence of discontinuities:

(1) Stable and unstable curves can be arbitra rily short. Points that come arbitr arily close to the singularity set S in forward (respect ively backward ) timehave their local stable (resp. unstab le) curves cut arbitrarily short. By the ergodicity of ip , which is proved in ISi5] and taken for granted in thi s argument ,points with arbit rarily short stable or unst able curves are dense in M, The bestth at we can do, therefore, is to choose A to be a posit ive Lebesgue measure setthat is homeomorphic to a prod uct of two Cantor sets .

(2) Connec ted components of unstable curves m ay remain short for a longtime. Imagine the following worse-case scenario: Suppose a piece of unstableleaf 'Y is, under the action of ip , magnified by a factor of rv ~ and cut into twopieces of rough ly equal lengths. Suppose also that each one of these segments isdealt the same hand, i.e. it is again magnified by rv ~ and cut into two equalpieces. Suppose this goes on indefinitely, so that after n iterat es, ipn ("() has 2n

components each of length rv ( ~ ) n . Thi s is det riment al to our const ruct ion ofipR : A ---7 A, for in order for a segment to make an acceptable return, it mustgrow to the size of A. While this scenario exactly as described is ext remelyunlikely, long stretches of time during which the images of'Y are cut fast er thanthey have a chance to grow may have an effect on the tai l of the return t imes.

Decay of Corre lat ions for Lorent z Gases and Hard Balls 109

Of these two concerns, (2) has an obvious bearin g on {L{ R > n} ; it will bethe cente r of our attent ion in the rest of this sect ion. It turns out that once (2) isresolved, (1) can be handled as well. We first discuss briefly some problems thatarise from (1): Since it is virtually impossible to track the evolut ion of Cantorsets, we track instead the rectangle Q spanned by A. An immediat e question is:when the q.n-image of an s-subrectangle QS of Q is mapp ed to a u-subrectangleof Q, does q.n(A n QS) coincide with A n q.n(Qs)?

It turns out that in the unstable direction, q.n(A n QS) is bigger, and bitsof this Cantor set fall through the gaps of A witho ut "ret urn ing". T his leads tomore complicated estimates on return times. We remark that technical as theymay seem, these prob lems are far from purely technical in nature: The maindifference between q. and an Anosov map is the presence of discontinuities , andthe gaps in A are precisely the hand iwork of the discontinuity set.

4.3 Growth of Unstable Curves. To prevent the phenomenon in (2)above from happening, consider first a condition of the following typ e:

(*) There exists N E 7l+ and J > 0 such that for all unstab le curves,with €b) < J , the number of connected components of q.Nb ) is < ANwhere A > 1 is the minimum expansion on unstab le curves.

Condit ion (*) has the following inte rpretation: Thinking of cutting as introducing a form of local complexity, (*) says th at the growth in local complexityis dominated by the rate of expansion. When thi s condition is sat isfied , we haveimmediat ely that on average, each of the components of q.nb ) grows exponentially in length.

We first explain why Sinai billiards have exponent ial decay of corre lat ionsassuming that q. satisfies Condit ion (*). Ju stification of this condition is postponed to later. We will maintain throughout that the numb er J in (*) is therelevant length scale to consider, referring to segments shorte r than J as "short "and those longer than J as "long".

Let , be an unstable curve. On "I we introd uce a sto pping t ime

T( x) := the smallest n such that the component of q.n, containing q.nxhas length > J.

Here is what we propose to do: We run q. unti l t ime T , tha t is to say, we runq. on each component of q.n, (for as long as it takes) until it becomes "long".Th en we stop. The following is an est imate on the distribution of T .

Lemma 4.2 Let m, denote the Lebesgue measure on "t- Assuming (*), thereexists 0: < 1 independent of, such that

Proof 4.3 Let us assume for simplicity that N = 1 in (*), and let K be themaximum number of components into which a "short " unstable curve can be cutby 1> . By (*), K < A. Th e number of components that have remained "short"up until tim e n is :::; CKn where C = [£(r) /8] + 1. Thu s the total measure ofthe pull-back of these short components is :::; CK" A- n. Taking 0: = K A- 1 < 1completes the proof.

When a component of 1>i f' becomes "long", i.e. when it reaches a length > 8,we start up the process again. This defines on f' a sequence of stopping times

with T1 = T and Tn is th e first time after Tn- 1 when th e component in quest ionbecomes "long". The key idea is to look at the images of f' under 1>T",not 1>n. Observe that 1>T1 (r) is th e union of curves all of which are "long", andthat th e same is true of 1>T2(r ), 1>T3(r ), and so on. Moreover , pretendi ng for themoment that T1 < T2 < T3 < ... represents real time, one would conclude that f'grows exponentially, and there are never any short curves around. The situation,therefore, is entirely similar to that for the cat map (see Sect. 3.1). Using th emixin g property of 1> (proved in [Si5]) in the place of topologi cal t ransit ivity,we see that the proof in Sect . 3.1 cont inues to work, giving /1{R > n} < con.

To reconcile th e stopping times Tn with real time, observe that the distribut ion of (Tn - Tn- d ITn- 1 is essent ially th e same as that ofT, which is esti matedin the Lemm a above to have exponentially decaying tails.

To summarize, then, we first confuse Tn with rea l time, and conclude exponential decay of correlations for reasons similar to those for the cat map . UnderCondi tion (*), the distributions of Tn - Tn- 1 have exponent ially decaying tails.A little bit of work shows that th ese two est imates together give exponent ialdecay of corre lat ions with respect to real t ime.

It remains to explain why Condit ion (*) holds for 1> , th e billiard map inquestion. Let S(1>n) denote th e singularit y set of 1>n. It is not hard to seethat S(1)n) has the same st ructure as S(1)), except that it contains more curvesegments and becomes denser as n increases. For Z E M , let N(1)n , z) denotethe numb er of smooth segments of S(1>n) meeting at z. If z is not a multiplepoint of S(1)n), then N(1)n , z) = 1. Also, let N(1)n) = SUP zEM N (1)n, z ). Theverification of Condition (*) is a rephrasing of th e following observation due toBunimovich:

Lemma 4.4 (IBSCI]) . There exists K depending only on n sucli that forall n 2: 1,

Verification of Condition (*) assuming Lemma 4.4 Recall that unstable curves are t ransversal to S(1)n), so tha t for each n , if an unstable curve f' is

Decay of Correlations for Lorentz Gases and Hard Balls III

sufficient ly short , th en it will not meet s(.pn) in more than N(.pn) points. Thisimplies th at .pn, will have not more than N (.pn) + 1 components . We chooseour parameters in th e following order: First choose N so that K N + 1 < ANwhere A > 1 is the minimum expan ion along unst able curves . Th en we choose8 > 0 such th at if , has length < 8, th en , meets S (.pN) in ::; K N points. This8 is th e relevant length scale for Condition (*).

We conclude this sectio n with an explanation for Bunimovich's observation.Consider a (stra ight) billiard trajectory with multipl e tangencies, starting atz E AI , ending in z' E AI, with no regular collisions and at least one ta ngencyin between. We st ress th at z and z' are point s in the phase space lvI , not on thebilliard table.

Starting from a small neighborhood of z, there is a finite numb er of waysof reaching a neighborhood of z': it is possib le for the billiard traj ectory not totouch any of the scatterers in between, or to bounce off the first but not th esecond , or the second but not th e first , and so OIl. From this one deduces th efollowing picture (see Figure 7):

- there are small neighborhoods U of z and U' of z' such that U is the disjointunion of a finite number of sectors VI , · . . , Vk and U' is the disjoint unionof a finite numb er of secto rs V{, · . . , V~ ;

- for each i , th ere exits n j such that .pnj maps \0 diffeomorphically ont oVj.

Here each \0 repr esent s one "type" of trajectories from U to U', and n j is thenumb er of (nontangenti al) bounces in between. Clearly, the numb er of sectors ,k, depends only on the configurat ion of scatters.

An upp er bound for N( .pn, z) can be est imated as follows. Let N (.pi IVj ,z')denote the number of smoot h segments of S( .pi) passing through z' that lie in

r ~- --,V, ,, ,.· ,

• II Z '· ,, ,, ,,

(n )

.. 'z' "V', ,

I ,

I r II Z I. ., ,,, ,... _.....

--_.. x

Figure 7: Neighbor hoods of multiple point s(.p(n) = .pnj when restricted to the sector \0)

112 N. Chernov an d L. S. You ng

Vj. T hen pulling back the picture from z' to z , we have

N(n ,z) ::; k + L:N(n-n jlVj ,z') .j

Since N(n- njlVj ,z') ::; N(n-llVj ,z'), which we assume induct ively to be::; K(n - 1), we have argued that N (n ,z) grows linearly with n , completingthe proof of Bunimovi ch's lemma and the hence the proof of exponent ial decayfor this class of billiards.

§5. Correlation Decay in Related Billiard Models

Here we describ e other physical models with elast ic collisions for which estimateson corre latio n decay rates have been proven or conjectured. In Sects. 5.1 - 5.5,the discussion pertains to the billiard map (or relevant sect ion map) . Resultsfor the corresponding flows are considerably more delicat e and are discussed inSect. 5.6, under the headline "Real tim e dynamics".

5.1 Sinai Billiard Tables. Imagine a Lorent z gas whose scatterers are solarge th at they overlap and t rap the par ticle in a bounded diamond-like regionas shown in Fig. 8. Thi s defines a billiard system on a ta ble whose sides areconvex inward. The resulting models are called Sinai billiard s. T hey differ fromthe Lorent z gases discussed earlier in two ways.

A. Traps at the corners. If a trajectory comes close to a corner point (wheretwo scatterers inte rsect ), it may experience two or more rapid collisions withina very short time. Between those rapid collisions, the unstable curves do nothave a chance to grow. More precisely, the expansion factor along unstablecurves under the billiard map is

D = 1 + r B (5.1)

where r is th e time between consecut ive collisions and B the geomet ric curvat ureof the outgoing wave front made by the unst able curve . T he quantity B ispositive and usually bounded; since r can be arbit rarily close to 0, D can bearbitrarily close to 1. As a result, is not uniformly hyperbolic. For more details,see [BSC1, BSC2, C6]) .

Here is one way around this problem. It is a simple geomet ric fact (see e.g.[Re]) that if two scatterers meet at an angle (X > 0, then there can be at most1+Jr / (X rapid collisions of the type described above. After that the par t icle mustleave the corner. Hence if all the angles (X l , • •• , (Xk made by the scat te rers attheir intersections are positi ve, then the maximum number of consecutive rapidcollisions is bounded above by m = max,{1 + Jr / (Xi} . The map m is thereforeuniformly hyperbolic, and if one proves exponent ial decay of corre lat ions for

Decay of Correlatio ns for Lorentz Gases and Hard Balls

Figure 8: Sinai billiard s in a diamond-type table

113

 m, then the same property for will follow. Thi s is done in [C5, C6! (seepar agraph 13 . below).

On the other hand , if one of the angles (Xi is zero, i.e, if two of the scat terersintersect tangent ially, form ing a cusp, then the numb er of consecut ive rapidcollisions is easily seen to be unbounded. In this case, the hyperbolicity of isvery nonun iform, and correlat ions are believed to decay slowly, at the rate

There is a strong numerical and analyt ical evidence of this asymptot ic behavior[MR], but a rigorous proof is not yet available.

B. Condition (*) . The proof of exponential decay for the periodic Lorentz gasgiven in Section 4 relies on a property of that expresses the fact that expansion domin at es local complexity along unstable curves. Thi s prop erty, which wecalled Condition (*), has not been verified for general Sinai billiards with cornerpoints. Indeed, Lemma 4.4 is likely to be false. In [BSC2, C6], Condition (*) isassumed in order to obt ain bounds on correlat ions. We refer the read er to [Bu]for recent advances in this direction.

5.2 Lorentz Gases with Infinite Horizon. Infinit e horizon in a periodicLorentz gas refers to the prop erty that there is no finite upp er bound for thelengths of free runs between collisions. Thi s is equivalent to the existence of

114 N. Cherno v and L. S. Young

Figur e 9: The per iodi c Lorentz gas with ou t horizon:long free runs generat e infinitely many singularity lines

corridors along which the particle can move indefinitely without colliding witha scatterer (see Fig. 9).

In contrast to the finite horizon case, the singularity set here is the union ofinfinitely many smooth curves. Figure 9 shows how they are generated. It is nowpossible for an unstab le curve, however short, to be cut in one iterate by thesingularity set into an arbitrarily large, possibly infinite, number of disconnectedcomponents . Hence Condit ion (*) in Sect ion 4 fails. Nevertheless, the map <II isknown to have exponent ial decay of correlat ions IC6]). T his can be explainedas follows. T he regions in the phase space where the singularity curves accumulate correspond to long intercollision flights (Fig. 9). Dur ing those flights, theunst able curves expand very st rongly, according to (5.1), where T in now verylarge (in fact , B is also large, as B ~ T 1/ 2 ; see IC6]). This strong expansionis suffic ient for overcoming the effect of the cut t ing. Careful analyses of theseissues are carried out in [BSC2, C6].

5.3 Lorentz Gases under External Forces. Consider the situation wherethe par ticle in a periodic Lorentz gas is subjected to an external force F. Theequat ions of motion (between collisions) are now

q =p, p = F (p,q)

where q = (x,y) is the posit ion and p the velocity vector of the particle. Weassume there is an integral of motion [ (p ,q), so that the dynamics can bereduced to mot ions on 3-D surfaces [ (p ,q) = const. We assume that the forceF is small, so that the particle tra jectories between collisions are nearly stra ight .For simplicity, we assume also the finite horizon condit ion.

If the force F is given by the gradient of a potential, i.e. if F = - 'VU(q) forsome U, then the dynamics is Hamiltonian . In particular , total energy U(q) +~ IlpW is preserved, as is a measure compatible with volume called the Liouvillemeasure.

Decay of Corre lations for Loren tz Gases and Hard Balls 115

For more genera l forces, the system admits no smoot h invariant measures,and one looks for the existence of a physically meaningful invariant measur e, i.e.an SRB measure (see Sect . 2). The techniques described in Sections 3-4 applyto this model to provide an SRB measure which also has strong statist ical propert ies: exponential decay of corre lat ions and the CLT. For a detailed analysis ofthis model, see [C7] . A separate argument [C7] shows that thi s SRB measure isunique and its support is the ent ire phase space. Moreover, the system has theBernoulli prop erty in both discrete and cont inuous t ime.

5.4 Multi-Dimensional Lorentz Gases. Consider the motion of a particle in lRd

, d 2: 2, bouncing off a periodi c array of fixed convex scatterers . Thi sis the d-dimensional version of the period ic Lorent z gas. We again assume finitehorizon. It is shown in IC2] that the correlat ion function decays at least fasterth an a st retched exponent ial funct ion, i.e.

for some constants a, b > 0. It is believed tha t the act ual rate of decay IS

exponent ial, and the work to prove that is underway.

5.5 Multiple Correlations. A natural generalization of the correlat ionfunction (1.2) is:

where h ,.. . ,!k are functions on M and t 1 , . .. , t k are moments of time. Thi sis called a multiple correlat ion funct ion.

Certain physical constants are expressed in terms of mult iple correlat ionfunction s. For exampl e, the so-called super-Burnett coefficient (arising in higherorder expansions of the diffusion equation) is given by an expression t hat involvesthe sum

00

L [Ca ,b,c,d(O, m ,n , k) - Ca,b(O, m)Cc,d(n, k)m ,n ,k = - oo

(5.2)

where a,b,c, d are certain funct ions.The facts about correlat ion functions obtained in [BSC2, C3] allow us to

prove, in the case of Lorentz gases, that th e sum in (5.2) converges absolute ly forany four Holder continuous functions a,b,c and d. For a proof and the discussionof the Burnett coefficient , see [CD].

5.6 Real Time Dynamics. The discussions in Sects . 5.1-5.5 are for discrete t ime dynamics; more precisely, the st at ement s there pertain to certainsection maps corresponding to the flows in question. Physically interest ing tim eis, of course , real (or continuous). In finite horizon situations, the cent ral limit

116 N'. Cherno v and L. S. Youn g

theorem and invarian ce prin ciple for real t ime follow from the correspondingresults for discrete ti me; see IBS2, BSC2, DPJ. Th e correlat ion function Cf ,g(t ),however, may behave quite differently in discrete and cont inuous times.

In a periodic Lorentz gas without horizon, one has exponent ial decay of correlations in discrete time as we have seen in Sect . 5.2. Th ere is strong evidence,however, that corre lat ions decay slowly in real t ime [FM1, F.tvI2]:

(5.3)

T he reasons behind this est imate are as follows. For a par ticle in a corr idoras shown in Fig. 9, a collision-free flight of length 2 t occurs with probabili ty'" l it . Hence, over a period of t ime t , a fraction of the phase space of measure'" l it does not have the chance to mix with the rest of the phase space. Fromthis one deduces the est imate (5.3). In short , the slow-down in correlat ion decayis caused by the presence of long corr idors, which act as traps for the otherwiseuniformly hyperbolic dynamics.

In contras t to the previous situa tion, in Sinai billiard s with cusps (caused byta ngent ial intersection s of the scatterers), even though corre lat ions in discretetime seem to decay like l it as we have discussed earlier, it is reasonable toexpect that their decay rat e in cont inuous time is much fast er , possibly as fastas exponent ial. The difference between discrete and cont inuous tim es can beexplained by the fact that in a succession of rapid collisions near a cusp, thetotal time elapsed is very short even th ough the number of collisions may belarge.

In a Lorentz gas with finite horizon, corre lations in real t ime are expected todecay exponent ially as in discrete tim e, but the situation is considerably moreinvolved. Mathematical proofs are not yet available, but research is underway.

5.7 G ases of H ard Balls. As dynamical systems, hard balls are considerably more complicated than Lorentz gases. Full hyperbolicity for an arbit rarynumber of balls in a torus has only been proved very recent ly 1882]. Rigorousst udies of corre lat ion functions are not within reach at the present time. Theonly facts th at are clear are th at there are many traps in the phase space, theyare of various kinds, and tha t it is possible for trajectories to be caught in themfor very long periods of time without achieving full hyperbolicity.

The following t rap is well known: certain phase tr ajectories may interactonly with a proper subset of the balls. Even though th e set of initial conditionswith th is prop erty is of zero Lebesgue measure , nearby phase trajectories takearbitrarily long t imes to get to the rest of the phase space, slowing down therate of correlat ion decay. We remark that this phenomenon of cluste rs is one ofthe main obstacles in the proofs of full hyperbolicity [882] and ergodicity forgases of hard balls.

For infinite gases assuming that the par ticles are at local equilibrium (i.e.they are distributed randomly and uniformly in space) , one can heuristicallyest imat e the correlat ion funct ion Cf(t) as follows [PRJ:


Let the function f in Gf (t) be th e x-component of the velocity vector ofa selected (tagged) ba ll. We fix that component vx(O) at tim e t = O. The taggedparticle interacts with its neighbors and, after tim e t , its init ial velocity vx(O) isshared by all the particles in a volume Vi around it . Hence the average velocityof th e tagged particle at tim e t is (vx(t )) '" vx(O) /Nt where N; ~ pVi is th eestimated numb er of particles in th e volume Vi and p is the density.

Fur th er decay of the functi on Gf (t ) ~ (vx (t) ) can only occur because Vigrows with t. For simplicity, assume that Vi is a round ball of radius Rt . Thenit is standard in hydrod ynamics th at Rt rv ,Ji. This gives the est imate

(5.4)

where d is the dimension of physical space, and the coefficient a depends on thedensity p.

The estimate (5.4) was arr ived at more accurate ly by a vari ety of empiricaland th eoreti cal methods in statist ical mechanics. B. Adler and T . Wainwrightdemonstr ated it by molecular-d ynamics calculat ions. R. Dorfm an and E. Cohen established thi s est imate using kinetic th eory. M. Ern st , E. Hauge, andJ. van Leeuwen , then K. Kawasaki , and then Y. Pomeau derived (5.4) fromhydrodynam ic mode-coupling theory. We refer the reader to lEW , PRJ for references and for furth er discussion of the topic .

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(1999), 153-1 88.

Entropy Values and Entropy Bounds

N. Chernov

Contents

§1. Entropy: General Formulas .§2. Entropy of Lorentz Gases: Asymptotic Estimates .§3. Mean Free Path . . . . .§4. Entropy of Infinit e GasesReferences. . . . . . . . . . .

122128131135141

Ab stract . We describe rigorous mathematical results on the KolmogorovSinai entropy for Lorentz gases and hard ball systems (bot h finite and infinit e) .Exact formulas and asymptotic estimates of the entropy are discussed for variousmodels.

122 N. Chernov

§1. Entropy: General Formulas

Entropy is an important numerical characterist ic of dynamical systems. It , ina sense, measures the amount of chaos, or complexity, in the system.

Two different versions of entropy are widely used in the study of dynamical systems. The measure-theoretic entropy (called also Kolmogorov-Sinai entropy) is associated with a measurable transformation T : X ~ X preservinga probability measure u. By contrast, the topological entropy is associated witha cont inuous transformation T : X ~ X of a topological space X not equippedwith any measure. Generally, the topologic al entropy htop(T) is greater than theKolmogorov-Sinai entropy hJ1(T) , and any invariant measure fL with the property h11(T) = htop(T) is called the measure of maximal entropy. For continuoustime dynamical systems (flows) the entropy is defined as that of the time-onemap . More detailed discussions of entropy, its definition, properties, and historyof the subject, can be found , for example, in [ME].

We will primarily work with the Kolmogorov-Sinai entropy (also referred toas KS entropy). Throughout , we denote by h(T) the KS entropy of the billiardmap T : n~ n with respect to the smooth invariant measure v and by h(<pt

)

the KS entropy of the flow <pt with respect to the Liouville invariant measure

fL·There is a remarkable relation between the two entropies, h(T) and h(<pt ) ,

that follows from a more general Abramov's formula for the entropy of suspension flows [Ab]:

Proposition 1.1 We have

(1.1)

(1.2)

(1.3)

(1.4)

where f is the mean value of the free path T(X) on n:

f = l T(x)dv

The value of the KS entropy is closely related to those of Lyapunov exponents:


h(T) = LI:+Xi(X)dv(x)

where the sum 2:+runs over all positive Lyapunov exponents Xi(X) > 0 at everyx E n (counting multiplicity). We also have

h(<pt) = JM I:+x7(x) dfL(X)

where the sum 2:+ runs over all positive Lyapunov exponents x7(x) > 0 of theflow <pt at every x E M (counting multiplicity) .

Entropy Valu es and Entropy Bounds 123

The above formulas (1.3) and (1.4) are known as Pesin identities. Theywere originally found by Ya. Pesin in the context of smoot h hyperboli c systems with smoot h invariant measures [PI. Lat er these formul as were proved forsmooth systems with singularit ies (including billiards) [KS], assuming only partial hyperbolicity, and for invariant measures that only have smooth conditionaldistributions on unstable manifolds. Such measures are now called Sinai-RuelleBowen measures (also, SRB measures). Ergodic SRB measures in hyperbolicsystems are the only physically observable measures, in the sense th at theycharacterize space distributions of typical phase t ra jectories. It is interestin gthat SRB measures are the only measures for which the Pesin identity holds, sofor all the other measures the entropy is strictly less than the average sum ofpositive Lyapunov exponents . For more discussion of thi s topic see an excellentsurvey IY] .

SRB measures correspond to nonequilibrium steady st at es in statistical mechanics. If one perturbs a Hamiltonian system (that has a smooth invariantmeasure by the Liouville theorem) by an external force or a bound ary condition, then generally the perturbed system does not have any smooth invariantmeasure. Then physically interestin g invariant measures are those that describethe evolution of typical phase points, and such measures are, in many cases, SRBmeasures. More precisely, if the original system is hyperbolic and the perturbation is small, then an SRB measure is very likely to represent a nonequilibriumsteady state.

Various perturbations of hard ball gases and Lorentz gases under exte rna lfields or boundary condit ions have been st udied in the literature. In many casesnonequilibrium steady states in the form of an SRB measures have been observednumerically and somet imes invest igated mathematically [CELS, C41. See alsothe surveys [Bu, CYj in this volume for more details. In all those cases, Pesin 'sidenti ty for the entropy is very likely to hold as well, but there is no mathematicalproof of that fact in such a generality.

We now get back to our hard balls and Lorentz gases.Th e relation of the ent ropy to Lypunov exponents may not be practically

very useful, because the Lyapunov exponents are not easy to compute . Theycharacte rize the asymptotic rate of expansion of unst able vectors. One can simplify this relation noti cing that, due to the Birkhoff ergodic theorem, the averageasymptot ic rates of expansion are equal to the average one-step rates of expansion. This is stated below.


h(T) = lIn [JU (x)1dv(x) (1.5)

Here J U(x ) is the Jacobian of the different ial map DT restricted to the uns tablesubspace E~ c TeD. (the latter is spanned by all the tangent vectors with posit iveLyapunov exponents) .

124 N. Chernov

Note that JU(x) is t he factor of expansion of volume in the space E~ und erthe map DT : E~ ---+ Erx'

The advantage of th e last entropy formul a (1.5) over the previous one (1.3)is act ually qui te deceptive. To find the unstable subspace E~ c Txn, one essentially needs an asymptot ic procedure pract ically equivalent to the computationof all posit ive Lyapunov exponents .

There is, fortunat ely, an explicit characterization of the unstable subspaceE~ and an explicit formula for the entropy h(T) in terms of the so called curvature operator Bx . That operator was introduced by Ya. Sinai in the seventieslSI , S3], and it has been the main tool in Sinai 's pioneering works on Lorent zgases and hard ball systems. The operator Bx is given in terms of an infinitecont inued fraction defined below.

For any point x = (q,v) E M we denote by dx = (dq,dv) tangent vectorsin TxM, so that dq E IqQ and dv E T"Sd-l . Note th at dv ..l v, because Ilvll =const. Denote by Jx the hyperplane in IqQ orthogonal to the velocity vector v.It can be naturally identified with T"Sd-l, since both are perpendicular to thevector v . We will define a linear operator Bx : Jx ---+ Jx = T"Sd-l , with the helpof a few auxiliary linear operators.

Let Xt = (q(,Vt) = tx. If there is no reflections at 8Q between x and Xt,then the velocity vectors v and Vt are parallel, hence the spaces Jx and Jx , areparallel and can be naturally ident ified by parall el t ranslation.

Let t be a moment of reflection at 8Q, i.e. assume qt E 8Q. We have aninst ant aneous transformation of th e velocity vector at t ime t given by

vi = vt - 2(n(qt) . vt)n(qt)

Here vt and vi are the velocity vectors before and after the reflection, respect ively, and n(qt) is the unit norm al vector to 8Q at the point qt pointi ng inwardQ. We have two hyperplanes in the tangent space Iq,Q, perpendicular to vtand vi, we call them J;, and J;;" respectively.

Denote by U : Iq,Q ---+ Iq,Q the reflector across the hyperplane Iq,(8Q)tangent to 8Q at the reflection point qt. The reflector U is obviously given by

U(w) = w - 2 (n(qt) . w)n(qt)

for all w E Iq,Q. It is easy to see that U(vt) = vi and U(J;,) = J;;" and U isan isometry. The operator U may be used to identify J;, with J;;" and thus wecan identify the hyperplanes Jx , for all t , but we will not pursue this goal.

Denote by 8 : Iq,Q ---+ Iq,Q the unique linear opera tor specified by twoconditions:(i) 8(vt )= vi ;(ii) for any vector w E J;, we have

8(w) = 2 (vi ' n(qt ))V+Kq, V_(w) E J:'

Here V_ is the project ion of J;, onto Iq,(8Q) parallel to th e incoming velocityvector vt , and V+ is the proj ect ion of Iq,(8Q) onto J;;, parallel to the normal

Ent ropy Values and Ent ropy Bounds 125

vector n(qt ). Also, K q, is the curvature operator of the bound ary hypersurface8Q at the point qt defined, as usual, by

n(qt + dq) = n(qd + K q, (dq) + o(lldqll)

for vectors dq E 'Tq,(8Q). Note: since K q, is a self-adjoint posit ive-semidefiniteoperator, then so is 8 U- 1

.

Assume now that the past tra jectory of x is completely defined. Let 0 >t l > t2 > ... be all the past moments of reflection (note that i , ~ - 00 asi ~ 00). At each reflection moment i , we denote by U, and 8 i the two linearoperators introduced above. Let TO = - t l and Ti = t, - tHI > 0, i 2': 1, be theintercollision tim es. Then

(1.6)

where *means A- I . Note that the terms 8 i Ui and T;I alterna te as the fract ion cont inues downward . In a sense, these two alternating terms describe thecont ribut ion of reflections and free paths as they appear on the tra jectory .ptx ,t < O.

Note thatI

Ex, = --....,,---tI + IIf;

if there is no reflections between x and Xt . At each moment of reflection t i , theoperator Ex, changes discont inuously, and we have

(1.7)

Hence, the opera tors Ex, are naturally relat ed to each ot her along the trajectory.ptx .

If x = (q,v) E n, i.e. x is a reflection point , we define

Th en it follows from (1.6) and (1.7) that

Here 0 = t l > t 2 > ... are the past moments of reflect ions.One can easily check that Ex maps Jx into itse lf. In all that follows we

restr ict Ex onto the hyperplane Jx'

126 N. Chernov

Proposition 1.4 The operator- valued continued fraction (1.6) conve rges atever y point x E n with an infinite past trajectory. Moreover, if Bx,n is a fin itecontinued fraction obtained from (1.6) by trun cation at the n -th reflec tion, then

1liB -B 11 <-x x,n - Itnl

The operator B x is self-adjoint posit ive semi-d efinite.

The proof of the convergence is based on the fact that all the operators in(1.6) are self-adjoint positive semi-definite, i.e, r. > °and 8 i 2: 0. The firstproof was pub lished in 18Cj, see also [LW] . In a weaker form the statem ent wasgiven without proof earlier in [84] . For 2-D Lorent z gases the convergence wasproved earlier in [811 .

Remark 1.5 . Th e past trajectory cI>t x , and hence the operator B x , is welldefined unl ess the follo wing anomalies occur :(i) Th e traj ectory cI> t, t < 0, hits a "corner point" in the configurati on space. Nosuch point s exist in the Lorent z gas model where all the scaiierers are sm ooth.In the hard ball model, corner points in the configuration space correspond tomultiple collisions of balls (where three or more balls collide simultaneously) .Th e dynamics is discontinuous at such points.(ii) Th e trajectory is tangent to the boundary in the configurati on space (thi ssituation is called a grazing collision, it is possible in both Lorentz gases and hardball gases) . At such points the dynamics is continu ous but not differentiable, i.e.these are singular point s for the dynamics.(iii) Th e traj ectory experiences infinitely many collisions within a finit e intervalof tim e. Th is sort of disast er is possible for some billiard systems . However, asG. Galperin (Gal! and L. Vaserstein (V! showed, this never happens in gases ofhard balls or Lorent z gases (more generally , this is impossible in any semidispersing billiards) .As a result , the operator B x is defin ed at all regular (nonsing'ular) phas e points.Moreover, it depends on x continuously.

The operator B; explicit ly describes the unstable subspace E~ at every pointx E M:

Proposition 1.6 If Lyapunov exponents exist at a point x EM, then theunstable subspace E~ c TxM (th e subspace spanned by all the tangen t vectorswith positive Lyapunov exponents) satisfies

E~ = {(dq,dv) : dq E J~ , dv = Bx(dq)}

Here J~ c Jx is the subspace spanned by the eigenvectors of B ; with positiveeigenva lues.

Entropy Values and Entropy Bounds 127

We note that since B x is self-adjoint and positive semi-definite, the spaceJx is the orthogonal sum J;: EB J~ of two Bx-invariant subspaces so that B x ispositive on J;: and zero on J~ . Note also that dim E;: = dim J;: .

Th e entropy can also be explicitly given in terms of the operator Bx :

Theorem 1. 7 We have

h(T) = in Indet(I +T(x)B;)dv(x)

and

(1.9)

(1.10)

The formula (1.10) has a long history. It was first established for 2-D dispersing billiards by Sinai [SII in 1970. Its multidimensional version for semidispersing billiards app eared in 1979 in a preprint by Sinai [S3], with an outlineof a proof. A complete proof of both (1.9) and (1.10) for semi-dispersing billiards was provided by Chernov [C2] in 1991. He also extended both formulas tomore general classes of billiard tables in [C21 and later in [C3] . In fact , Chernovproved [C3] that (1.9) holds for every billiard table, in any dimension, as longas unstable bundles of t rajectories do not focus right on the bound ary. He alsofound a necessary and sufficient condit ion on a billiard table for the formula(1.10) to hold. The condition is th at unst able bundles of t raj ectori es do notfocus between collisions [C3] (we note that if they do, th e integral in (1.10)diverges) .

The proof of the above theorem is based on the following ideas. The formula(1.9) follows from (1.5) provided we can establish

J"( x) = det(I + T(x)B;) (1.11)

Thi s is not true in the Euclidean met ric (dq)2+ (dV)2 on n , but there is theso called pseudo-metric on n (also called the p-metri c) in which JU(x) is indeed given by (1.11) . In the p-metric, the distance on unstable manifolds inn is induced by the Euclidean metri c on the orthogonal cross-section of thecorresponding outgoing bundles of trajectori es in the configuration space . Thi sconst ruction of a pseudo-m etric goes back to Sinai [SI] and is commonly usedin other pap ers on billiard dynamics . The verificat ion of (1.11) is then quit e elementary, see, e.g., [C2] . One should also note that by changing metric in nonechanges the function JU(x) but its integral entering (1.5) remains unchanged , asit follows from the invariance of the measure u. Lastl y, the formula (1.10) followsfrom (1.9) and (1.1) by rather st and ard and simple calculat ion, see [C2, C3].

128 N. Chernov

§2. Entropy of Lorentz Gases: Asymptotic Estimates

Estimation of the entropy and Lyapunov exponents of Lorentz gases have beendone by physicists since early eight ies. One motivation was to explore the quantitative characterist ics of chaotic dynamics and observe transition from a regularmotion (on a torus without scatterers) to chaos (starting when a small scatteris placed on the torus) .

For a 2-D periodic Lorentz gas with a single circular scatterer of radius r > 0on a unit torus the entropy was est imated [FOK] by

h(T) ~ -2lnr (2.1)

as r -+ O. Since the mean free path was long estimated to be f ~ (2r)-1, wehave by (1.1)

(2.2)

Since h(tI>t) -+ 0 as r -+ 0, one obtains an asymptotic behavior of the entropynear the transition point (between the "regular motion" at r = 0 and "chaos" atr > 0). The above est imates have been proved, see below.

It was also conjectured in [FOK] that for any d-dimensional periodic Lorentzgas with a spherical scatterer of radius r > 0 one should have h(T) ~ - d In r ,which turned out to be wrong , see below. In the analy sis of h(T), the followingimportant quantity is involved:

In in T(X)dv(x) - in In T(X) dv(x) (2.3)

It was numeri cally estimat ed [FOK] that this quantity remains bounded andhas a posit ive limit (~ 0.44 ± 0.01) as r -+ O. The first part of this conjecture(boundedness) was lat er rigorously proved, see below. The convergence to a limitis still an open problem.

In the 2-D case, the only positive Lyapunov exponent coincides with theent ropy. For multi-dimensional periodic Lorentz gases with a single sphericalscatterer of rad ius r , individual positive Lyapunov expon ents for the billiardball map T have been st udied in [BD] . It was estimated that every positiveLyapunov exponent Xi > 0, as a function of r , increases like const -lln r ], asr -+ O. Moreover, every positive exponent but the maximal one was conjecturedto be ~ - 1/4 In(r/2). The maximal Lyapunov exponent was conjectured to be~ - (3d+ 2)/4In r . The last two conjectures turn out to be wrong , see (2.9) and(2.10) below. The first one is correct , see (2.10) below.

P. Baldwin IB] gave a theoretical argument support ing the following sharpening of the formula (2.1):

h(T) = -21n r + const + O(r) (2.4)

Entropy Values and Entropy Bou nds 129

His argu ment is not a mathematical proof, and so his prediction st ill remainsan open problem.

The following theorem was rigorously proved by Chernov.

Theorem 2.1 ([C2]). The entropy of the d-dim ensional periodic Lorentz gas(d ~ 2) with a single spherical scatterer of radius r > 0 in a unit torus is givenby

h(T) = - d(d - 1) In r +0(1)

and

as r -+ O. The mean free path is given by

(2.5)

(2.6)

(2.7)1-IBd

l rd 1T = = + 0(1')IB d- I l r d- 1 IBd- 1Ir d- I

Here IBk l is the volume of the k-dimensional uni: ball, see (3.4) below. Thedifference (2.3) is always positive and lLnifo1'mly bounded in r for every d.

The proof in [C21is based on the approximation of the operator R%in (1.9)by 8 1UI I

, see (1.8). The norm of the error is bounded

d. Proposition 1.4. Therefore, the substitution of 8 1UIl for Bt in (1.9) can

only change the integral in (1.9) by a uniformly bounded amount .Next , for small r the operator 8 1 has eigenvalues of order 1'-1, which can

be found by an elementary calculation for spherical scatterers, details may befound in [C21. As a result , the integration in (1.9) gives

h(T) = (d - 1) ( -In r + Lin T(X) dV(X)) + H(d) + 0(1) (2.8)

The term H(d) here comes from the substitution of 8 1UIl for Bt in (1.9). lts

value was computed explicitly in [C21 : H(2) = 2, H(3) = In 4, and for d ~ 4 wehave

H(d) = (d - 1) In 2 - (d - 3) ISd-

2111

td-

2 ln JI=t2 dt

Here ISkI is the k-dimensional volume of the unit sphere Sk in IRk+! , see (3.3)below.

Last ly, the boundedness of (2.3) that was proved in [C2] gives (2.5). Theest imate (2.6) then follows from (1.1) and (2.7). The formula (2.7) is quiteelementary, see (3.2) below.

130 N. Chernov

It also follows from (2.8) that the exist ence of the limit of the quantity (2.3)is equivalent to the following asymptotic formula :

h(T) = - d(d - 1) In r + const + 0(1)

Both remain, however, open questions, as well as the more refined prediction(2.4).

All the open questions involving the entropy h(T) can be equivalent ly resta ted for the entropy h(if>t) , in view of (1.1) and (2.7).

As for the Lyapunov exponents for T , it follows directly from (2.5) that themaxim al one is bounded by

- d in r + 0(1) :::; Xm ax :::; - d(d - 1) In r + 0(1) (2.9)

By using again the approximation of R%by 8 1UtI , and the asymptotic eigenvalues of the lat ter , see [C2] for details, it is easy to est imate every positiveLyapunov expon ent from below: Xi ~ - d In r + 0(1). Together with (2.5) thisgives an asymptotic formula

Xi = - dln r + 0(1) (2.10)

for every positive Lyapunov expon ent Xi > O.Therefore, all positive Lyapunov exponents have the same asymptotics as

r ---+ O. It was also conjectured in [C2] that all the positive Lyapunov exponentsshould be actually equa l. This conjecture is still open . However, it was shownrecently [Dr , LBD], both analytically and numerically, that in a 3-D randomLorent z gas (with a random configurat ion of scatterers) the two positive Lyapunov exponents are dist inct!

Two more general results were proved in [C2].Consider a periodic Lorentz gas with m disjoint spherical scatterers with

radi i rl , . . . , r m in a unit torus . Put

Zo = rt-1 + . .. + r~-l

andZ d-l l + d-1 l1=r1 nrl + '" r m n rm

The entropy of such a Lorentz gas was proved [C2] to be

h(T) = -(d - 1)[lnZo + ZI/Zo] + 0(1)

and

(2.11)

h(if>t) = -(d - 1) IBd- l l [Zo In Zo + Zl] + O(Zo)

as rl ,' " ,rm ---+ 0, while the distances between the scatterers remain boundedaway from O. The mean free path is

Lastly, consider a periodic Lorentz gas with m disjoint convex scatterers ina unit torus, which are homotetically shrinking with a common scaling factorE> O. Let SI be the total surface area and VI the total volume of the scattererswhen E = 1. Then we have [C2]

h(T) = -d(d - 1) InE + 0(1)

and

as E ---7 O.

§3. Mean Free Path

Recall that the mean free path

f =L7(x)dv (3.1)

relates the entropies of the map T and the flow t by (1.1). It is interestingthat the mean free path can be exactly computed in terms of the geometriccharacteristics of the Lorentz gas:

_ IQI ·lsd-

11

7 = 18QI . IBd-ll (3.2)

Here IQI is the d-dimensional volume of the domain Q available to the movingparticle, 18QI is the (d - I)-dimensional area of the boundary of Q. Also,

(3.3)

is the (d - Ll-dimensional volume of the unit sphere in md. Here f(x) is the

gamma function , f(n + 1) = n!, I'(z + 1) = xf(x) , and f(I/2) = V7i. Lastly,

(3.4)

is the volume of the unit ball in lRd- l .

It is also interesting that the expression (3.2) holds for any billiard system,in any dimension. In particular, for planar billiard tables, d = 2, we have

_ 7r IQI7= 18QI (3.5)

132 N. Chernov

and for 3-D billiard tables we have

_ 4 1QIT = 18QI (3.6)

The formul a (3.2) follows by a simple calculat ion involving the invariantmeasure f1 of the flow t and the invariant measure v of the map T , see [C3].

The formulas (3.2)-(3.6) are known in integral geomet ry and geomet ric probability, see, e.g., Eq. (4-3-4) in [MI. Eq. (3.5) is often referred to as Santalo'sformula, since it is given in Santalo's book [Sa].

Next we discuss the free path in the system of hard balls. It is relat ed tothe mean intercollision tim e, one of the basic characteristics of the gases of hardballs.

Consider a gas of hard balls in ffi.k of diameter CJ and uni t mass, meannumbe r densit y n (the average numb er of balls per unit volume) and the meantemperature T . T he temperature is related to the mean kinetic energy by theclassical formulas

and

(3.7)

Instead of the mean numbe r density, one can use the mean "volume density"(the fraction of volume occupied by the balls):

p = IB kl • CJk n

In physics, th e classical Boltzmann mean free tim e formulas ICC, EW]:

_ Jr1 /2 CJ 1tSoltz(k = 2) = 8E1/2 P = 2CJ nVJrkBT

and

_ Jr 1/2 CJ

tSoltz(k = 3) = 8 (6E)l /2 P1

(3.8)

give the mean free t ime l between successive collisions for each ball , on theaverage. The Boltzmann formulas hold in the so called dilute mode (or gasmode) when n -+ O. For larger densities (dense mode, or fluid mode), there isclassica l Enskog's correcti on to the Boltzmann formula, which we give only inth e 2-D case:

_ Jr 1/2 CJtEnskog(k = 2) = 8 E1 /2 PX

1

2CJ n X V JrkBT(3.9)

see, e.g., [Gas], where X is the Enskog scaling factor

X ~ 1 + 0.782 · 2p + 0.5327 · (2p)2

see, e.g., ICL] .

It is remarkable that the Boltzmann equat ion can be derived mathemat icallyfrom th e billiard free path formula (3.2). Thi s was done in [C3] in the followingsetup.

Consider a system of N hard balls of diameter a and unit mass in thek-dimensional torus T 1whose linear dimension is L > O. The k-dimensionalvolume of the torus']' 1is i) , The number density is n = N / tr ,and the "volumedensity" is p = IBklakn.

The balls move freely and collide with each other elast ically. Let qi,1, ... , qi,kand Pi,1 , ' " ,Pi,k be the coordin at es of the position and velocity vector, respectively, of the ith ball. T he configurat ion space Q of the system is a subset of thekN-dimensional torus T 1N

, which correspond to all feasible (nonoverlapping)positions of the balls. T he total kinetic energy of the system is preserved in tim e,and we fix it : pi ,1 + ...+ P7v,k = 2EN, where the const ant E > 0 is the mean

kinet ic energy per particle. The phase space is th en M = Q X S~N-1 whereS~N-l is the (kN - I)-dimensional sphere of radius (2EN)1 /2.

The dynami cs of the hard balls with elast ic collisions correspond to thebilliard dynamics in the configurat ion space Q with specular reflection s at theboundary 8Q . The billiard par ticle in Q will move at the speed (2EN)1 /2 ratherth an the convent ional unit speed. The boundary 8Q consists of N (N - 1)/2cylindrical surfaces corresponding to the pairwise collisions of the balls. Denoteby Ci,j the open solid cylinder corresponding to overlapping positions of theballs i =J j . It is given by the inequality

k

2:)qi,r - qj,r)2 < a2 (mod L)r=1

The configurat ion space is th en Q = T 1N\ UihCi, j , and its boundary is 8Q =

Q n (Uih8Ci,j) 'In order to esti mate the mean free path by using Eq. (3.2) one needs to

compute the volume IQI of the space Q and the surface area 18QIof its boundary8Q . This is a difficult problem, very hard to solve exact ly, since the cylindersCi,j have plenty of pairwi se and multiple intersections. However, one can findthe asymptotic values of both IQI and 18QI at very low densities, as n -+ O.

Some simple calculations [C3] yield

IQI = LkN(1 - 0(1))

N(N - 1)18QI = 2 ·18C1,21· (1 - 0(1))

A little tri ckier is the est imation of 18C1.2I. Cert ain geomet ric considerationsIC3] yield

134

This gives the following:

N. Chernov

N - 1 2k kp kN18QI = J2 . - a- . L (1+ 0(1 ))

Now, according to (3.2), the mean free path of the billiard par ticle in the domainQ is

fIQI · lskN- I I · (kN - 1)

18QI·ISkN-21

J2 a(kN - 1) ·I SkN- 112kkp(N _ 1) .I SkN-21 . (1+ 0(1 )) (3.10)

f o =

Now comes a somewhat surprizing observation. Fir st , the billiard systemin Q is not ergodic. Indeed, the tot al momentum P = (PI , . . . , Pk ) , whereP; = L iPi,r, is invariant under the dynamics. Those phase trajectories whosetotal momentum P is large will display slow relative motion of the balls , andthus the mean free path between reflections in 8Q along such trajectories will belarger than f in (3.10). On the contrary, the mean free path along trajectorieswith zero or small P will be below f . The value of f in (3.10) only gives thephase space average of the mean free path s taken over individual t rajectories.

Physically interestin g regime is the one at equilibrium, where the total momentum is zero, P = O. Let fo denote the mean free path on the surface P = 0in the phase space. A lit tle more computat ion [C3] gives

J2a (kN - k - 1) . ISkN-k- 112k kp (N _ 1) . ISkN-k-2 1 . (1 + 0(1))

V'h a . r ( kN-;k+I)2kp .r (kN-;k+2) ·(1+0(1))

One can 't ranslate ' thi s result into the physically sensible mean free t ime [as follows. The speed of the billiard par ticle in Q is (2EN) I/2, and so the meanintercollision time (in the whole system) is [ SYS = fo (2EN)-1/2. The meanintercollision time for every individual par ticle is simply [par = [sys . N/2 , sinceevery collision involves two particles. Thi s gives

JrI /2 . r ( kN-k+l) . Na[par = kN k ; . (1 + 0(1)) (3 .11)

2k+ 1 . r ( -; + ) . (EN)l /2 P

Now taking the limit in (3.11) as N -+ 00 and using a hand y formular(N )/r(N - 1/2) = 1N(1+0(1)) yields

_ Jr l / 2 atpar(N -+ (0) = 2k+l (Ek/2)l /2 P . (1 +0(1 ))

In par ticular , for k = 2 and k = 3 we recover the Boltzmann mean free t ime forhard disks and hard balls (3.7) and (3.8).

§4. Entropy of Infinite Gases

135

Estimation of the entropy and Lyapunov exponents for systems of hard balls hasbeen always difficult , on both numerical and theoretical levels. Relatively littleis proved rigorously, and the issue is still pretty much open . For recent est imatesbased on kineti c theory and numeri cal experiments we refer the reader to thesurvey [BZDJ .

One interestin g theoretical esti mate of the entropy for a system of two harddisks on a torus was proved by Wojtkowski in 1988 IW] . He showed that as th edisks are so large that they always nearly contact each other the entropy of th eflow approaches infinity.

Here we concent ra te on the entropy of infinite systems of particles. We describe three rar e mathematically proven theorems in thi s direction. First weneed to describe basic facts about infinite systems. We avoid some technicalitieshere, a complete account of the issue may be found in [SCI .

Infinite particle systems. We will consider infinitely many particles in lRd

interacting via a pair potential U(llq - q'll) where q,q' E IRd are the centersof the interacting particles. The potenti al U has hard core, i.e. U(r) = 00 for0 < r ~ ro and finite range, i.e. U(r) == 0 for r ?: rI . Here 0 < ro ~ rI are someparameters. If ro < rI , then for ro < r < rI the pot enti al U(r) must satisfycert ain conditions of regularity and smoothness [SCI . Th is model somewhatgeneralizes the model of hard balls, which corresponds to the case ro = rI .

T he configuration space of an infinite system consists of countable subsetsQoo c IRd such that Ilq - q'll ?: 2ro for every q -I q' E Qoo . The phase spaceMoo consists of pairs X = (Q00, p 00 ) where Q00 is a configurat ion and P00 is aIRd-valued function on Qoo . The value p = p(q) for q E Qoo is the momentumof the particle at q.

The definition of dynamics on Moo is not a trivial task. For systems withpot ent ial, one might run into unsolvable probl ems of integrating infinitely manycoupled differenti al equat ions. Even for hard balls, some weird development smay occur . For exampl e, the system may "collapse" when infinitely many ballswith arbitrary large velocities are coming down "from infinity" into a finite domain of IRd, where they experience infinitely many collisions on a finite intervalof tim e.

The formal definition of dynamics requires a special const ruct ion. Let Vn bea sequence of increasing cubes in lRd with a common center and parall el facessuch tha t u, Vn = lRd. For every X = (Poo ,Qoo) E Moo and each Vrt we definea special dynamics cI>k (X) as follows. We freeze the particles out side Vn andtho se whose hard core intersects 8Vn . Hence, only the particles x = (q, v) with

can move. We regard the boundary 8Vn as consist ing of rigid walls at which themoving particles inside Vn bounce off elasti cally, as hard balls of radius ro.

136 N. Chernov

Observe th at since the potenti al U has a finite range, th e moving par ticlesinside Vn only feel the influence of finitely many frozen particles outside Vn .

Therefore, the dynamics te (X) is well defined for every X E Moo . The flowk is called a partial flown (or partial dynamics) in the cube Vn. For eachx = (q, v ) E X denote by xn (t ) th e t ra jectory of x in the par tial dynamics tr

n•

Key assumption. Assume that for each x E X and s > 0 there is an no =no(x, s) such that the t ra jectory xn(t ) for ItI < s is the same for all n > no.

If the Key Assumption holds, then the tra jectory x(t) of every particle x E Xfor all t E lR is well defined by simply taking the limit of xn(t) as n ---+ 00 .

We denote by t the resultin g dynamics on the part of Moo where the KeyAssumption is satisfied.

Gibbs measures. Next , we define a family of the so called Gibbs measures fl>. ,eon the phase space Moo . Consider again the sequence of cubes Vn ---+ lRd

, and ineach Vn a finite system of Nn particles with the total energy En. Assume thatthe walls of the container Vn are rigid again, so that the particles in Vn bounceoff 8Vn elastically. The dynam ics in Vn preserves the total energy En and theLiouville measure fln on the surface of constant energy (this measure is calledthe microcanonical distribution) .

Consider a sequence of such finite systems so that Nn/Vn ---+ >. > 0 andEn/Nn ---+ e > 0 as n ---+ 00 . The parameters>. and e characterize the meannumb er density and the mean kinetic energy per particle, respectively. The limitas n ---+ 00 is called a thermodynamic limit . The weak limit of the sequence ofmeasures fln (if one exists) is a measure fl>. ,e on Moo called the Gibb s measure.

Theorem 4.1 (Sinai [S2]). If the potential U satisfies certain regularity andsmoothness assumptions and the mean number density>. is low enough, then(a) the Gibbs measure fl>. ,e exists;(b) the set of phase points X E Moo satisfying the Key Assumption has [ullfl>. ,e-measure, i.e. the dynamics is p'>. ,e-almost everywhere defined;(c) the measure fl>. ,e is preserved under the dynamics t . It is also preservedunder the d-dimensional group of space translations.

Inaddition, the Gibbs measures fl>. ,e are invariant under the partial dynamicsk for each cube Vn ·

The proof of the theorem is based on the const ruct ion of the so called clusterdynamics. Let r > 1'1 ' For any configura tion Qoo consider the union of balls ofradius r cente red at all the points q E Qoo . A connecte d component of th atunion is called an r -cluster. Sinai proved th at with fl>. ,e-probabili ty one, eachparticle x = (q,p) belongs in a finite r-cluster that does not interact with anyother cluster during a certain interval of tim e. Of course, withi n a finite clusterthe dynamics is well defined. This observation allows the const ruct ion of thedynamics in the ent ire system.

Entropy Valu es and Ent ropy Bounds 137

For infinite syste ms of hard balls the above existence theorem was provedby Alexander [AI], without restrictions on the density A. In this case the Gibbsmeasure f-L>. ,e can be characterized more explicitly:

1. For f-L>. ,e-almost every configuration Qoo the condit ional distribution onthe space of momenta p EP00 is a direct product of Gaussian distributions withdensity

where (3 = d/(2 e).2. The marginal distribution on the space of configurat ions is a d-dimensional

Poisson measure with density A. Thi s means th at for any bounded set B C ]Rd

the numb er of points q E Qoo n B is a Poisson random variable with parameterA·Vol(B) .

The parameter (3 is relat ed to the temperature T by (3 = (kBT)-l , where k»is Boltzmann 's constant . The temperature is then relat ed to the mean kineticenergy by e = (d/2)kBT.

Space-time translation group. Consider the d-dimens ional group SU , u E

lRd, of space translations on Moo . The translati on Su shift s all the part icles by

the vector u and leaves their momenta unchanged. Space translations commutewith the dynamics t and together they genera te a (d+ I)-dimensional abeliangroup r t,u = t 0 SU on Moo of space-time translations.

Th e Gibbs measure f-L>. ,e is invariant und er the group rt ,u. Denote byh>. ,e(rt ,U) the measure-theoretic entropy of the group r-- with respect to themeasure /-lA ,e' For the definition and basic prop erties of the entropy of multidimensional groups of measure-preserving transformat ions see Conze [Col. Onecan consider h>. ,e(rt ,U) as the natural entropy characterist ic of the Gibbs measure f-LA ,e, it is called the space-time entropy.

Th e following est imate for the space-time entropy was proved by Chernov.

Theorem 4.2 ([CI]) . Assume that the density A is low enough, i.e. A <AO(e) for some AO(e) > 0 (the system is in a dilute mode). Then the space-timeentropy h>. ,e(rt ,U) is finit e and satisfies the following estimate:

h>. ,e(rt ,U) < A' const (e)

The proof is based on Sinai 's construct ion of clust er dynamics.We note that this theorem does not ensure that h>. ,e(rt,U) > 0, even though

thi s seems very likely. For now, this remains an open problem.

Lyapunov spectrum. The second theorem due to Sinai deals with th e Lyapunov spectrum of infinite systems of particles. In order to state the result weneed to describe an algorithm for computation of Lyapunov exponents for finitesystems.

138 N. Chernov

Let M = MV,N be the phase space of a system of N par t icles in a cube Vwith rigid walls (with no rest rictions on the energy so far). Let Tx(M) denotethe tangent space to M at a point X E M. Denote the dynamics on M by \lit,.It generates the family of Jacobi maps (derivatives of \li t,)

Denote by Tlk)(M) the kth exterior power of Tll )(M). It is the space of all

exte rior products e l 1\ ez 1\ . . . 1\ ek where e, E Til )(M) . The Jacobi maps JJrgenet are the maps

Now let us fix the total energy E . Then the dynamics \lit, rest ricted tothe energy surface MV,N,E C MV,N preserves the Liouville measure J-LV,N,E(the microcanoni cal distribut ion). Thi s measur e has m = 2dN - 1 Lyapunovexponents, which we write down in t he decreasing order

A version of an idea of Benet tin et al. [BGGS] implies that

k

lim ! r In tr JJr(k) [JJr(k)]* dJ-Lv N E(X ) = 2I>1N) (4.1)

H oo t }M ' ,V, N ,E ;= 1

Here * denot es the adjoint transform ation.Now, consider the t hermodynamic limit as N --+ 00 , N IV --+ A > 0 and

E IN --+ e > O. The measure J-LV,N,E weakly converges to the Gibbs measureJ-L>. ,e on M oo · We would like to characterize the Lyapunov spectrum of the Gibb smeasure J-L>. ,e by a function cp(p) = CP>. ,e (P) for 0 < P < 2d such th at

I. (N) ( )im X[ N] = cp PN-+oo P

(4.2)

(here [PN] is the integral part of pN) . The function ip would describ e the dist ribution of Lyapunov exponents in many particle systems. Obviously, cp(p) mustbe a decreasing function. Of course, the above formula (4.2) is just a conjectureat present .

In terms of the function cp(p), we can st ate another conjecture :

1 [pN ] p

N L X1N) -+ h(p) :=1cp(s) ds

; = 1 0

(4.3)

as N --+ 00. We note that since cp(p) is a decreasing function , h(p) must bea concave funct ion.

Substituting (4.1) , we can rewrite (4.3) as

139

lim N1

lim ~ r In trJ1:([pN]) (J1:([pN]))* d/.Lv,N,E(X) = 2h(p) (4.4)N--+ oo t --+oo t } MV, N ,E

Instead of proving (4.4) as such, Sinai argues as follows. It is not really finitesystems that are physically interesting, but rather an infinite system of particles.So, the thermodynamic limit N -7 00 should be taken first , and then the timelimit t -7 00. This would better fit the concept of a Lyapunov spectrum of theGibbs measure j1.>-. ,e. So, Sinai changes the order in which the limits are t akenand conject ures that

lim ~ lim N1 r In trJ1:([pN]) (J1:([pN]))* dj1.v,N,E(X) = 2h(p) (4.5)

i -e-ee t N --+00 } M V , N . E

Furthermore, since we now take the limit N -7 00 first , we can as well replacethe finite dynamics 1lJt,r in V by the partial dynamics t,r in the same cub e V asdefined earlier. Accordingly, the maps J1:(k) must be defined in terms of t,r ,and X be a point in Moo . This is yet another ste p closer to working directlywith an infinit e system .

Theorem 4.3 (Sinai [S5]). Let j1.>-. ,e be a Gibbs measure on Moo . Assumethat the density A is low enough and the temperature (i.e. the mean energy e) ishigh enough. Then for every t > a and j1.>-. ,e-almost every point X E M oo thereexists

lim A ~ IV In tr J1: ([PN]) (J1: ([PN])) * = 2ht (p)v--+IRd

• 0

where ht(p) is independent of X. Furthermore, there exists

. 1lim -ht(p) = h(p)

t --+oo t

The function h(p) is continuous and concave.

The proof of the theorem is based on the cluster dynamics const ructed inthe earlier paper by Sinai [S21 .

The ent ire function h(p) can be regarded as an entropy-like characteristi c ofthe Gibbs measure j1.>-. ,e. But particularly importan t is its maximum

hm ax = maxh(p)p

Note th at hm ax = h(po) where Po is select ed so that <.p(Po) = O. Hence, hm ax

corresponds to "t he sum of all posi tive Lyapunov exponents". The followinggeneralization of Pesin 's identity (1.4) was also proposed by Sinai [S5] :

140 N. Chernov

Conjecture 4.4 (Sinai's conjecture ). The value hm ax coincides with thespace-time entropy of the Gibbs measure {.L>-. ,e:

hm ax = h>-. ,e(rt ,U)

Another intri guing quest ion is the asymptotic behavior of the largest Lyapunov exponent Xl as V -+ IRd

, either for the finite dynamics 'l1~ or the par t ialdynamics ~ . Note that the value of the function <p(p) at p = 0 only givesa lower bound for Xl . Sinai remarks in [S5] tha t the largest exponent Xl eit herremains bound ed or grows slowly (e.g., logari thmically) with the volume of V .Numerical est imates of Xl indicate a very slow growth, but do not rule out theboundedness of Xl '

Entropy per particle. Now we turn to the third, earlier theorem by Sinai andChernov [sq. It deals with anot her entropy-like characteristic of an infinite gasof hard balls.

Let {.L>-. ,e be a Gibbs measure on Moo . Pick a sequence of cubes Vr, witha common center and parallel faces and once again consider the partial dynamicsk. Note that under k the balls inside Vn move freely, collide with each otherand bounce off the walls of th e cube Vn and the frozen balls sti cking out of thewalls (those balls act like bumps).

For each n th e partial dynamics ~n preserve the Gibbs measure {.L>-. ,e' It isnot ergodic, though , for the numb er of moving balls N, their total energy E , andthe positions of the frozen balls are all the obvious integrals of motion. Fixing N,E and the positions of the frozen balls intersecting the walls (the balls outsideVn can be ignored altogether) yields a finite hard ball syste m in a cont ainer,though with somewhat peculiar bound ary. T he boundary is composed of the flatfaces of Vn and the fragments of spherical surfaces of the frozen balls st ickingout .

Let h>-. , e(~J be the entropy of the flow k with respect to the measure{.L>-. ,e. It can be computed with the help of (1.10) as follows. For every cube Vr,and phase point X E Moo let Bx,vn be the operator defined in Sect ion 1 forthe trajectory of the phase point X und er the dynamics k. Of course , onlythe coordinat es and moment a of the moving balls in Vn are included in theconstruct ion of B x,vn' the frozen balls are either a par t of the boundary orignored complete ly (if outside of Vn ) . Int egrating the equation (1.10) over thephase space Moo gives

(4.6)

Clearly, the entropy h>-. ,e increases as the cube Vn grows, because more andmore moving balls are capt ured in the cube Vn. We are interested in the entropyper unit volume

1 tVolVn h>-. ,e(vJ

The related quantity (,\ VolVn)-lh>-. ,e(Vn) can be called the entropy per particle.

Theorem 4.5 ([SC]). Let the cubes Vn have sides i; = 2nLo, where t.; > 0is a constant. Assume that the density is low enough, i.e. A < AO (e) for someAo(e) > 0 (the syst em is in a dilute mode). Then there is an h = h(A, e) > 0such that(a) We have

lim ~ r II TT hA e(~) = hn ----+ oo vO vn ' n

(b) For J.1A ,e-almost every phase point X E Moo

. Ihm U I TT tr Bx ,v = h

n --+oo vO Yn

A weaker version of thi s theorem was obtained by Sinai in 1978 [S3], wherehe proved that liminf(Vol Vn) - lhA,e(~J > o.

Sinai and Chernov conject ured th at the quantity h = h(A, e) actually coincides with the space-time ent ropy:

If this is true, it would imply that hA,e(rt ,U) > 0 solving the open problem statedaft er Theorem 4.2. If this is not true, th en hA,e can be regarded as yet anotherentropy-like characteristic of the Gibbs measure J.1A ,e. It would be interesting tofurther investigate its properties, in particular its asymptotics as A -1 O.

References

[Ab] L.M. Abramov , On the entropy of a flow, Dokl. Akad . Nauk SSSR 128 (1959),873-875.

IAI] R. Alexander, Time evolution for infinit ely many hard spheres, Commun.Math. Phys. 49 (1976) ,217-232.

IB] P.R. Baldw in, The billiard algorit hm and KS entr opy, J . Phys. A 24 (1991) ,L941-L947.

IBZD] H. van Beijeren , R. van Zon, and J . R. Dorfman, Kinetic theory estim at esfor the Kolmogorov-Sinai entropy, and th e largets Lyapunov exponent fordilute hardsphere gases and for dilute random Lorent z gases . A survey inthi s volume.

[BGGS] G. Benettin , L. Galgani , A. Giorgilli , and J .-M. Strelcyn , Lyapunov characteri stic exponents for smooth dyn amical systems and for Hamiltonian systems; A method for computing all of them. Pa rt 1: Theory. Meccanica 15(1980), 9-20.

[BD] J .-P. Bouchaud and P. Le Doussal, Numerical study of a d-dimensional periodi c Lorentz gas with universal properti es, J . Statist . Phys . 41 (1985),225-248 .

142

[Bu]ICC]

[CELS]

[CL]

[C11

[C2]

[C3]

[C4]

[CY]

[Co]

[DP]

[EW]

[FOKI

[Gal]

[Gas]

[KS]

[LBD]

[LW]

[ME]

[M]

N. Chernov

L. A. Bunimovich, Existence of transport coefficients, a survey in this volum e.S. Chapman and T.G. Cowling, The mathematical theory of nonuniformgases, Cambridge U. Press, 1970.N. t. Chernov, G. L. Eyink, J. L. Lebowitz, Ya. G. Sinai , Steady-state electrical conduction in the periodic Lorentz gas , Commun. Math. Phys. 154(1993) , 569-601.N.r. Chernov and J .L. Lebowitz, Stationary nonequilibrium states in boundary driven Hamiltonian systems: shear flow, J . Statist . Phys. 86 (1997) ,953-990.N. Chernov, Space-time entropy of infinite classical systems, In: Mathematical Problems of Statistical Mechanics and Dynamics, 125-137. Math. App!.(Soviet Ser .) , 6 , Reidel , Dordrecht-Boston, Mass. , 1986.N.r. Chernov, A new proof of Sinai 's formula for entropy of hyperbolic billiards. Its application to Lorentz gas and stadium, Funet. Ana!. App!. 25(1991) , 204-219.N. Chernov, Entropy, Lyapunov exponents and mean-free path for billiards.J . Statist . Phys. 88 (1997) , 1-29.N. I. Chernov, Sinai billiards under small external forces, submit t ed. Themanuscript is available at www.math.uab.edu /chernov/pubs.htmlN. Chernov and L.-S. Young, Decay of correlations for Lorentz gases andhard balls, a survey in this volume.J. P. Conze, Entropie d 'un Groupe abelien de Transform ations, Z. Wahrsch .Verw. Geb. , 25 (1972) , 11-30.Ch . Dellago and H.A. Posch, Lyapunov spectrum and the conjugate pairingrule for a thermostat ted random Lorentz gas: numerical simulations, Phys.Rev . Lett. 78 (1997) , 211.J .J . Erpenbeck and W .W. Wood, Molecular-dynamics calculations of th evelocity-autocorrelation function . Methods, hard-disk results, Phys. Rev. A26 (1982) ,1648-1675.B. Friedman, Y. Oono and I. Kubo, Universal behavior of Sinai billiardsystems in th e small-scatterer limit, Phys. Rev. Lett. 52 (1984) , 709-712.G. Galperin, On systems of locally interacting and repelling particles movingin space, Trudy MMO 43 (1981) ,142-196.D. Gass, Enskog theory for a rigid disk fluid , J . Chern . Phys. 54 (1971) ,1898-1902.A. Katok and J .-M. Strelcyn, Invari ant manifolds, entropy and billiards;smooth maps with singularities, Leet . Notes Math., 1222, Springer, NewYork, 1986.A. Latz, H. van Beijeren, and J .R. Dorfman , Lyapunov spectrum and theconjugate pairing rule for a th ermostatted random Lorentz gas: kinetic th eory, Phys. Rev. Lett . 78 (1997) , 207.C. Liverani, M. Wojtkowski , Ergodicity in Hamiltonian systems, Dyn amicsReported 4 (1995) , 130-202.N. Martin and J . England, Mathematical theory of entropy, Encyc!. Math.Its App!. 12 , Addison-Wesley, Reading Mass ., 1981.G. Matheron, Random sets and integral geometry, J . Wiley & Sons , NewYork, 1975.


[P] Ya.B. Pesin, Charac terist ic Lyapunov exponents and smoot h ergodic theory,Russ . Math. Surv. 32 (1977), 55-11 4.

[Sa] L.A. Santal6, Integral geome tr y and geometric probability, Addison-WesleyPub!. Co., Reading, Mass ., 1976.

[SI ] Ya.G . Sinai, Dynami cal systems with elastic reflect ions. Ergodic propertiesof dispersing billia rds, Russ . Math. Surv. 25 (1970), 137-189.

[S2] Ya.G . Sinai , The const ruct ion of cluster dynami cs for dynami cal systems ofstat ist ical mechanic s, Vest . Moscow Univ. 1 (1974) , 152- 158.

[S3] Ya.G. Sinai , Entropy per particle for th e syste m of hard sph eres, HarvardUniv. Preprint , 1978.

[S4] Ya.G. Sinai, Development of Krylov 's ideas , Afterwa rds to N.S. Krylov,Works on the found ation s of stat ist ical physics, Princeton Univ. Press, 1979,239-281.

[S51 Va. G. Sinai, A remark concerning the thermodynamic al limit of th e Lyapunov spectrum, Int ern . J . Bifurc. Chaos 6 (1996) , 1137-1142.

[SC] Ya .G. Sinai and N.!. Chernov, Entropy of a gas of hard spheres with respect to the group of space-t ime translations, Trudy Seminara Pe trov skogo 8(1982) ,218-238. English t ranslation in: Dynamical Systems, Ed . by Va. Sinai,Adv . Series in Nonlin ear Dyn amics, Vo!. 1, 1991.

[VI L. N. Vasers tein , On systems of particles with finite range and/ or repul siveinteraction, Commun. Math . Phys. 69 (1979) , 31- 56.

[WI M.P. Wojtkowski , Measure theoretic entropy of the system of hard spheres ,Ergod . Th . Dynam . Sys. 8 (1988) , 133-1 53.

IY] L.-S. Young, Ergodic th eory of chaot ic dynamic al systems, XII th Int ern at ional Congress of Mathematical Physics (ICMP'97) (Brisbane) , 131-143,Internat. Press, Cambridge, MA, 1999.

Existence of Transport Coefficients!

L. A. Bunimovich

Contents

§1. Int roduc tion .§2. Lorent z Gas .§3. Transport Coefficients for Periodic Fluids§4. Viscosity for Two Hard Disks . . . . . . .§5. Some Facts from the Theory of Billiards§6. General Outline of the Proofs§7. Concluding RemarksReferences . . . . . . . . . . . . . .

146148153156162165176177

Abstract. We discuss the rigorous results on the existence of transport coefficients in the (deterministic) dynamical systems. Th is fundamental problem ofthe nonequilibrium statist ical mechanics has been so far solved only for somemodels of the Lorentz gas and for some systems of hard spheres. Th e naturalhierarchy of models based on a number of moving particles in the system allowsto determine the simplest models where the corresponding transpor t coefficientsmay exist. It also suggests a st rategy for the fut ure studies in thi s area .

1 This work has been partially support ed by th e Nation al Science Foundat ion GrantDMS-9970215.

146 L. A. Bunimovich

§1. Introduction

One of th e fundamental prob lems in nonequilibrium statistical mechanics is toderive the laws of macroscopic motion from the laws of microscopic motion (e.g.,from Newton 's equation). Macroscopic dynamics is essentially generated by theconservat ion laws obeyed by microscopic dynamics. Conservation laws forceda system to ret urn to a local equilibrium state if a relative ly small fluctuat ionoccurred. Transport coefficients, loosely speaking, characte rize the rates of theseprocesses of approaching an equilibrium state in the system after it was (slight ly)perturbed. Each transport coefficient corresponds to some conservat ion law ofthe microscopic dynamics.

There are three classical (mechanical) conservat ion laws for a system of interacting particles governed by Newton's equat ions. Those are the laws of conservation of mass, of a momentum and of an energy. T he corresponding transportcoefficients are called coeffic ients of diffusion, of viscosity and of thermal conductivi ty. In nonequilibrium statistical mechanics there are several approaches todefine transport coefficients (see e.g., [Ba], [ML [Spl). The clearest and least ambiguous approach is via Green-Kubo formula. Th ese formulas express transportcoefficients via time integrals of some functions on the phase space of corresponding dynamical system. The physical meaning of these phase functions isthat they are currents of the corresponding conserved quantities (mass , moment a, energy, etc .). Transport coefficients enter into the (time-noninvertible)equations of macrodynamics as coefficients. Therefore, a proof of existence ofnondegenerate transport coefficients is a necessary part of the der ivation of (tim enoninvertible) macrodynam ics from (tim e invertible) microdynamics.

Green-Kubo formulas allow to reduce the probl em of existence of transportcoefficients to the proof that time correlat ions of some phase functions decaysufficiently fast , so that the corresponding time integrals do exist . The importantfeature is that one must prove it for some concrete phase functions rather thanfor "typical" ones. This requires the development of some new techniqu e to showthat some concrete function is a "typical" one.

The machinery to prove such resul ts for hyperbolic dyn amical systems withsingularities has been developed in [BS2L where a Markov partition with infinite number of elements has been constructed for two-dimensional dispersing billiards. It allowed to derive a diffusion equation for a macrodynam ics ofa two-dimensional periodic Lorentz gas with bounded free path from its microdynamics [BS3]. This result was the first where time-non-invertible (hydro dynamic) equat ion has been derived for a determinist ic syste m. In the courseof this derivation it was also proven the existence of nondegenerat e diffusioncoefficient for two-dimensional periodic Lorentz gas with bounded free path. Itwas also natural to expect that thi s new powerful machinery would allow toprove for some models the exist ence of another transport coefficients. However ,it did not happen. The reason for that , as it has been seen at that time, wastoo complicated machinery for the construction of Markov partition in [BS2].

Existence of Transport Coefficients 147

Ten years later this machinery of construction of some Markov approximations of the dynamics of hyperbolic system with singulari ties has been essenti allysimplified in [BSCl], [BSC21. Chernov [Cli has developed this machinery fartherand generalized the results of [BS3] to a periodic Lorentz gas in any dimension(with bounded free path).

Another important achievement was the proof of Ohm 's law for a periodictwo-dimensional Lorent z gas in a weak elect ric field [CELS]. It is well knownthat weak perturbations do not destroy hyperbolicity and good stat istical propert ies of an unp erturbed system survive them. However , even a weak electricfield does accelerate a (now charged) moving particle and therefore its energyis growing with time. To account for this nonphysical and noninteresting situation, it has been introduced in [CELSI a Gaussian thermostat along theway discussed in several papers of this volume. The resulting syst em becamenon-Hamiltonian but ret ained conservation of energy and kept its hyperboli cproperties.

Still there were no models of mechanical systems where one would be ableto prove the existence of other, than diffusion, t ransport coefficients.

To resolve this probl em it was crucial to realize that the Lorentz gas is thesimplest model that one could imagine, where a diffusion may exist . In fact,a mass must move to have a diffusion of mass. Therefore, at least one particlemust move. So it is exact ly a Lorent z gas where this happ ens.

Hence, it was very natural that the first results on the existence of diffusionwere proved for a Lorentz gas. It is just the simplest model where diffusion mayexist .

With thi s idea, the hierarchy of mechanical models, which provides e.g., forthe simplest models where viscosities and thermal conduct ivity may exist , hasbeen considered in [BSp].

This hierarchy is formed by periodic N disk fluids. Mathematically it is anunfolding of a gas of N hard disks on a torus. The simplest system on this hierarchy, i.e., N = 2, obviously, provides for a simplest model where viscosity mayexist. In fact , a par ticle needs at least one other particle to exchange momentumwith. The existence of shear and bulk viscosities for a periodic two disk fluidhas been proven in [BSp] und er the condition th at a fluid should be sufficient lydense, which (exactl y!) corresponds to the boundedness of free path in a periodic Lorent z gas. Certainly the models in thi s hierarchy are well known. Thereis Boltzmann hypotheses on their ergodicity. These models are also in the heartof a molecular dynamics [H] where one of the main problems is to compute thereal values of transport coefficients .

The approach developed in [BSp], [B2] allowed to determine (see also [HI) thesimplest models where some transport coefficients may exist and also to relat ethe problem of existence of transport coefficients with the types of singularit ies ofcorresponding systems. In other words, "how nonuniform" could be hyperbolicityin a system where some transport coefficient st ill may exist . It allows to outlinethe clear program of future studies in this area .


For instance, a periodic three disk fluid is the simplest model where a nondegenerat e coefficient of heat condu ctivity may exist. (Th is fact was known inmolecular dynamics [H] .)

These models may seem too naive (too few particles) from th e physical pointof view. However , a periodic fluid does have an ensemble of states, which allowthe general machinery to go on, even though that this ensemble has measurezero, i.e., the corresponding states are "nontypical" for a nonperiodic fluid. Moreover, the studies of periodic fluids form the essential par t of molecular dynamics[HI, IEWI·

In this paper , we discuss the corresponding results. Mostly we give the outlines of the proofs referring the reader to the original papers . However , an effortwas made to make the ideas and some techniques of the proofs und erstandableto nonspecialist s.

The pap er is organized as follows. In Sect . 2 we consider the Lorentz gas.Sect . 3 deals with the general (formal) derivat ion of transport coefficient s forperiodic fluids . Viscosities for two hard disks periodic fluid are discussed inSect . 4. The next Sect. 5 contains some basic and necessary facts from the theoryof billiards. The proofs are outlined in Sect . 6. The last Sect . 7 is devoted topossible directions of the future studies in thi s area .

§2. Lorentz Gas

Lorent z gas is a dynamical system generated by the free motion in d-dimensionalEuclid ean space ]Rd, d 2: 2 of a point particle in an array of infinit ely heavy(immovable) particles which are called scat terers. A moving particle interactswith scatterers via the law of elast ic reflections . Therefore a modulus of thevelocity is conserved and we set Ilvll = 1.

Thus, Lorentz gas is a model of a two-component fluid where two types ofparticles, moving and fixed, are present. (Lorentz considered scat terers to bed-dimensional spheres. Now it is usually assumed that fixed particles in theLorent z gas are strictly convex with smooth (of class c- , k 2: 3) boundarieswhose sectio nal curvature is uniform ly bounded away from 0 and 00.)

We assume that the scatterers are situa ted in space periodically and do notintersect each other. The first assumpt ion has been made because the highlyinteresting case of Lorent z gas with randomly distributed scatterers is (far) outof reach of the existing mathematical techniques. Indeed, the st udies of theLorent z gas with random distribution of scatte rers lead to a model of a randomwalk in random environments which is much more complicated than the oneswhich are currently under attack in this theory. The second assumpt ion will bedropped in the next section where we discuss periodic one-component fluids.

A free path of a moving particle is defined as the length of its passagebetween two consecut ive reflections. There are two classes of periodic Lorentzgases which may have quit e different properties.

Existence of Transport Coefficients

o~

oa)

b)

Figure 1: Periodic Loren tz gas with an infinite a) and a finite b) horizon .

149

Definition 2.1 A Lorentz gas has finit e (infinit e) horizon if the length ofa free path of a mo ving particle is bounded (unbounded).

By proj ectin g the particle tra jectory of the periodic Lorent z gas to a suitabletorus ']['or d one gets a dynamical system with a compact phase space. This phasespace M = Q X Sd-l is a product of the unit (d - l.j-dimensional sphere, thespace of velocities, and of a configurational space Q, which is the torus ']['or d witha finite number of scatterers removed from it. The projection of the motion ofa point particle to Q generates a dynamic al system with continuous time (flow)on M , which we denote by {st} . It is called a billiard flow , or simply a billiard. Itpreserves the Liouville measure dp. = cJ1-dq d'l3 , where dq and d'13 are the Lebesguemeasures in Q and Sd-l , respect ively, and cJ1- is a normalizing factor .

A billiard system is one of a few, which have a global Poincare section. Itallows to represent a billiard flow as a flow unde r a function (special flow) (seee.g., [CSF]). To perform this reduction one should follow a t rajectory only at themoment s of reflections of the particle from the boundary. To be more precise,let us consider a cross section of the phase space M = {x = (q,v) E M, q EoQ, (v,n(q)) 2 O}, where n(q) is the inward unit norm al vector to oQ at q, and


(" .) is the inner product. Thus M consists of unit vectors with base points atthe boundary oQ and pointing inside Q.

Let x E M . We denote by r (x ) the first posit ive moment of t ime when thetra jecto ry of x reflects from the boundary. We call r (x ) a free path of a par ticleat the point x. The first retu rn map T : M -+ M is defined via T x = S T(X)+ ° X.

T he map T generates a dynamical system with discrete t ime (a billiard map).It preserves the measure du, which is the projection of the Liouville measur e dJ.1onto M . Both the map T and the flow {st} are known to be ergodic, mixing,K- and Bernoulli systems [SI], IGO]. However, in order to prove the existence oftransport coefficients for the Lorentz gas one needs to study much more delicateproperties of this dynamical system.

The general st rategy of the proof of the existence and nondegeneracy oftransport coefficients for a dynamical system is the following one. At first , wemust restrict ourselves to a suitable class of functions on a phase space (observabies). Then one should show that these functio ns enjoy sufficiently fast rat eof t ime correlat ions decay. By sufficient ly fast we mean here an integrab le (fora system with cont inuous t ime) or a summable (for a system with discrete time)rat e of time correlat ions decay. T hen one should prove for these phase funct ionsa cent ral limit theorem.

T he observables from this class must have some continuity (smoot hness)properties. Indeed, it is well known (see e.g., ICC]) that t ime correlat ions forsome ("fast ly oscillat ing") phase functions may decay very slowly even for theuniformly hyperbolic dynamical systems (i.e., for the syst ems with the bestpossible stochast ic prop erties). In par t icular the corresponding time correlat ions("tails") could be non-integrable.

It turns out that even for some phase funct ions from the chosen class a cent ra llimit theorem may fail. Th ose are the so-called functions homological to zero.T he reason for that is that a sum of covariances of values of these funct ionsalong the trajectories of a dynamical system tends to zero. (T his fact is wellknown in the general theory of limit theorems for stationary rand om processes[ILl·)

Therefore, on top of the standard technique of est imat ing a rat e of tim e correlations decay and proving a cent ral limit theorem, in order to show the existenceand nondegeneracy of t ranspor t coefficients one must verify that some specificphase funct ions which define the transport coefficients (rather than generic ones)are nonhomological to zero and thus sat isfy to a cent ral limit theorem. T hesefunctions are, in fact , the currents of the conserved under the dynamics quant it ies (see below). In the out line of the proofs we will concentra te on this problembecause the general techniques for proving of ergodicity, mixing, a cent ral limittheorem and for esti mat ion of a rate of corre lat ions decay have been discussedin ot her papers in this volume.

Now we proceed with the statement of the main theorems for the period icLorentz gas with a finite horizon. To do so we need first to specify a classof phase funct ions under st udy. The custo mary choice is to consider piecewise

Holder continuous functions . We recall that f(x) is a Holder continuous functionon M if If(x) - f(y)1 :::; C(f) llx - yllQ for some a > 0, which is called a Holderexponent, and a const ant C(f) . (If a > 1 then f(x) is constant.) A functionf(x) is piecewise Holder continuous if M can be partitioned into a finite numberof subdomains separated by a finite number of smooth hypersurfaces , and f(x)is Holder continuous on each element of this partition. For inst ance, a free pathT(X) is piecewise Holder continuous function .

As has already been mentioned, each transport coefficient corresponds tosome conservation law in a system. There is the only one transport coefficient fora Lorentz gas. It is a diffusion coefficient which corresponds to the conservationof mass. Indeed, a momentum is not conserved in the Lorent z gas because thescatterers do not move. Besides, the conservation of energy is equivalent tothe conservat ion of mass because in this system there is no redistribution ofenergy between particles. Thus the only transport coefficient is the coefficientof diffusion D which, according to Einstein formula , can be expressed as

2 roo r 2 rooD = d Jo J

M(v(x(O)),v(x(t)))df-t(x(O))dt = dJo ((v(O),v(t)) )dt (2.1)

where v(x(s)) is the velocity of the particle at the moment s, x(s) = (q(s), v(s))and (-, .) denotes the expectation with respect to the invariant measure f-t . Inthe relation (2.1) we assume that a mass m of the particle equals one.

Observe that th e diffusion coefficient D in (2.1) is proportional to the integralof tim e autocorrelation function of the mass current T(s) = mv( x(s)) = mv(s) .Einstein formula (2.1) is the first one in the hierarchy of Green-Kubo formulasfor transport coefficients (see Sect . 3).

T heorem 2.2 Let f(x) be a piecewise Holder continuous function with zeromean, i.e., (I) = O. Then the variance

00

(2.2)n = - oo

exists, i.e., a 2 < 00. If moreover a -I- 0 then the sequence

f( x) + f(Tx) + ... + f(Tn -lx)(a2n)1/2

(2.3)

converges in distribution to the standard normal probability distribution as n -+00 .

It is the general result of the theory of limit theorems for st ationary random processes th at the sum (2.2) equals zero if and only if a function f( x)is homological to zero, i.e., there exists a function g E L2(M, v) such thatf( x) = g(Tx) - g(x) .


T heorem 2.2 was proven in [BS3] for d = 2. Chernov proved it for an arbitrary(finite) dimension [C11 .

The next theorem deals with the existence of nondegenerat e diffusion coefficient for the periodic Lorentz gas with finit e horizon. One can st udy t imeautocorrelat ions of a displacement of the par ticle instead of its velocity, becausethe part icle keeps a constant velocity between the reflections from the scat terers.Displacement is easier to deal with and it is somewha t more visible.

First , we lift the dynamics back up to the Euclidean space jRd from a subsetQ of the torus 1l'0rd . T hus the particle moves in an infinite periodic array ofscatterers in jRd. One can assume that the init ial posit ion q(O) of the par ticlebelongs to the unit cube which contains an origin. Let q(t ) be the vector ofcoordina tes of the par ticle at t ime t and qn the point of its nth reflection fromthe scatterers.

The major (and only) assumpt ion that brings nonreversibility into the problem is that the initi al position (q(O) or qo ) and th e init ial velocity v(O) are chosenaccording to some absolutely cont inuous (with respect to J-L or to 1I) measure.

Consider the cont inuous and discrete displacement vectors

respectively.

A(t) = q(t ) - q(O )q Jt and

A qn - qoqn = - - -

y'n(2.4)

Theorem 2.3 The vectors ij(t ) and ijn both converge in distribution to ddimensional nondegenerate normal probability distributions with zero mean.

The covariance mat rices of these limit ing normal distributions are, in fact ,the diffusion mat rices for the Lorentz gas. T hey can be expressed by th e GreenKubo formula as (2.1) and

001

Qdisc = 2(T(X)) L ((ql - qof 0 (qn+ l - qn)),n = - oo

(2.5)

respectively, where 0 stays for a tensor product of d-dimensional vectors andx1I' is the transposit ion of a vector x.

The existence of a nondegenerat e diffusion matri x for a periodi c Lorent zgas with a finite horizon suggests that the dynamics of this system should begoverned at large space and t ime scales by the diffusion equation.

The next theorem proves this fact . To make a transit ion from micro- tomacro-dynamics one needs some space-t ime rescaling. Let t > O. We rescaletra jecto ry of the par t icle as qt(s ) = q(st )/ Jt , where 0 :::; s :::; 1. T he measureJ-L induces the probability distr ibut ion J-L t on the set of all possible trajecto riesqt(s), 0 :::; s :::; 1. Each t rajectory qt(s), 0 :::; s :::; 1, is a broken line in jRd . Henceqt(s) can be considered as a continuous mapping from the unit interval [0, 1] tod-dimensional Euclidean space. In other words each trajectory qt(s), 1 :::; s :::; 0

of the Lorentz gas can be represented as a point in the space qO,I) (lRd ) ofcontinuous vector functions on [0,1] which assume their values in lRd .

Theorem 2.4 The measures J.1t converge weakly to a Wiener measure.

Theorem 2.4 provides the desired derivation of the diffusion equation for theperiodic Lorentz gas with finite horizon because the fundam ent al solution ofthis equation is a transition function for a Wiener process .

The Theorems 2.2-2.4 were proven in [BS3] for d = 2. Chernov IC1] provedthese results for any d < 00 . Th e Green-Kubo formula (2.5) for discret e t imealso has been proven in IBS3! (for d = 2) and in [C1] (for any d < (0) . TheGreen-Kubo formula for continuous tim e was proven very recently [C3] .

Th e technique to prove Theorem 2.2 is rather general and has been developed for nonuniformly hyperboli c systems of general nature. (This technique isdiscussed in the various papers in this volume.) The same is true for the proof ofexistence of nondegenerat e transport coefficients for the dynamical systems ofinteracting particles. All these proofs so far were based on Green-Kubo formulas(even thou gh in statistical physics there are oth er approaches to the derivationof transport coefficients , see e.g., IBa], ISpJ). The next section provides the (formal) derivation of Green-Kubo formulas for the systems with a finite numb erof interacting particl es on (finite dimensional) tori.

§3. Transport Coefficients for Periodic Fluids

Consider an infinitely extended one-component fluid in thermal equilibrium.In d dimensions the fluid has five locally conserved fields: the particle densityp(O)(x ,t) , the d compon ents of the momentum density p(i)( x ,t) , i = 1,2, .. . ,d,and the energy densit y p(d+I) (x, t), which depend on location x E lRd and timet E R They are distributions (generalized functions) on phase space indexed byx , t.

The local conservation laws in a fluid are given via the relations

(3.1)

i = 0,1 , . .. , d + 1, where j (i ) are the corresponding local currents. Since the interaction between particles has some range, the local currents are not uniqu elydefined. However the space averaged currents always are (see below). The average with respect to the equilibrium thermal distribution of the fluid is definedby (.). Then the Green=-Kubo formula for the t ransport coeffi cients have theform

r~~) = 2k~TJdtJdxo ... dxd((jii)(X,t)j~)(O,O)) - (jii ) (O, O))(j~) (O , O)) )(3.2)


where Q , (3= 1, 2, . . . , d; i , j = 0, 1, . . . , d + 1, k» is the Boltzm ann constant andT is the temperature of the fluid. To derive (3.2) one uses the stationarity of theequilibrium dist ribut ion in space and t ime. To simplify the not ations we assumed = 3 thus dealing with the case which is the most important from the point ofphysics.

Then the relations (3.2) define 225 = (15 x 15) coefficients . However, thanksto the rot at ion invariance and to the time reversal symmet ry of the local (microscopic) dynamics all but three transport coefficients defined by (3.2) vanish.These th ree coefficients can be expressed by a linear combination of shear andbulk viscosity and by th e thermal conductivity ISp].

We consider a part it ion ~ of jRd into the hypercubes and assume that a fluidis periodically repeated over all jRd with N part icles of mass m in an element of~ . Let the particles interact via a central force - V'V.

Denote by qj and Pj coordin at es on the d-dimensional torus A = [0,£]d andmomentum of the jth par ticle, j = 1,2 , . . . , N, respectively. T he pair force ofinteraction can be written as f(q) = - L: V'(q - n£), where the summation

n EZd

is taken over all elements of the parti tion ( We consider here th e short rangepotenti al and therefore the sum contains only a single non-zero term .

For all smooth (test) functions f on A the momentum density is defined (indistributional sense) by

N1ddXf (x )p(o)(x , t ) = L f(qj (t ))Pjo(t )A j = l

(3.3)

where a = 1, 2, . .. ,d.The stress tensor To /3 (X, t ) is defined through the moment um conservat ion

law

Making use of the relation F(q) = - F( - q) one gets

Consider now any pair i f:. j of particles. Let A -+ 'Yij (A) E A be an arbitrarysmooth curve with 'Yij (O) = qi, 'Yi j (l ) = qj , if:. j . Then we have

(3.6)

Exist ence of Tr ansport Coefficients 155

One gets the expression for the stress tensor by inserting (3.6) into (3.5) and bycomparing the resulting expression with (3.4). The result reads as

(3.7)

It can be seen from (3.7) that the st ress tensor depends on the choice ofa curve "tij , what should be the case because the interaction between particles isnot local. We need, however, only the total st ress tensor , which is indep endentof the choice for curves "ti j provided "ti j 0 "tji has zero winding number. Usuallythe most useful and easy to compute convent ional choice is to take the shorteststraight line joining qi with qj .

Then

TQ{3(t) = f ddxTQ{3(x , t)

NIl N (3.8)= L m Pj Q(t )Pj {3 (t ) + 2" L (qj(t) - qi(t))QF{3(qj(t) - qi(t)),

j =l i# j= l

where qj - qi is the shortest distance on the torus A.Let us fix the total momentum to be zero and the total energy to be E .

We denote the expectat ions with respect to the corresponding (microcanonical)distribution (measure) by (-)E ,N . The flow on phase space is stationary under( .) E ,N . If the dynamics is mixing and the rat e of mixing is sufficiently high thenone expects that the time integrated stress tensor, normalized by 0 ,

(3.9)

satisfies a central limit theorem.The corresponding variance is formally given by

where a, (3,"t, t5 = 1, . .. , d. Up to conventional factors, V is (if it exists) theviscosity tensor for the periodi c fluid. It should be proportional to the volume

IAI = ed for d ;:: 3. Moreover V Q {3" s/ 2kBTIAI should converge to r~~') in theinfinite volume limit .

It is easy to see th at

(3.11)

since F is centra l force and thus

(3.12)

Now we use that our periodic fluid is invariant under rotations by right angles.Consider a linear operator V act ing on the space of d x d matrices as (VA)"'13 =

L~,8=1 V ", I3 ,, 8A y8. Then for all matrices A and R which rotate by 7r/2 we have

(3.13)

(3.14)

It follows from (3.12) and (3.3) that the only nonvanishing entries of the viscositytensor V are '0"'13 ,"'13 = '013"',"'13 and '0", ,,, ,1313' Furthermore

'0"'13 ,"'13 = '0 , 8,,8 ,V ", ,,, ,,,,,,, = '01313 ,1313

'0", ,,, ,1313 = '0",88 if ex =I- (3, 'Y =I- <5

The relations (3.11) and (3.14) imply that in fact the viscosity tensor D containsonly three nonvanishing ent ries. Traditionally they are expressed through theshear viscosity

1 1TJ = 1A12k

BT'012,12

and the bulk viscosity

1 lId~ = 1A12k

BTd2 L D", ,,, ,1313

"',13=1

The third viscosity coefficient is in fact an art ifact of the periodi c approximation. Indeed, und er full (on contrary to discrete) rotation invar iance (3.13)implies 2'0"'13 ,"'13 = V ", ,,, ,,,, ,,, - '0",,,,,1313 , ex =I- (3, in addit ion to (3.14). Thus TJ and~ are the only viscosity coefficients for nonperiodi c fluid.

§4. Viscosity for Two Hard Disks

We consider in thi s sect ion a periodic fluid with only two particles (N = 2)in a cell [BSp]. It certainly is th e simplest periodic fluid for which a nontrivialstress tensor may exist.

We perform the following canonical transformation for two hard disks syste m

q = ql - q2 ,

1qc = 2" (ql + q2), P = PI - P2,

1Pc = 2" (PI +P2)

(4.1)

Exist ence of Transpor t Coefficients

Then equat ions for the relati ve motion r ead as

157

dm dt q(t ) = p(t ),

ddt p(t ) = 2F(q(t )), (4.2)

where q E [- ~ , ~ ] are coordinates on d-dimensional to rus, i.e., with periodicboundary condit ions. By set t ing t he cente r of mass momentum Pc = 0 we getthe st ress tensor

(4.3)

where a , {3 = 1,2, . . . .d.We need our system to have st rong chaot ic proper ties. In fact, for central

limit th eorem to hold we need sufficient ly high rate of mixing. To ensure it weassume that our particles are hard spheres, i.e., V is a hard core potential. Weass ume also that d = 2.

So the syste m that we are going to analyze is a period ic fluid in the planeconsist ing of two hard disks of diam eter R per uni t cell. These disks interactthrough elast ic collisions .

Let p = mv and t he mass m = 1. To have the relati ve velocity Ivl = 1 weset E = 2~ (pi +p~ ) = 1/ 4. Thus the relat ive velocity is specified by an an gle{) , with 0 ::; {) ::; 21l' . We set e= 1 and t hus have a un it two-t orus . Therefore t heonly param et er left is t he hard core radius, which can be used to cha racterizea density. For inst an ce, t he closed packing density is p' = V3R2

.

It is known from the theory of dispersing billiards that there are two cases,which give rise to dynami cal sys tems wit h qui te different stat ist ical behavior .

Infinite Horizon. If 0 < R < 1/2 (Fig. 2a)) then the fluid par ticl es canente r the next cell. Moreover in this case a par ticl e can move for arbit rarilylong distan ce without experiencing collisions with ot her par ticles. As we willsee soon, this case corresponds to the periodi c Lorentz gas with unboundedfree path (infinite hori zon ). On t he ot her hand, if 1/2 < R < 1/ v'2 t he disksare confined and cannot pass each ot her (F ig. 2b)). The corresponding billiardsystem is confined to a bounded domain which has a "diamond" shape (Fig. 3b)). The closed packing cannot be reached because of the imp osed symmet ry withrespect to rotations to the multiples of 1l'/ 2.

We will discuss first th e infinite hori zon case. The corresponding billiard liveson th e two-dimensional torus [-1/2,1 /2]2 with a boundary formed by a disk ofradius R cente red at the origin (Fig. 3a )). Let tn be the time of nth collisionof the billiard particle with the boundary and v' (tn) be the post collisional,v(tn) = v' (tn- d the precollisional velocity.

Then the t ime integrat ed st ress tensor reads


a)

b)

Figure 2: Periodic two disk fluid with an infinite a) and a finite b) horizon.

where x( ·) is the Heaviside function. The invariant measure I-" of this systemreads dl-" = cll- dq1dq2dfJ where ell- a normalizing factor. We denote by (-) theaverage with respect to 1-".

Then

(4.5)

Here PD is the quantity which is called the dynamical pressure . It coincides withthe thermodynamic pressure in the infinite volume limit [Pl . For the system of

a)

b)

159

Figure 3: Billiard systems corresponding to two disk periodic fluid: a) an infinite horizon, b) a finite horizon.

two disks we havePD = (4(1-1rR2))- 1.

We will simplify now the expression (4.4). Let us approximate a hard corepotential by a very steep but smooth potential which not vanishes only on thedisk of radius R. Consider a function g(y) which is smooth on the unit circle[-1/2,1 /2) and g(y) = y for Iyl < ~ - E. Then

Taj3(t) = ~ (~g(qa(t))Vj3(t)) + ~ (1- g'(qa(t)) Va (t)Vj3(t)) (4.6)

Let ma(n) = sign(va(tn )) x [number of crossings of the line qa = 1/2 between

160 L. A. Buni movich

tn and tn+l, to = 0, tn+1 = 1 if t.; < t < tn+d. Then

(4.7)

To establish an existence of the shear and bulk viscosities one needs to provethe central limit theorem for the function (4.7). It is widely believed though (butnot rigorously proved so far ) tha t the cent ral limit theorem fails for the periodicLorentz gas with infinite horizon [BI, Bu1, BS3, BSC2, C1]. Thi s belief is basedOIl the existence in the phase space of this system of the corr idors formed by theneighborhoods of th e (parallel) families of trajectories which have no reflectionsat all with the bound ary. The velocit ies of the par ticle are st rongly corre late dduring its passage through such corridors.

Finite Horizon. We consider now the case 1/2 < R < 1/12. Let us shiftthe coord inate system by the vector (1/2 ,1 /2) . Then the corresponding onepart icle billiard table is formed by the complement of the union of four disksof radius R cente red at the points (±1 /2 , ±1 /2) . T he corresponding region issometi mes referred to as a "diamond" (Fig. 3b)). One should point out th atits bound ary has singularities. Therefore, although we refer to this case as tothe finite horizon it requires more sophisticated analysis (see Sect. 6) th an theperiod ic Lorentz gas with a finite horizon.

The dynamic pressure [PI in this case is given via the relation [BSp]

PD = - (1- vi4R2 - 1) (4 (1- vi4R 2 - 1- 1rR2 (1 - ~ arccos 2~) ) ) - 1

(4.8)

It is easy to see that PD diverges at R = 1/12 as (4 (1 - 12R)r 1 and decreasesto zero at R = 1/2. (The last is somet imes considered as a precursor of the gassolid transit ion for the infinite hard sphere syste m.)

Again the expression for the dynamic pressure can be simplified by subt racting a bound ed function . One can write

(4.9)

where F is a smoot hened hard core (billiard ) force and the function g is definedas

_ { -1/2g(y)-y = 1/2 ~f y > 0 } = (~ _ 0) ()

If Y < 0 2 Y(4.10)

where 0(·) is Heaviside function. T he funct ion g(y) appears because of the shiftof the origin. To derive (4.9) one uses that q is a smooth funct ion in case ofa finite horizon.

Exis tence of Transport Coefficients 161

To establish the existence of viscosities the central limit theorem has to beproven for the function

1 _Vi Ta{3 ([O, t]) =

~ [~X(tn :S t) (~ - ()) (qa(tn))~(VI(tn) - V(tn)){3 - oa{3 tPv]We introduce the new variables T = (T1' T2 ,T3) as

T1 ([0, t]) = ~ (711 ([0, t]) + 722([0, t])) - Pvt

T2([0, t]) = ~ (711 ([0, t]) - 722([0, t]))1 _

T3([0, t]) = 2T12( [0 , t])

(4.11)

(4.12)

By making use of Ta{3 = T{3a and of (3.11) we obtain that the covariance matrixfor T has the form

(4.13)

It is customary to write physical quantities in the dimensionless form. In particular, it allows to compare the results of experiments conducted in somewhatdifferent conditions or conducted with different but , in a sense, similar systems,e.g., for the syste ms of different number of particles. To do so, let us consideragain N hard (hyper-) spheres with mass m and diameter a on a d-dimensionaltorus of linear dimension £ (see (3.10)) . The kinetic energy, E, of this system isidentified with the temperature by E = N ~ kBT. The physical viscosity tensorV is given via

-1 dNV a{3,,8 = V 4E V a{3 ,,8 ,

where V = fd is the volume of the torus.We define the reduced density by

(4.14)

(4.15)

where Pcp is the density of close packing . It is easy to see that V depends(nontrivially) only on p" and N. By scaling space, tim e and mass we obtain

V a{3"f,(m,E,V,A;N) = (4.16)

a-(d-1)Jm: (~) [(p*)(d-l) /dVa{3"f, (I ,N,V3/ 2dN ,(p*)1/d;N)].

By symmetry, the term in the square brackets depends only on the dimensionlessviscosities 'f/* , C, fj * , which are functions of the reduced density p* and thenumb er N of part icles.

Thus, up to dimensional factor s, 0'1 = ~ , the bulk viscosity, 0'3 = 'f/ , the shearviscosity, and 0'2 = fj which becomes equal to 'f/ for a rot ation ally invar ian t fluid.

We need to prove that the covariance mat rix V is positive. Because V isa diagonal matrix to establish V > 0 it is enough to show that

(4.17)

exists for i = 1,2 , 3, and is stri ctly positive. (The average in (4.17) is taken withrespect to dJ-L = C/-l dqldq2d1'J.)

The following stateme nt [BSp], [B21 establishes the existence of a st rictlypositive viscosity for a periodic two disk fluid.

Theorem 4.1 (Central limit theorem for the stress tens or) For any R , 1/2 <R < 1/V2, there exist (positive) variances a ; : (1/2 , 1/V2) -+ (0 , 00], i = 1,2, 3,such that for any real number z

{I } 1 jZ 2lim J-L fi Tl ([0, t]) < z = ~ due: " /2<7,t--+oo y t Y 2 1l'0' i - 00

(4.18)

In particular, 0 < 'f/ , ~ < 00. For every bounded open set A C ]R3 with boundaryof zero Lebesgue measure one has

lim J-L ({ ~T([O,t] E A}) = rIT du; ~e-u7 /2<7it --+oo y t } A . Y 21l'0' i,=1

(4.19)

This theorem has been first proven in [BSp] for a dense open subset ofthe disk's rad ii R E [1 /2 , 1/ V2]. To prove it for all values of R in the openinterval (1/2 ,1 / V2) we obtain below a subexponential est imate for a numb erof singulari ty curves that can intersect in a point of a phase space of a two-diskperiodic fluid. Th eorem 4.1 establishes the existe nce of a strictly positive shearand bulk viscosities for this model.

§5. Some Facts from the Theory of Billiards

In this section we give the exact definition of billiards and describ e some of theirprop erties needed in what follows. We mostly restri ct ourselves to the case oftwo-dimensional dispersing billiards . It will allow us to simplify the exposit ionand besides to make it more visible.

Let Q be the closure of a bounded connected domain in two-dimensionalEuclidean space ]R2 or on a 2d torus Tor2 with a piecewise C3 boundary. A point


of the boundary 8Q is called singular if it belongs to the intersection of someregular components of 8Q. All other points of 8Q are called regular. If theboundary 8Q is strictly convex inward at every regular point , then the billiardin Q is called dispersing (or Sinai) billiard.

Regular components of the boundary 8Q are the non-self-intersecting curvesf i , 1 :::; i :::; k, which are closed or intersect each other only at their endpoints .

, kWe denote S = Uijij= 1(fin I'j) the singular part of the boundary. In case of

two disk periodic fluid the set 5 consists of four points. Correspondingly 8Q\5is the regular part of the boundary.

At any regular point q E 8Q there exists the unique internal normal vectorn(q). Thus the sign of a curvature k(q) of th e boundary at such point is uniquelydefined. In dispersing billiards the curvature k(q) is positive in all regular pointsof the boundary.

Definition 5.1 A billiard in Q is a dynamical system generated by an uniform motion of a point particle inside Q and with elastic collisions at the boundary 8Q.

A periodic Lorentz gas generates a dispersing billiard on the appropriated-dimensional torus 'll'ord . Moreover , a boundary 8Q in this billiard system isjust the union of boundaries of scatterers and thus 8Q does not contain singularpoints.

A periodic two disk fluid (see Sect . 4) generates a dispersing billiard aswell. However , in case of a finite horizon the boundary 8Q has singular points(Fig . 3b)) . A per iodic N-disk (N 2: 3) fluid generates not a dispersing butrather a semi-dispersing billiard . In semi-dispersing billiards a boundary haszero curvature along some directions.

A billiard defines a piecewise smooth flow {st} on its phase space M =Q X Sl = {x = (q,v) : q E Q, Ilv ll = I}. Then flow {st} preserves the Liouvillemeasure d/l = c!-, dqd'l3 , where dq and d'13 denote the Lebesgue measures on Qand Sl respectively and the constant c!-' is a norm alization. For billiards onecan naturally pass from the dynamical syste m with continuous tim e {st} to thedynamical system with discrete time, where one counts tim e by a number ofcollisions with the boundary.

Let M = {x = (q,v) : q E 8Q, (v,n(q)) > O} where (".) is the standard innerproduct. Denote by M the closure of M in the space M . The boundary 8M =

M\M consists of two parts, i.e., 8M = So U Vo = Ro where So = {(q,v) : q E8Q, (v,n(q)) = O} , which corresponds to the "grazing" collisions with the boundary 8Q, and Vo = {(q,v) : q E 5}, which corresponds to the singular point s ofthe boundary 8Q. We denote by T the map of M induced by the flow {st} .

We introduce in M the coordin at es (1', cp ), where r is the parameter of thelength of an arc on the curve 8Q and ip is the angle between vectors v and n(q) ,- 7[ / 2 :::; ip :::; 7[/2 . In case of two disk periodi c fluid M is, in these coordinates, theunion of four rectangles. The map T preserves the measure dv = c; cos ipdrdsp ,

where c.. is the norm alization. Let T+(X) = T(X) and L(X) be the first positiveand the first negat ive moment of collision of the t rajectory of x E M with theboundary. Thus T ±lx = ST±Cxl+Ox .

T he maps T and T- 1 are piecewise smooth. They have singularities on theset T - 1Ro and T Ro respectively. We denote R i = i- Ro and Rm,n = U~=m n;- 00 :::; m :::; n :::; 00. It is easy to see that a map T" (T-n) , n ~ 1, has singularit ieson R_n,o(Ro,n)' T he set R- oo ,oo consists of a countable numb er of smooth curveswhich we call discont inu ity curves.

We call a smoot h curve r E R- oo,oo a maximal discontinuity curve if thereis no smooth curve I" E R- oo,oo such that I" :) r . It is easy to see that anymaximal discontinuity curve belongs to some set Rm , where m is an integer.

The following lemm a (see [SI], [BSl]) gives the important information onthe st ruct ure of the set of discontinuity curves.

Theorem 5.2 Let a maximal discont inuity curve r belong to R m (R_ m ) ,

m ~ 1. Th en the endpoints or of r belong to the set Ro,m (R_ m,o).

A point x is called a multiple poin t if it belongs to more than one discont inuitycurve . A rank of a multiple point x is defined as min{n : x E R-n,n}'

Consider a curve, C iII of class C 1. A curve, is called in creasing (decreasing) if it satisfies an equation 'P = 'P(r ) such tha t d'P/dr > 0 (d'P/dr < 0). Theproperty to be an increasing (decreasing) cur ve is preserved under th e act ionof the map T (T - 1

) . A curve, is called m- in creasing (m-decreasing) , m ~ 1,if T - m , (T m , ) is an increasing (decreasing) smooth curve. All discontinuitycurves which belong to the set R1,oo (R-oo ,- l ) are th e increasing (decreasing)curves.

Let , be an increasing (decreasing) curve given by the equat ion 'P = 'P(r ).We introduce the p-Iength of , via the formula p(,) = I/' cos ipdr . All increasing(decreasing) curves are stretched (with respect to th eir p-Iength) und er theaction of the map T (T- 1

) . T his property implies hyperboli city of the map T.Hyp erboli city means that for almost every point x E M t here exists two

C1-curves ,Cul(x) and ,Csl (x) such that Tn ll'(s)Cxl (T - nll'(u)C x)) is a smooth

map for any n ~ 1 and limn---+oop(Tn, Csl(x)) = 0 (limn---+oop(T - n, Cu)(x)) = 0).Such curves ,CUl (x) and , Csl (x) are called local unstable and stable manifold ofa point x , respect ively. We will refer to them as to LUM and LSM of a point x .

The coefficients of expansion (contraction) may essent ially vary along LUM(L8M) in a sense that th e ratio of these coefficients at two point s x and y of LUM(L8M) could be unbounded. To avoid this we introduce a homogeneous LUM(L8M) and denote them by HLUM (HLSM) as such pieces ofLUM (L8M) wheresuch ratios can be cont rolled. (See the exact definition in [BSC2], Def. 5.3.) Inwhat follows we always mean homogeneous local manifolds when referring tolocal manifolds.

A parallelogram is a subset U c iII such that for any two points x , y E U theintersect ions ,Csl (x)n,Cul(y) and ,Cul(x)n, Csl(y) each contain exactly one point


and both these points belong to U. Thus a parall elogram is a set with a structureof a direct product. Indeed to obt ain a parallelogram choose a point x E M suchthat , Cs)(x) and ,Cu)(x) exist . Then take all "first" intersections of "sufficientlylong" LUMs and LSMs which inter sect ,Cs) (x) and ,Cu) (x) respectively.

Let A be a measurable subset of M. Denote ,~u)(x) = ,Cu) (x)n A, ,~) (x) =,Cs)(x) n A. We shall call a parallelogram U O-homogeneous if for any point

x E U the set ,~) (x) h~)(x)) is contained in HLUM (HLSM). This notionmeans that the coefficients of expansion (contraction) at different points of suchparallelogram do not differ too st rongly from each other. (Again we refer to[BSCl ,2] for the exact definitions .)

We will call a domain K c M a rectangle if it is bounded by two LUM and bytwo LSM such that the ends of each of these LUMs (LSMs) belong to the pair ofthese LSMs (LUMs). The corresponding LUMs (LSMs) are called u-boundaries(s-boundaries) of K . For any parallelogram U c M one can find the minimalrectangle K(U) :::> U. We will call K(U) the support of parallelogram U. Byu-boundary (s-boundary) of U we mean u-boundary (s-boundary) of K(U).

§6. General Outline of the Proofs

In this section we discuss the genera l ideas and techniques which are behind theproofs of the result s formulat ed in the previous sections.

Th e other papers of thi s volume (see also references therein) provide theanalysis and the proofs of hyperbolicity for a hard spheres gas, and for theLorentz gas as well as the detailed discussion of est imates of the rate of correlat ions decay. Therefore we will emph asize here another aspect of the proofs ofthe existence of nondegenerat e t ransport coefficients .

T he first one has to do with a number N of particles in a fundamental domainof a fluid. New and old singularities appear in a system when a number of movingparticl es increases. It result s in a "greater nonuniformity" of the correspondingnonuni formly hyperbol ic billiard system.

All systems considered in this volume belong to the class of hyperboli c systems with singulari ties. These singularities cut the images of the unst able manifolds which are st ret ched by the dynamics. T he key issue is which one of theseprocesses, stretching or cutting, will prevail, The st retching always occurs withexponent ial rate. Therefore, the crucial question is whether or not the processof cutting can also go at an exponential rat e. If it can do so then whether ornot this exponent is smaller th an the hyperboli c one.

Thi s question was first formulated in [BSC11. It was also shown there (seealso [V]) that in dispersing billiards with smooth boundaries (e.g., in a Lorentzgas) cutting is going on much slower (actually, linear ) pace than st retching does.

Both processes, stretching by hyperbolic ity and cutting by singularities,are local ones. Therefore, the corresponding condit ion that stretching prevails

over cutting must also be formulated locally. (Recall that we discuss only twodimensional ergodic hyperbolic systems with a single positive Lyapunov exponent >.. Therefore, we will formulat e here the corresponding condition only forsuch systems.)

Condition of a Moderate Cutting (CMC). Maximal numb er of smoothcurves in R -m,m, m ;::: 1, which intersect or terminate at any point x E M doesnot exceed g(m) < K5.m for some constants K = K(Q) > 0,0 < 5. < >. where >.is the only positive Lyapunov exponent of a system. The CMC condition dealswith the prop erties of the dynamical system und er study. In our case, it is theshape of a billiard table that defines these prop erties. However, to prove theexistence of nondegenerate transpor t coefficient s it is also needed to show thatthe phase functions which enter into the corresponding formulas satisfy to thecondit ions of Theorem 4.1 (central limit theorem) . Thus one must verify th atthese functions are piecewise Holder cont inuous and are not homological to zero.We will st art with the CMC condit ion and then turn to the conditions on phasefunctions.

A billiard map T in a diamond (F ig. 3b)) has two typ es of singularit ies. Oneappears because of tangencies of t rajectories with a boundary. As the resultinitially close points get reflected from different regular components of 8Q. Thistyp e of singularity is peculiar for dispersing billiards . Another type of singularity stems from the singular points of the boundary 8Q. We will discuss thissingularity in some more details.

Consider a point x = (q,v) such tha t a ray initiated in q with direct ionv has its first intersection with 8Q in its singular point ij (Fig . 4a)) . Then,formally speaking, a point x does not have an image T x . However , in th is caserather two images of x can be constructed (Fig . 4). One image corresponds toa set of t rajectories, with velocit ies parallel to v , which reflect from one of theregular components of 8Q intersect ing at ij , while another image correspondsto the analogous family of parallel t rajectories which get reflected from anotherregular component of 8Q which contains ij. Therefore, it is possible to constructtwo outcoming trajectories from ij which correspond to a single incoming to ijt rajectory. (It is easy to check that only one outcoming traj ectory from a singularpoint ij E 8Q corresponds to an incoming trajectory if the angle between regularcomponents of 8Q inter sect ing at ij equals 3600 Inwhere nis an integer . Actually,in this case two outcoming from ij trajectories coincide.) At the first sight itseems that this process of branching of the singularity curves leads to theirexponential growth which may violat e the condition CMC. However, it is notthe case.

Fir st , we formulat e the statement which shows th at singularity curves ofbounded ranks have rather simple structure in any sufficient ly small neighborhood . The following lemma can be easily "ext racted" from [BS1! (see also [BSp]).

Theorem 6.1 For any multiple point x E M \So and for any integer m > aif rank( x ) ~ m then there exists a neighborhood U(x) such that

Existence of Transport Coeffici ents

a)

b)

Figur e 4: Two images of a point mapped to a singular point of t he boundary.

167

(i) the closure U(x) does not contain multiple point s, besides x itse lf, withranks that do not exceed m.

(ii) there exists an unique discontinuity curve I'+(x) c Rm 1 (x), 0 < ml ::;m (I'_(x) c R _m 2 (x)), 0 < m2 ::; m, max(ml, m2) = m , among alldiscontinuity curves which pass through x, such that I'+(x) (I'_(x)) divid esU(x) into two semi -neighborhoods, U{(x) and Ui(x) (U1(x) and U2-(x)) .

(iii) all passing through x decreasing (increasing) discont inu ity curves, besidesI'+(x) (I'_(x)), in tersect only one of semi-neighborhoods U{(x) or Ui (x)(U1(x) or U;(x)) (Fig. 5).

We return now to singularit ies of the boundary 8Q . Let x E M and itstrajectory has first intersection with 8Q at a singular point g. Consider a smallneighborhood U(x) 3 x. It is easy to see that its image TU(x) becomes "broken"

r-(x)

Figure 5: Local st ruc ture of discontinu ity curves.

(along a curve consist ing of points whose trajectories also have th eir first intersection with 8Q at q) into two semi-neighborhoods. Consider now two points Yland Y2 which have the following properties

(i) a tr ajector y of a point Yl (Y2) has its first intersection with 8Q at q

(ii) one (of two) out coming directions for a (generalized) image TYI coincideswith one (of two) out coming directions for T x , and one (of two) out comingdirections for TY2 coincides with the second outcoming direction of T x(Fig. 6).

It is possible to show [B2] th at for a two-dimensional hyperbolic billiard withsingularit ies of th e boundary 8Q, but without tangent singularities CMC holdsand g(m) < K(Q)m.

Therefore, a number of singularity curves which intersect at any point x E Mdoes not grow faster th an linearly if 8Q does not contain singularit ies or if 8Qdoes not have dispersing components. However , the boundary of a billiard table(diamond) , which corresponds to a periodic two disk fluid has the both.

Even though for each of these types singularit ies g(m) cannot grow fasterthan linearly, these two singular ities can "help" each other to fight hyperbolicity.(We believe th at the following scenar io of "interaction" between these two typesof singularit ies, while being logically possible, in fact cannot occur . However, weare unable to prove it.)

Suppose that a trajector y of a point x E M first becomes tangent toa boundary 8Q at Tk , x and then (at some higher iterat e Tk, +k2 of T) hitsa singular point q of th e boundary. Then after tangency all singularity curvesget moved to a semi-neighborhood of the point Tk , x (Fig. 7). Suppose now th at


a)

""'.r •, ,, .t •

: 't '

yj Ix

b)

Figure 6: "Glueing" of semi-neighbo rhoods of points Yl and Y2 totwo semi-neighborhoods of x .

169

the images (under T k 2) of all these singularity curves occur at the same semineighborhood which corresponds to one (of two) outco ming from Ii directions ofTk 1 +k2x . Let y be such point that 7r(Ty) = Ii and one of two corresponding toTy outcoming from Ii directions coincides with one of the outco ming direct ionsof the point Tkl+k2X. Then, it is logically possible, that all singularity curvesfrom a neighborhood U1 of T kl+k2- 1X and all singularity curves from a neighborhood U2 of y will appear at the same neighborhood of one (of two) images ofT kl +k2- 1x (Fig. 7). Thi s process of glueing semi-neighborhoods, which containall singularity curves from the corresponding (complete) neighborhoods, maylead to an exponent ial growt h of singularity curves which pass through a pointx E M . We will show though that it is not the case in the following Lemma onnon exponential proliferation of singularities.


Theorem 6.2 Let Q be a two-dimens ional billiard region such that DQcon tains at most a fin it e numb er k of singular points and a fin it e num ber ofdispersing componen ts. Th en

g(m ) :S exp(a (Q )m/ logk m) (6.1)

where a = a( Q) is a cons tant.

Proof. We will show that the sequent ial events of tangency with the boundary DQ and then hitting a singularity of DQ cannot occur very often alonga trajectory of any point x EM.

Let x be a periodic point of T , i.e., T 'r« = x for some n > O. We will callx a branching periodic point if r-« E Va for at least one k = 0,1 , . . . , n - 1.Otherwise x is called a nonbranchin g periodic point .

For any (nonnecessary periodic!) point x E M and for any integer n wedenote by # {Tn x } the cardinality of the set {T nx }. It is enough to consider

~T X

. ,

' [·.· . Tx• I· .· ,

Aa)

iTY- 2" ------------

.,.

riTY="2 -----------:-

b)

Figur e 7: a ) Singular ity cur ves in a neighborh ood of x get mapped (under T or T (2) =T oT) to the sa me semi -neighborhood of T (2)X; b) "Glueing" of semi-neighborhoods(after hit t ing singularity of f)Q) in a ph ase space.

Exist ence of Transport Coefficients 171

n > o. Observe, that #{Tnx} can be greater than 1 only if Tk x E Vo for somek = 0,1 , .. . , n - 1. (However, in th is case # {Tnx } st ill can be equal 1 if theangles of intersection of corresponding regular components have a form 1r[m ,where m is an integer.)

Obviously, a numb er of singularity curves passing through x E M may inprinciple grow faster than linearly (see [BSC2]) only if its trajectory branchesi.e., hit the set Vo .

Suppose that T kx E Vo for some k 2: 0 and T£x (j. Vo for some 0 :::; e< k .Then a total number of singularity curves from R_n,o , n > k, which pass throughx equals to the sum of corresponding singularity curves along both branches oftrajectory {T mx} .

We have#{Tnx} :::; #{Tn-Ix} + #({Tn-IX} n Vo).

We call a set of points X,YI E {Tx} , . . . ,Yn E {T nx} a branch of trajectoryx ,T x , .. . ,Tnx if TYk = Yk+1 for all k = 1, . . . , n - 1. It is easy to see thata number of singularity curves from R_n,o passing through x is a sum of numbersof singulari ty curves taken over all branches of trajectory x ,T x , . . . .T":«.

Some branches of t rajec tory {Tmx} could be finite if {T kx } contains a periodic point . All other branches of a branching trajectory are infinite .

The first simple but important fact is that a numb er of singularity curveswhich pass through any nonbranching periodic point z E M grows at mostlinearly. Indeed, let T'tz = z. Then there is a finite number rz of singularitycurves from R_n,o which contain z, Therefore a number of singularity curvesfrom R_np,o which pass through Tnpz does not exceed rzp . Finally, for anym = np +nl , 0 < ni < n , we have that a number of singularity curves from theset R_m,o which contain T'" z = T"? Z does not exceed rz (p + 1).

Let now x E M be a nonperiodic point. Take on the trajectory {Tn x } allsuch segments (n ,n + €) which start with a tangency to 8Q and end with theclosest to n moment n +£ > n of hitting of some singular point of 8Q. We shallcall such segments the T S-segment s.

The crucial fact is that such T S-segments cannot occur along the trajectoryof a point x very often unless x is a periodic point. But for a periodic branch wehave a linear est imate for a number of curves from R_n,o passing through it.

Let N(n) be a number of (nonp eriodic) orbits of length n that go througha tangency to 8Q from a singular point of 8Q to a (possibly the same) singularpoint of 8Q. It is not difficult to see that

N(n) < b(Q)kn (6.2)

for some constant b(Q) > O.Indeed, let us consider all orbits that are originat ed from some singular

point q E 8Q. Such beam gets par titioned into regular beams by trajectories which meet at the next reflection singulari ties of the boundary. Therefore,a numb er of different orbi ts of length n which go through the shortest possi-


ble TS-segments cannot grow exponentially fast . Clearly it could be not morethan (2k)k TS-segments of length 2 on any nonperiodic orbit. In fact, such orbit must be periodic otherwise because it contains at least two times the sameT S-segment of length 2.

Hence, the lengths of T S-segments must grow along any nonperiodic branchor periodic branches appear along a trajectory but their periods must grow if#{Tnx} --+ 00. Furthermore, obviously a number of possible TS-segments of

n-+oo

length m cannot grow faster than b(Q)km , where b(Q) > 0 is some constant. Thisfact together with (6.2) gives (6.1). It is easy to see that the "worst" possible caseis when along a trajectory we have first all shortest (of length 2) TS-segments,then all TS-segments of length 3 etc . Subexponential estimate (6.1) then followsfrom the fact that a number of T S-segments of a given length m cannot growfaster than exponentially (in m).

Corollary 6.1. The condition of moderate cutting (CMC) holds for a periodic two disk fluid.

We now turn to the proof that phase functions in the expression for thestress tensor (4.11) (4.12) are not homological to zero.

The machinery to prove that some phase function is not homological to zerohas been developed in [BSC2] (see also [BSp]). Here we describe this machineryand formulate the corresponding statements.

Let F(x) be a function on a phase space of some hyperbolic dynamicalsystem which satisfies to a central limit theorem. We would like to show thatthere is no such phase function H (x) that

F(x) = H(Tx) - H(x) (6.3)

First, we assume that on contrary the relation (6.3) holds for some phase function. Then we find such periodic trajectory Zo,T Zo, t2 Zo , ... , Tn-l Zo, T" Zo = Zo,that

n -l

L F(TkzO) =J: O.k=O

(6.4)

As the next step we choose a sufficiently small O-homogeneous parallelogramU« ::1 Zo which is also "sufficient ly dense" in its support K(Uo).

The third step has to do with a function H(x) rather than with a pointzo 0 This function is measurable and integrable. Therefore one can find a 0homogeneous parallelogram U1 , which is sufficiently dense in its support, andsuch that H(x) is "almost" constant on U1 .

The major idea of the proof is to show that there is a contradiction between(6.4) and the existence of a parallelogram U1 constructed on the third step.

The step number four brings together a periodic point Zo and U1. At thisstep we construct a "first return" map T1 : Ui ---+ Ui on a subset Ui c U1

of all such points of U1 which return to U1 and besides do visit before that

Exist ence of Transport Coefficients 173

a parall elogram Uo. It can be proven that one can choose U1 so small th at themap T1 would be invertibl e and preserves the measure t/ , (See Lemma 5.7 andits proof in [BSp].)

The final step of the proof is to show that the existence of a "first return"map T1 , which "passes" through a neighborhood of a periodic point Zo , contradicts to the fact th at a function H (x) is "almost constant ' on U1• It essent iallyuses the fact that our billiard system is hyperboli c.

To make the exposition more self-consistent we present here the exact st atements which correspond to each step of the proof. Besides we outl ine the proofof the final step. A reader can find the details in [BSC2] and [BSp].

It is important to mention that, having in hands th e machinery developedin these papers, one does not need to go through all of it aga in in order to provethat some phase function F(x) on a phase space M of a hyperbolic dynamicalsystem is not homological to zero. The only thing which is needed is to finda periodic point Zo E M such that the relation (6.4) holds. Then the resultsof [BSC2] and [BSp] imply th at F(x) is not homological to zero (or, in otherwords, F(x) is not a coboundary) .

We have for i = 1,2 ,3 ,

(6.5)

where the expressions for function ~i (X) are given via (4.11), (4.12).It is easy to see that the periodic point of period two (Fig. 8a)) sat isfies

(6.4), where instead of F (x ) one must plug ~1 (X) and ~3 (X) . The existence ofsuch two-periodic points is obvious.

For ~2 (X ) the sum (6.4) along the trajectory of this periodic point vanishes.Therefore we should pick anot her periodic point (Fig. 8b)). The existence of thisperiod four tra jectory follows from cont inuity. It is not difficult to check thata sum (6.4) for ~2(X) is st rict ly positive.

The next lemma (see [BSC2]) makes the second step in the proof.

Lemma 6.3 Let Zo be a periodic point of a discrete dynamical system generated by a dispersing billiard. For any EO > a there exis ts a a-homogeneousparallelogram Uo :1 Zo such that Zo rt 8K(Uo) and v(Uo )/v(K(Uo)) > 1 - EO·

Certainly this statement holds for any hyperboli c dynamical system whichallow a Markov partition (or a Markov approximation through, e.g., a Markovsieve (see [BSC2], [C2])).

Th e third step of the proof is ensured by the following lemma [BSC2], [BSp].

Lemma 6.4 For any E1 > a one can find a a-h om ogeneous parallelogramU1 such that v(Ud /v(K(Ur) ) > 1 - E1 and some real number h such that

v{ x E U1 : IH(x) - hi > Edv(Ur) < E1

where ins tead of H (x), h one should substit ute ~i (X ) and Oi, i = 1,2 ,3 , respectively.


a)

b)

Figure 8: Periodic t raj ectories of diamond billiard .

Before formulatin g the statement that takes care of the step four of th eproof, we must give the exact definition of the "almost" first return map T1 . Fora point x E U1 we take on its posit ive semi-t rajectory the first point T ":» E

Uo. Then we ta ke the first point Tn+k x on the positive semi-t rajectory of thepoint T":« such that T n+kx E U1 . Moreover , we require that Tix tf- U1 fore = 1,2 , ooo,n - 1.


We define now Tlx = Tn+kx . Hence, the map T, : Ul ---+ Ul is defined onthe subset of all such points of Ul which return to Ul and before that they visitti;

Lem m a 6.5 (See [BSp]). One can choose Ul so small that the map T,U1 ---+ Ul is invertible and preserves the measure u,

The last part of the proof of Theorem 4.1 goes in the following way. First,we define two maps , ¢> and ¢>l, on Us , Let x E Ul and Tnox E Uo be thefirst point on its trajectory which belongs to Uo. We have T'n « = Ts«, wherenl > no. We denote j; = " Cul(Tno+kx ) n "Csl(Tnox) and define ¢>x = T- no-kj;and ¢>lX = Tn,-noj;.

It is easy to see that ¢>x belongs to the same HLUM as x , and ¢>lXbelongsto the same HLSM as Tc x , Moreover, ¢>x and ¢>lX belong to the same trajectory,namely Tn,+k(¢>x) = ¢>lX.

We denote

and

i= 1, 2, 3.

Lemma 6.6 IS' (¢>x ) - S(x) 1;:::: So - 1':2, where 1':2 ---+ 0 as diam Uo ---+ O.

Indeed , the trajectori es of the points x and ¢>x are close dur ing the firstno iterations. Also, the tra jectories of the points T'": » and Tno+k(¢>x) = i: areclose to each other during the first nl - n iterations of T . The piecewise Holdercontinuity of ~i(X), i = 1,2 ,3 implies tha t the corresponding subsums is thesums S, (x) and SH¢>x ) are close to each other.

Therefore, there remain exactly k terms in the sum S:(¢>x) not accountedfor, coming from the images of Tn°( ¢>x) = T -kx . Observe th at these imagesapproximate the per iodic orbit of zoo Thus the corresponding sum of values of~i is close to So, which proves Lemma 6.6. This lemma implies that

IH (Tlx ) - H (x )1 + IH(¢>l X) - H( ¢>x) I ;:::: So - 1':2 (6.6)

It remains to prove that for the majority (with respect to the measure v)of points x E Ul their images ¢>x and ¢>l X coincide in Ul , and that the setsof th ese images, {¢>(x)} and {¢>l (x)} , have relatively large measure (> constv(Ul)) . Then for such points ¢>l (X) = Tl(¢>x) and together with (6.6) it givesa contradiction with the second estimat e in Lemma 6.4. We refer to [BSC2] and[BSp] for further detai ls.

Thus, it is proven that all three nonzero entries of the matrix D are strict lypositive, and therefore shear and bulk viscosities for two disk periodic fluid doexist and do not vanish .

176 L. A . Bunimovich

§7. Concluding Remarks

T he general derivat ion of (formal) expressions for transport coefficients in a periodic N-disk fluid (see Sect . 3) allows to construct the hierarchy of the modelsfor which it is natural to try to prove the existence of nondegenerate t ransportcoefficients. T he models in th is hierarchy are naturally parametrized by a number N of particles in a fundamental domain of such model. Obviously, one needsat least one moving particle to have a diffusion in a system. However , this moving par t icle must have a possibility to change a direction of its moti on. Thusone needs some scatte rers and the Lorent z gas (which is a two component fluid)naturally arises as the first level in this hierarchy.

Again , obviously, it is necessary to have at least two particles to ensurea nontrivial momenta exchange. However, it is already not obvious (but simple)to realize th at one needs at least three particles to ensure a nontrivial exchangeof energy (thermal conduct ivity ).

Indeed, let us follow the same routine as in Sect. 3 to derive the expressionfor a coefficient of thermal conduct ivity. The total energy current is given by

N I l 1 N 1je(t ) = ~ - P·(t )2- p(t) + - ~ (q(t) - qi(t))- P·(t )F(q·(t ) - qi(t))L..J 2m J m J 2 L..J J m J J

j = 1 i # j =1

(7.1)

As for viscosit ies the goal is to prove a cent ral limit theorem for the t imeintegrated current

1 rt

y't io ds(j e(s) - (jE (O)/E,N)

The corresponding covariance mat rix reads as

(7.2)

The invariance under discrete rot ations implies V~~) = k<5o:(3 . Convent ionallythe thermal conduct ivity is defined by

1 1 (E)k = !Ai 2k

BT2 Dll

Suppose that N = 2. If we assume PI +P2 = 0, then

(7.4)

Thus for two particles periodic fluid the thermal conduct ivity vanishes. To havea nontrivial thermal conduct ivity one must consider a periodic fluid with at least

Exist ence of Transpor t Coefficients 177

three par ticles (N ~ 3) per cell. (T his result is well known in computationalstat istical mechanics [H] .)

T herefore, to prove the existence of a nontri vial coefficient of thermal conduct ivity one must est imate a rate of correlat ions decay and to prove the CMCfor the semi-dispersing billiard corresponding to a 3-disk periodic fluid. T henone must prove that the phase function in (7.3) is nonhomological to zero. (Itis easy to see that it is piecewise Holder continuous.)

Such results would complete the program to prove for the simplest mechanical models th e exist ence of (nondegenerate) transport coefficients whichcorresponds to three fundamental conservat ion laws (conservation of mass, momentum and energy) .

Another challenging question is to understand why the formal expressionsfor the st ress tensor for a periodic two disk fluid work very well in a range oflow densiti es (where they are supposed not to work). Indeed, small densiti escorrespond to the case of infinite horizon. However , the results computed according to the formal expressions (4.14) fit very well to the experimental results(see the Appendix in [BSp]). It would be very interest ing to underst and why ithapp ened.

Also, it was shown in the Appendix to [BSp] that the shear viscosity of theN = 2 periodic hard disk fluid compares surprisingly well with the Enskog theory (see e.g., [Ba], [Sp]) and with numerical results for the N = 108, 500, 4000periodic hard sphere fluids IAGW]. Such lit tle N-dependence means that dissipat ion is essent ially local and that to a good approximation small subsystemsare statistically independent. It would be nice to obt ain some rigorous resultson this matter.

References

[AGW] Alder , B. J ., Gass , D. M., Wainwright , T . E.: St udies in molecular dynam icsVIII. The transport coefficients for hardspheres fluid. J . Chern. Phys. 53,3813-3826 (1970).

[Ba] Balescu, R.: Equilibrium and Nonequilibrium Statistical Mechanics. NewYork , J. Wiley and Sons, 1975.

[B1] Bunimovich, L. A.: Decay of correlat ions in dynamical systems with chaoticbehavior. J . Exp. Theor. Phys. 89 , 1452-1471 (1985).

[B2] Bunimovich, L. A.: Singularit ies of billiards and transport coefficients (inpreparation).

[Bl] Bleher , P. M.: Statist ical propert ies of two-dim ensional periodic Lorentz gaswith infini te horizon. J . Stat . Phys. 66 , 315-373 (1992).

[BS11 Bunimovich, L. A., Sinai, Ya. G.: T he fund am ental t heorem in t he t heoryof dispersing billiard s. Mat h. USSR Sb. 19 , 407-423 (1973).

IBS2] Bunimovich, L. A., Sinai, Ya. G.: Markov par t it ion for dispersing billiards .Commun. Mat h. P hys. 78 , 247-280 (1980); Err at um 107, 357- 358 (1986).

178

[BS3]

[BSC1]

[BSC21

[BSp]

[C11

[C2]

[C3]fCC]

[CELS]

lEW]

[GO]

[HI

[IL]

[M]

[PI

[Sll

[S21

[Sp]

[VI

L. A. Bunimovich

Bunimovich, L. A., Sinai, Va. G.: Statistical pro perties of t he Lorentz gaswith periodic configuration of scatterers . Commun . Math. P hys. 78 , 478-497(1981) .Bunimovich, L. A., Sinai , Va. G., Chernov, N. 1.: Mar kov partitions for twodimensional hyperbolic billiards. Russ. Math. Surv. 45, 105-152 (1990) .Bunimovich, L. A., Sinai , Va . G., Chernov, N. 1.: Statistical properties oftwo-di mensional hyp erbolic billiards. Russ . Math. Surv. 46, 47-106 (1991) .Bunimovich, L. A., Spohn, H.: Viscosity for a periodic two disk fluid: anexistence proof. Commun. Math. Phys. 176, 661-680 (1996) .Chernov, N. 1.: Statistical properties of th e periodic Lorent z gas . Multidimensio nal case. J . Stat . Phys. 74, 11-53 (1994) .Chernov, N. 1.: Limit t heorems and Markov approximations for chaotic dynamical systems. Prob. T h. ReI. Fields 101, 321-362 (1995).Chernov, N. 1.: Private communication.Crawford , J . D., Cary, 1. R. : Decay of correlations in dynamical systems.P hys . 6D , 223-232 (1983).Chernov, N. 1., Eyink, G. L., Lebowitz , J . L., Sinai , Va. G.: St eady-stateelectrical cond uction in the period ic Lorentz gas . Commun. Math. Phys.154, 569-601 (1993).Erpenbeck, .J. .J., Wood, V. V.: Molecular dyn am ics techniques for hard coresyste ms. In: Berne, B. J . (ed.) Statistical Mechanics. Modern Theoretic alChemistry, Vol. 6., New York: P lenum P ress, 1977, pp . 1-40.Ga llavotti , G., Ornstein, D.: Billiards and Bernoulli schemes. Commun .Math . Phys. 38 , 83- 101 (1975)Hoover , W . G.: Molecular Dynamics. Lect. Notes in P hys. 258 , Heidelberg,Springer-Verlag , 1986.Ibr agimov, 1. A., Linnik, Yu. V.: Indep endent and Stationary Seque nces ofRandom Variables. Groningen: Wolters , Noordhoff, 1971.McLennan, .J. A.: Introduction to Nonequilibrium Statistical Mechanics.Englewood Cliffs: Prentice Hall, 1989.Presutti, E. : A mathematical definit ion of the thermodynamic pressure. J .Stat . Phys. 13, 301-314 (1975) .Sinai , Va . G.: Dynamical syst ems wit h elastic reflections. Ergodic propertiesof dispersion billiards. Russ. Math. Surv. 25 , 137-189 (1970) .Sinai, Va. G.: Ergodic properties of the Lorentz gas . Funct . Anal. Appl. 13,46-59 (1979) .Spohn, H.: Large Scale Dynamics of Interact ing Particles. Berl in, Heide lberg, New York: Springer-Verlag, 1991.Young , L.-S.: Statistical propert ies of systems wit h some hyperbolicity including certain billiards. Ann . Math . 147, 585-650 (1998) .

Interacting Particles

C. Liverani!

Contents

§1. Int roduction . ..§2. A General Setting

2.1 Flows .2.2 Maps .

§3. How to Establish Hyperbolicity3.1 Cone Families . . . . . . .

§4. How to Establish Ergodicity . .4.1 Alignment of Singularity Sets4.2 Noncontract ion Property4.3 Sinai-Chernov Ansat z . .4.4 Sufficient Conditions . .

§5. Stronger Statis t ical Properties§6. Interacting Par ticles . . . .

6.1 Two Particles on ']['2 .

6.2 Repelling Potential .6.3 Attr acting Potentials6.4 Smooth or Leinard-Johns Type Pot enti als6.5 Non-Ergodicity . . ..6.6 Billiard s with Gravity.6.7 Magnetic Billiards . . .

§7. Non- Hamiltonian Systems .7.1 External Electri c Fields .7.2 Internal Degrees of Freedom

181182183184186187188189189190190191192192196197198199201205206206207

1 I would like to thank Victor Donn ay and Mad ej Wojtkowsk i for helpfull suggest ionsand acknowledge the suppor t of th e ESF Programm e PRODYN

180 C. Liverani

7.3 Stochast ic Dynamics 208§8. Final Considerations . 210References. . . . . . . . . . . . 211

Abstract. We discuss a manifold of systems th at generalize, in var ious ways,the stand ard billiard s. To be able to present many of the results in a unifiedfashion we start by summarizing a general theory of the st at ist ical propertiesof Hamil tonian systems with discontinuities. Then the actual examples are presented.

Interacting Par ticl es

§1. Introduction

181

(1.1)

The Hard Ball System s (HBS) have been developed as a simplified model for thedynam ics of a gas of molecules. Neverth eless, the interaction among molecules ismuch more complex than elast ic collisions. It is then natural to try to study theergodic prop erties of more genera l situ ations. Here we will review some knownresults in this direction.

Let us start by noticing that , once potentials are introduced, the problem ofergodicity becomes quit e a delicat e one.

Consider n E N points, of mass m i , moving on the dEN dimensional torus']['d and interacting via a smooth pairwise potential. More precisely, if the sizeof the torus is norm alized to be one, let V : [O ,l]d -+ JR, V E C(oo)([O,l]d) ,V(x) = °for all llxli 2: R, R ::; 1. Consider then the Hamiltonian

nIl nH(x,p) = L 2m.llpil12 +"2 L V(Xi - Xj)

i =1' i,j=1

and the associat ed equat ions of motions

{Xi = :::

Pi = - ~j=1 V'V( Xi - Xj ).

The possible obst ruction to ergodicity is best understood at high energiesE . In such a regime it is convenient to rescale the t ime. Let us introduce thenew time T = Eh and the new Hamiltonian

_ n 1 2 1 n

H(x ,p) = L 2m.llpill + 2E L V(Xi - Xj )'i =1 ' . i,j=1

It is easy to check th at the equat ion of motions associated to H(x,p) , the (1.2),are exact ly the ones obtained by (1.1) by setting X(T) = X(T), p(T) = E-~p(T). 2

(1.2)

In other words, the motion with energy E und er the potential V is equivalentto the motion with energy one subject to the potenti al E- 1V .

The above discussion implies that the motion at high energies can be viewedas the free motion perturbed by a very small potenti al. Thi s is exactly thesituation to which KAM theory applies [1]' in particular all the tori, for whichthe entries of the frequency vector (m j 1pi) have "sufficient ly irration al" ratios ,will survive at energies high enough.

2 Now t he dot stand s for the derivative with respect to To

182 C. Liverani

In conclusion, t he motion at high energies is never ergodic.Nevertheless, this is far from being the end of the story. On the one hand,

if th ere is a large numb er of particles then the energy must be ridiculously highfor th e KAM theory to apply. On th e ot her hand, if the potential is not smoothor , possibly more relevantly, if it is not bounded ( presence of an hard- core) thenth e above arguments do not apply (at least not in the above naive version) .

To understand better the sit uation it is necessary to st udy carefully specificexamples. In particular , it seems relevant to consider

1. par ticles interact ing via some potential;

2. par ticles with internal degrees of freedom;

3. non isolated systems;

our attent ion will be concent ra ted mainly on th e first case, for which moreresul t s are available. Nevertheless, we will briefly comment on the oth er twosituations as well (crf. section 7).

Before st arting the discussion of the concrete examples some general considerat ions are called for .

Once we intend to consider more general models than HBS we immediatelysee the need for a generalization of the main properties of HBS to the newset t ing. In par ticular, we need tools to establish hyperbolicity and ergodicityfor the new mod els und er scru t iny. In essence, at least for poin t 1 above, it isnecessary to view HBS as an example of Hamiltonian flows with singularit ies.

Quite a bit of work has been done in this directi on , th e final result beingthat almost all th e instrumental properties and arguments valid for HBS havebeen generalized to the new set t ing. Here I will follow mainly the presentationdeveloped in [82], [87], [50], [14], [57J yet one must mention also [53], [13], [32Jwhere similar work has been done.

Section 2 to 5 are thus devoted to a brief review of some general results concern ing symplect ic dynamical systems with singularities, sect ion 6 and 7 reviewthe various examples that have been investigated , finally sect ion 8 contains a fewbrief remarks on th e relevance of the mentioned result s to statist ical mechan ics.

§2. A General Setting

Although th e syste ms we are originally interested in are flows with singularit ies,many results are more easily st ated in terms of the associated Poincare maps(especially the ones concern ing ergodic properties) . Yet , some t imes it is moreconvenient to work at the level of flows (e.g. while studying the hyperbolicproperties of certain syste ms). Whence the necessity to define smooth dynamicalsyste ms with singularities both in cont inuous and discrete time. Since thi s isa rather vast field and it is possible to consider ext remely genera l cases, we will

Interacting Part icles 183

rest rict ourselves to the minimal generality necessary to deal with the task athand , in particular we will consider only the symplect ic case.

2.1 Flows. Let X be a smooth manifold with a, possibly empty, piecewise smooth boundary ax. We assume that the manifold X is equipped witha symplectic st ructure w.3 Given a smooth funct ion H on X with non vanishingdifferenti al we obtain the non vanishing Hamiltonian vector field F = V'wH onX by imposing w(V'w H,v ) = dH(v) . The vector field F is tangent to the levelsets of the Hamil tonian X C= {z E XIH( z) = c}.

We distinguish in the boundary ax the regular part , aXn consisting of thepoints which do not belong to more th an one smooth piece of th e boundary andwhere the vector field F is transversal to the bound ary and the singular partofaXs which will contain the other points. The regular part of the bound aryis further split into the "outgoing" part , ax_, where the vector field V pointsoutside the manifold X and the "incoming" part, ax+,where the vector field isdirected inside the manifold. Additionally we have a piecewise smooth mappingr : ax_ ---+ ax+, called the collision map. We assume that the mapping rpreserves the Hamil tonian, H 0 r = H , and so it can be restricted to each levelset of the Hamiltonian. In addition we require r to be symplectic."

We assume that all the integral curves of the vector field F that end (orbegin) in the singular part of the bound ary lie in a codimension 1 submanifoldof X .

We can now define a flow ¢l : X ---+ X , called a flow with collisions, which isa concatenation of the cont inuous time dynamics <j>b given by th e vector field V ,and the collision map r . More precisely a traj ectory of the flow with collisions,¢/ (x ), x E X , coincides with the trajectory of the flow <j>b until it gets to theboundary of X at time tc(x) , the collision t ime. If the point on the boundary liesin the singular part then the flow is not defined for tim es t > tc (x) (the trajector y"dies" th ere) . Otherwise the tr ajectory is cont inued at the point r( <j>tcx) untilthe next collision time, i.e., for 0 ::; t ::; t; (r(<j>tc(xlx))

We rest rict the flow with collisions to one level set X Cof the Hamiltonianand we denote the resulting flow again by <j>t . Thi s flow is very likely to be badlydiscontinuous but we requir e that for a fixed t ime t the mapping <j>t is piecewisesmooth, so that the derivative d<j>t is well defined except for a finite union ofcodimension one submanifolds of X".

The symplect ic volume m := I\dw is clearly invariant for the flow, as will bethe measure J-Lc obtained by rest ricting the symplectic volume to the manifoldX c. It is also natural to define the measure J-Lcb obt ained by proj ecting, along the

3 T hat is a non-degenerate closed ant isymmet ric two form.4 We define a flow with collisions to be symplect ic, if for the collision map r rest ricted

to any level set X c of t he Hamiltonian we have r *w = w.

184 C. Liverani

flow direction, the measure J-le on the boundary aXe; such a measure is invariantwith respect to the associated Poincare map . Clearly for such an invariant measure all the trajectories that begin (or end) in the singular part of the boundaryhave measure zero. With respect to the measure J-le the flow q/ is a measurableflow in the sense of the Ergodic Theory and we obt ain a measurable derivati vecocycle dq/ : Txxe ---+ T</hxe. We can define Lyapunov exponents of the flowq/ with respect to the measure J-le, if we assume that

(cf.l55],[64]) .This set t ing suffices to discuss the hyperboli c properties of the flow. To inves

tigat e the ergodic properties it is necessary to int roduce further requirement son the systems. We will do so in the next sect ion by specifying the neededprop erties for a Poincare map .

2.2 Maps. In genera l, the phase space can be the finite union of compactsubset s of symplect ic manifolds, yet for our needs it suffices to consider compactsubsets of the st andard linear symplect ic space W = IRd X IRd equipped witha Riemannian metri c uniformly equivalent to the st and ard Euclidean scalarproduct and which defines the same volume element (measure) J-l .5 The measureJ-l is also equal to the symplect ic volume element . This setting is still too generalto be tractable , we need to put further conditions on the boundary of the phasespace.

By a submanifold of W we mean an embedded submanifold of W . Furtherwe define a piece of a submanifold S to be a compact subset of S which is theclosure of its interior (in the relative topology of the submanifold S) . A pieceX of a subman ifold has a well defined bound ary which we will denote by aX(it is the set of boundary point s with respect to the relative topology of thesubmanifold). Not ice that at every point of a piece of a submanifold, includinga boundary point , we have a well defined tangent subspace.

We will then consider only phase spaces made up of pieces of W which haveregular boundaries in the sense of the following definition.

Definition 2.1 A compact subset X c W is called regular if it is a finit eunion of pieces X i , i = 1, ... , k, of 2d - I -dim ensional submanifolds

The pieces overlap at most on their boundaries, i.e.,

5 Usua lly it is possible to reduce the general situ ation to the present setting by choosing judiciously th e Poincare sect ion and by using Darboux coordina tes, [31 ·

Interacting Par ticles 185

and the boundary 8Xi of each piece X i, i = 1, .. . k , is a fin it e union of compactsubsets of 2d - 2-dim ensional submanifolds.

To picture such set s one can think of the boundary of a 2d-dimensional cube.The faces are pieces of 2d - l-dimensional submanifolds and they clearly overlaponly at their boundaries. The boundary of each face is a union of pieces of 2d - 2dimensional submanifolds (actually it is a un ion of 2d - 2 dimensional cubes).

As a consequence of definition 2.1 the natural measures on the pieces X i , i =1, .. . ,k, of any regul ar subset X can be concocted to give a well defined measur eu.x on X (the 2d - 1 dimensional Riemanni an volume).

Definition 2.2 A compact subset M c W is called a symplectic box if theboundary 8M of M is a regular subset of Wand the int erior intM of M isconnected and dense in M .

We can now formulate th e precise requirements on the phase space of a discont inuous system.

Condition 2.3 (The space). The phase space of the system is a finit e disjoint union of symplectic boxes.

In the smoot h case the map T is a sympl ectomorphism of M .6 In the discontinuous case much more care is needed. We st art by assuming that th e sympl ecticbox M is par titioned in two ways into unions of equal numb er of symplect icboxes

M = M t U · · · U M ;t; = M 1 U · · · U M ;;, .

Two boxes of one partition can overlap at most on their boundaries, i.e.,

M tnM;c8Mt n8M; , i , j =I , ... , rn.

The map T is defined separate ly on each of th e symplectic boxes M t, i =1, .. . .rn. It is a symplectomorphism of the interior of each M t onto the inte riorM j , i = 1, . . . , rn and a homomorphism of M t onto M j , i = 1, . .. , rn . Weassume that th e map T is well behaved near th e boundari es of th e symplect icboxes.

Condition 2.4 (The map) . The second derivative of T and T - 1 increasesat most polynornially while approaching the boundaries 8M±.

Maps sat isfying th e above condi tions fall in the genera l setting of discont inuous maps discussed in [39] . Hence, the normal results of hyperbolic theoryapply.7

6 By "syrnplectomorphism" we mean th at the map is 0(2) and symplecti c.7 In par ticular Oseledec Theorem [551 and, if th e map is hyperbolic, the existence

of th e stable and unstable foliation s and th eir regulari ty properties (in particularabsolute continuity of th e foliations) [391.

186 C. Liverani

Let us introduce the singularity sets S + and S - .

S± = {p E M Ip belongs to at least two of the boxes M ; , i = 1, . .. ,m}.

The plus-singularit y set S + is a closed subset and T is conti nuous on itscomplement . Similarly T- I is cont inuous on the complement of S -. Note thatmost of th e poin ts in the boundary 8M of M do not belong to S - or S +.

We define for n 2: 1

S;; = S + U T - IS + U · · · U T - n+IS +.

andS;; = S - U TS- U · · · U T n- IS - .

We have that T" is continuous on the complement of S; and T - n is cont inuouson the complement of S;; .

Condition 2.5 (Regulari ty of singularity sets ). For every n 2: 1, both S ;and S ;; are regular.

If (M ,T,J-t) satisfies the above conditions we call it a smooth symplecticdynamical system with singularity (in discrete t ime).

§3. How to Establish Hyperbolicity

To establish the hyperbolicity (i.e. , the a.e. posit ivity of all the Lyapunov exponents) of a symplectic multipli cative cocycle A t with respect to a measurableflow (X , cPt,m) or a discrete t ime dynamical system (M ,T ,m) there exists a veryefficient technique mainly due to Wojtkowski [821 .8

Suppose th at there exists two Lagrangian t ransversa l subspaces VI ,V2 C

~2n , th en each vector v E ~2n can be uniquely writ ten as v = VI + V2 , Vi E Vi .We define then th e sector"

We say that a symplect ic mat rix L E Sp(~2n) is monotone if LC(VI , V2 ) CC(VI,V2 ) and that it is strictly monotone if LC(VI , V2 ) c Int(C(VI,V2 ) ) U {O}.

It is interestin g that the first case is equivalent to the matrices for whichQ(Lv) 2: Q( v) while the second is equivalent to Q(Lv ) > Q( v) for all v -=I- O. An

8 The use of cones and quadratic forms is also present in the work of Markarian [521based on [41], [42], [45] . The most general results in such directions, including thenon-symplectic case, are found in [32], while the special properties of the higherdimensional symplectic case are explored in [83]'[49\.

9 A sector is nothing else than a cone with a special shape. Of course, this is apparentonly in higher dimensions where different shapes are possible.

Interact ing Particles 187

important qu an ti ty associated to a monotone mat rix is the minimal amount ofexpansions , for vectors in the cone, measured according to the Q-form. Namely,

adL ) = in fvE int c

Q(Lv)Q(v) . (3.1)

The fund am ental t heorem in this setting is t he following.

Theorem 3.1 (Wojtkowski [82]). Given an eventually strictly mono tonecocyclelO L, : X -+ Sp(lR2n) (Ln : M -+ Sp (lR2n)) with

r sup In IIL t (x )ll/l (dx ) < 00}x t EIO,I]

all the Lyapunov exponents are different from zero almost everywhere.

3.1 Cone Families. We assume that two measurabl e bundles of t ra nsversalLagran gian subspaces are chosen in an open subset U c X (U e M ). We denotet hem by {V1(P)}PEU and {V2(P)}PEU respectively.

Condition 3.2 (Co nes I). We require that almost every point enters eventually in U and that the cone family C(p) = C(V1(p) ,V2(p)) is eventually strictlymonotone, that is, for almost every point p E U there exists a time t (continuous or discrete according to the situation) such that the point is again in U attime t and the image of the cone C(p) is stric tly contained in the cone C(Ttp)(C(eji(p))) .

For a smooth system with singularit ies (M, T , /1) sat isfying condit ions 1 to 4the differential DT gives r ise exactly to a symplect ic cocycle to which Theorem3. 1 can be applied directly.!!

Not so direct is t he application to flows with collisions. A ty pical problem isthat the flow may have conserved qu anti ties (a lways the Hamil tonian , but othermay be present , e.g . t he total momentum), and to each such conserved qu an titywill corres pond two zero Lyapunov exponents. In par ticular , if {Ii}f=1 are dind ependent integrals of moti on in involution.I'' then the Hamiltonian vector

lOByeventually strictly monotone cocycle we mean a multipl icative cocycle for which,for almost every x E X (x E M ) there exists a time t(x ) E IR (t(x) EN), such thatLt(x)(x ) (Ln(x)(x)) is strictly monotone.

11 Notice that it is always possible to make a symplectic change of coordinates thatsends VI(p), V2(P) into two preassigned lagrangian subspaces. Thus, by introducingthe right coordinates, by identifying all the tangent spaces with IR2n and by considering the return map to the set U, the present situation is reduced to the settingof Theorem 3.1.

12That is the Poisson Brackets {Ii ,1j } = 2:~,I=1 ~~ - ~M; = O.

188 c. Liverani

fields V'wIi corres pond to vectors with zero Lyapunov exponents. In such casesto study the ergodic properties of the syste m it is necessary to consider the newphase spaces X {a,} = {~ E X I I i(0 = a;} and verify that the factor systems(X{ a,} , ¢/, m{a,}) are still flows with collisions. The hyp erbolicity can then beinvesti gat ed by studying the behavior of a factor of t he derivative cocycle. Moreprecisely, let us consider the equi valence relation (in each TpX{ a,} ) v rv W iffv - w is a linear combination of {V'w,pI i } . Clearly, t he differenti al of the flowfactors nicely with respect to such an equivalence relation , yield ing the wante dfact or cocycle .

Following [84], to ada pt the present sit uation to the previous setting it suffices to chose at almost every poin t p E X two t ransversal Lagrangian subspacesVI(p),V2 (p) c TpX , with the further property V'w,pI i E VI (p).13

If there are d int egrals of the moti on and X is n dimensional t hen X {a,}is n - d dim ensional and the reduced system must st ill have d zero Lyapunovexponents. Not e that , by construc t ion , Q(v) = Q(w) if v rv w, thus the quadraticform quotients naturally on t he equivalence classes. T hus, if Q(¢/ v ) ~ Q(v) and,for some t > 0, Q(¢/ v) > Q(v) for each vE T X a i , v 1- 0, t hen we can say thatt he map dePt is event ua lly strict ly monotone since the same holds for t he factorof t he derivative cocycle.

In conclusion, if the cocycle determined by the derivat ive of t he flow is eventua lly strictly monotone, then, by Theorem 3.1, t here exists 2(n - d) non-zeroLyapunov exponents .

If a smoot h dyn ami cal systems with singularit ies is hyp erbolic then , by thework of Pesin [58], [591 in the smooth case and of Katok and Strelcyn [39] int he discontinuous case, through almost every point there are local stable andunstabl e manifolds of dimension d and the foliati ons into t hese manifolds areabsolutely cont inuous; moreover such syste ms have, at most , countably manyergodic components.

The sectors C(p) contain the unst able Lagrangian subspaces (tangent tothe unstable mani folds) and t he complementary sectors contain the stable Lagrangian subspaces (tangent to the stable manifolds). The secto rs can be viewedas a priori approximations to the unstable and stable subspaces.

§4. How to Est ablish Ergodicity

To establish t he ergodicity a more detailed knowledge of the system is required .While t he hyp erb olicity is just a measure t heoretical issue, ergodicity is wellknown to depend on some geometric propert ies of t he system as well. It t urnsout to be convenient to state t he needed condit ions in te rms of t he Poincare map .From now on we will require t hat the flows under consideration have a Poincar esect ion that sat isfies all the condit ions we are st ating for discrete syste ms.

I3The property \lw,pIi E V2(p) is fine as well.

Interacting Particles 189

First of all a st rengthening of the condit ions on the cone field is required.

Condition 4.1 (Cones II). Th e lagrangian bundles {V;(p)} described incondition 3.2 are required to be continuous.

Already Burns and Gerb er , in their st udy of geodesic flow 18], noticed theimport ance of cont inuous cone fields for th e st udy of ergodicity. In fact , thecondit ion could probably be relaxed to piecewise continuous by adding somerequirement s on the discontinuities, yet this is not needed for th e examplesst udied so far.

4 .1 Alignment of Singularity Sets. For a codimension one subspacein a linear symplect ic space its characteristic line is, by definition, the skeworthogon al complement (which is a one dimensional subspacej .l"

Condition 4.2 (Proper alignment of S- and S+). We assume that the tangent subspace of S - at any p E S - has the characteristic line contained stric tlyin the sector C(p) and that the tangent subspace of S+ at any p E S+ has thecharacteristic line contained stri ctly in the complementary sector C'(p) . We saythat the singularity sets S- and S+ are properly aligned.

Let us note tha t if a point in S ± belongs to several pieces of submanifolds th enwe requ ire that the tangent subspaces to all of these pieces have characterist iclines in th e interior of th e sector.

4.2 Noncontraction Property. In order to deal with the presence ofdiscontinuities it is necessary to talk of the "hyperbolicity of a finite piece oftrajectory, II since the concept of hyperbolicity is a global one, in general sucha concept has no meaning.l" To make sense of it at least two ways are known.On the one hand it may be possible to find a semi-norm (in tangent space) thatis never decreased. If such a semi-norm has sufficient ly nice properties, th en onecan use it to measure the hyperb olicity (this is the situation for semi-dispersingbilliards and it is discussed at length in other articles of the present volume, see[1 3] for an axioma tizat ion of th is set t ing). On the other hand one can use theQ form to measur e hyperboli city. In such a case it is necessary to have somecontrol on the amount of unwanted finite time effects .

Condition 4.3 (Non Cont ract ion). There is a cons tan t a, 0 < a ::; 1, suchthat for every n :2: 1 and for ever y p E M \ s;t

for ever y vector v in the sector C(p) .

14The followingcondit ions can be weakened [1 31 , yet the weaker conditions turns outto be very hard to check in concrete examples.

I 5The problem here is non-uniform hyperbolicity, if the system is uniformly hyperbolicthe following conditions are automatically satisfied.

190 C. Liverani

This means that vect ors in the unst able direction cannot shrink too muchbefore start ing their asymptotically exponent ial increase. Notably the abovecondit ion holds in all t he known examples.l"

4.3 Sinai-Chernov Ansatz. This is a property pertaining the derivativesof the iterates of T on th e singularity set itself, of T - 1 on S+ and of T onS - . Namely, we requi re th at , for almost every point in S - with respect to themeasur e J.Ls (J.Ls is the 2d - 1 dimensional Riemannian volume on S- US+) , alliterat es of T are differentiable and for almost every point in S+ all iterates ofT - l are differenti able. Moreover ,

Condition 4.4 (Sina i-Chernov Ansatz) . For almost every point p E S with respect to the measure J.Ls , the derivative cocycle is stric tly unbounded.Tha t is (cf. (3.1)) :

Analogous property must hold for S+ and T - 1 .

4.4 Sufficient Conditions. Under condi t ions 1 to 8 the following twotheorems hold .

Theorem 4.5 ([32], [50], Smooth case ). For any n 2: 1 and any p E U suchthat T np E U and a(DpTn) > 1 (i .e., p is stric tly monotone) there is a neighborhood of p which is contained mod 0 in one ergodic component of T .

It follows from thi s theorem that if U is connected and every point in itis st rict ly monotone then ut=~oo T iU belongs to one ergodic component . Sucha theorem was first proven by Burns and Gerb er [8] for flows in dimension3. It was later genera lized by Katok to ar bit rary dimension [31] and then toa non-s ymplect ic framework [32] .

Theorem 4.6 ([50], Discontinuous case ). For any n 2: 1 and for any p EU \ S :; such that Tnp E U and a(DpTn) > 3 there is a neighborhood of p whichis contained in one ergodic component of T .

Results of the same type , bu t under different hypo theses, can be found in[13] and [53].

16Interestingly enough, the less obvious case is constituted by Hard Balls Systems.In fact , through a tangent collision a vector in the unstable direction can shrinkby an arbitrary amount . To overcome this problem it is necessary to use a slightlysmaller cone that can be constructed thanks to a very deep result on the maximalnumber of collisions that can take place before having a fixed amount of time withno collisions [6] .

Interact ing Par ticles 191

Note that the condit ion 0" > 3 of the last t heorem is satisfied for almostall points p E M. Hence the theorem implies in par ticular that all ergodiccomponents are essent ially open. The theorem allows also to go further sincewe assume that only finitely many iterates of T are differenti able at p so tha tTh eorem 4.6 can be applied to orbits th at end up on the singularity sets bothin the future and in the past (e.g. p E S- and T'rp E S+). Though a specificamount of hyperboli city on this finite orbit is needed (O"(DpTn) > 3); note thatin the smooth case any amount of hyperboli city (O"{DpTn ) > 1) is sufficient.

Thi s theorem gives a fairly explicit description of points which can lie inthe bound ary of an ergodic component. By checking th at there are only fewsuch points (e.g. that they form a set of codimension 2) one may be able toconclude tha t a given system is ergodic (see the article on ergodicity of HBS inthis volume for more information on how to tackle such types of problems).

Although the techniques used in the proof make it unavoidable to requiremore hyperbolicity in the non-smooth case, it is not known if such a conditionis really necessary.

Note th at there is no need to formulate Theorem 4.6 separately for a pointp which has only th e backward orbit (p E S+) . One can apply th e theorem toT - np.

§5. Stronger Statistical Properties

It is natural to inquire about st ronger statist ical properties (mixing, K, Bernoulli) .These have been widely investigated as well. Let us ment ion some relevant results .

Theorem 5.1 (139]). If an hyperbolic smooth symplectic dynamical systemwith singulariti es (M ,T,f.l) satisfies conditions 1, 2, 3, it is hyperbolic and ergodic, then M is the union of finitely many sets {Ai}~o 1 such that T Ai = AH 1 ,

TAM - 1 = Ao. In addition T MIAi is a K automorphism .

To prove that the system is indeed K it thus suffices to prove the ergodicityfor all the powers of T.

Theorem 5.2 (114], 157]) . If an hyperbolic sm ooth symplectic dynamical system with singularities (M , T , f.l) satisfies conditions 1, 2, 3, is hyperbolic and K,then it is Bern oulli.

For the case of flows similar th eorems holds

Theorem 5.3 (139], 157]) . An hyperbolic ergodic flow with collisions ix ,« ,m) , with Poin care map satisfying conditions 1, 2, 3, is either a K flow or a Ksystem tim es a rotation.

192 C. Liverani

(6.1)

Th e second alternative means that the system can be viewed as a suspensionwith a constant ceiling. To prove that thi s cannot be the case, and thus th at wehave a K flow, it suffices to check that the st rong stable and st rong unstablefoliat ions are not joint ly integrable. This is generally true in the present contextand depends On the symplect ic (contact ) nature of our models, (see [321, sect ion3, for details).

Theorem 5.4 ([57]). An hyperbolic K flow with collisions, with Poincaremap satisfying conditions 1, 2, 3, is a Bernoulli flow.

Other statist ical prop erties th at can be investigated are, for example, therate of decay of corre lation and the Cent ral Limit Th eorem . For thi s subjectwe refer to the related art icle Decay of correlations for Lorent z Gases and HardBalls, by N.Chernov and L.-S.Young, in the present volume.

§6. Interacting Particles

There are several examples that have been studied. We will present them notnecessarily in historical order and we will explain their prop erties making useof the general results of the previous sect ions that, in many cases, where notyet available when th e original art icles appeared. Nonet heless, for historicalperspective it is important to mention that extremely relevant , although notconclusive, considerations On the ergodicity of particles interacting via a pairpotential can already be found in the seminal work of Krylov [371.

The very first possibility is to study a Sinai billiard in presence of a smallexte rnal field. Thi s is equivalent to st udying a Sinai billiard on a Riemanni anmanifold (where th e metric is the Maup ertuis metric associat ed to the potenti al,131). Such a program has been car ried out in 179], [80], [401 the final result beingthat if the metric is so close to the flat one that the billiard cannot have conjugatepoints, then the syst em is ergodic. T his result , although very interesting andrather general, is not so surprising since it is essent ially a perturbation result(the mechanism that produces ergodicity is exactly the same that operates indispersing billiards with finite horizon). If One wants to consider what happ ensin presence of larger fields, it is necessary to have more geometric structure inthe problem ; this can be justified by t he next very concrete example.

6.1 Two Particles on ']['2. We consider two particles moving on ']['2 andinteracting with a pair potential. The system is described by the Lagrangian

2

.C(X1,X2,X1,X2) = L ~i IIxil12 - V( X1 - X2 )i=1

where V is some sufficient ly short range potential.'?

17The problem here is that we are on a torus but we do not want the potential to"feel" the global properties of the manifold.

Interact ing Particles 193

6.1.1 Reduction to one Particle on 1f2. The most convenient way todescribe such a systems (following [71] and [19]) is to int roduce the new variables

~ = X2 - X1 ; y = X 1'

Remark that the energy and the total moment um

2

E = L ~i IIxi l12 + V(~) ;i=l

2

p= LmiXii=l

are conserved quantities for the flow. Moreover ,

By (6.2) it follows that ~ belongs to a circle, thu s the phase space is 1f4 x 8 1

and the equat ions of motions read

where /-l = Tn ! +rnz. Accordingly, the mot ion of Xl is determined by the motionmlm2

of ~ , while ~ moves as a point of mass /-l subject to a potenti al V : the flow ofthe two part icles is a isometric 1f2 extension of the flow of a single particl e, 1631 .

Note that , assuming very mild condit ions on the potential, such systems aresmooth flows with collisions 120] .

According to [71] if ml/m2 is irra tion al and the flow of the single particle isBernoulli , then the total flow is K and hence a result by Rudolf [63] implies tha tthe tot al flow is Bernoulli. If ml/m2 is rat ional, then the ergodicity of the totalsyste m depends on the components of the tot al momentum having irration alratio .

6.1.2 A Model System. The above discussion mot ivat es the st udy ofa particle in a radially symmet ric potent ial field. The most studied case is whenthe part icle moves on a torus, although the case of a part icle moving in a billiardtable inside which the potent ial is present has also been investigated [54] . Sincethe result s are essent ially similar we will concentrate on the first possibility.As already mentioned we will consider radially symmet ric potentials. In otherwords, there is a disk, in a torus, inside which a symmet ric potent ial is present .

T he dynamics of such a system consists of the composit ion of two easilyunderstood mot ions. Outside the disk, a particle moves in a straight line withunit speed. Inside the disk, the symmet ry of the potential implies that themotion is integrable.

A common assumption is that every parti cle that starts inside the disk andevery part icle that enters the disk will leave the disk. Thi s imposes certain

194 c. Liverani

rest rict ions on the potential V (r) and the energies considered, see (6.4). It isconvenient to introduce polar coordinates (r,B) , r E [0,R], BE [0,211"] on the diskD, and denote by <p E [- 11", 11"] t he angle a tra jectory makes with the boundaryof the disk.

T he symmetry of the potential implies that a par ticle tha t enters the diskat the point (B,<p) , <p E [0, 11"], will leave the disk at a point (B+ /:)'B(<p), - <p).The function /:)'B(<p), <p E (0,11") , is called the rotation function. In fact , therot at ion function determines complete ly the ergodic prop erties of the systemsince it determines uniqu ely the Poincare map from leaving D to entering D.From an abst ract point of view we have a generalized Sinai BjJ1iard [18] .

Definition 6.1 A generalized Sinai billiard is a composition of straight motion outside D and a generalized billiard reflection: the angle of reflection equalthe angle of incidence but, in addition, the particle is now rotated around theboundary of the disk a certain amount that depends on the angle of incidence.

Such genera lized billiard s have been exte nsively investigat ed [70], [38], [34],[4]' [35], [36] , [40] , [20], [541, [181, [19] we will sum up the results in the following.

We start by the following general result .

Theorem 6.2 ([201) . Suppose that the rotation function /:)'B(<p) is piecewisesmooth and there is a 0 > 0 such that for almost all 2or /:)'B' (<p) < 2 - o. Then, providing the time between returns to the disk issufficiently large (t > 2(2 - 0)0- 1), the billiard will have positive Lyapunovexponents almost everywhere.

To prove Th eorem 6.2 it is necessary to introduce a cone field in an appropriate set. The set U is a neighborhood of the bound ary of the disk withincoming velocity; the Lagrangian spaces are V1 (x , v ) = {(~,O)} C ]R4 andV2 (x ,v ) = { (o{,o}, where

. {(2- o)R }a = mill 0 , 0 .

A more useful, and more precise, formul ation of T heorem 6.2 is given by thefollowing.

Theorem 6.3 ([201) . Suppose that the rotation function /:)'B(<p) is piecewisesmooth and there is a 0 > 0 such that for almost all 2or /:)'B' ( 2(2 - 0)0- 1), the derivative cocycle is strictly monotone,with respect to the cone field described above, going from a collision to the next.In addition, conditions 1 to 8 are satisfied for some appropriate Poincare map.18

18The problem with t he Poincare map is tha t the obvious one const ruc ted on th e

Interacting Par ticles 195

Theorem 6.2 follows then by T heorem 3.1 and the fact that the trajectori esthat never collide form a zero measure set.

The exact nature of the rotation function depends on the potential V (r) inthe following way.

Let V(r), r E jR+ be a radial potential that sat isfies

{

limr-t or2V(r) = 0

supp V C [0, R)

V E C 2((0, R)) .

From now on we fix the energy equal to one half.Let h(r) = r 2(1 - 2V(r)) , before going further we

condit ion on our potential,

h'(r) > 0

(6.3)

impose an additional

(6.4)

for all except perhaps one value of r E (0, R) . This condit ion insures the absenceof "trapping zones": invar iant regions of phase space in which the motion iscompletely integrab le.

With this assumption, if the angular momentum l := r2 iJ =1= 0, then thereexists a time f E jR+ U {(X)} such th at r (t ) :::; 0 for t < i and r (t) = O. This is thetime at which the particle comes closest to the cente r of the potenti al. Denoteby f = r(cp ) this minimum radius. For potenti als satisfying (6.3) and (6.4), thefollowing expression for the rotation funct ion for cp E [0, 7l"/ 2) holds:

lR l

66 = 2 ~(cp ) f r Jh(r) - [2 ,l = R coscp = h(f ) ~ . (6.5)

An orbit that enters the potential field with angle 7l" - tp will rot ate clockwisearound the disk by the same amount that an orbi t entering the disk with anglecp will rot at e counterclockwise. T hus for cp E (7l"/ 2, 7l") , we can define

b.6(cp) = - b.6(7l" - cp ).

Thi s definition produces a rot ation funct ion that will typically be disconti nuousat cp = 7l"/ 2. In such a case, we ignore those trajectories that enter the disk withangle cp = 7l"/ 2. These points form a set of measure zero.

boundary of D does not immediat ely fall in t he setting of sectio n 2 since the symplectic form becomes degenerat e for configurat ions corresponding to tangent collisions. Alt hough it is possible to furth er genera lize th e setting to which the genera lresul ts apply, I find more convenient to choose a different sect ion made up of severa l linea r pieces with a const ra int on the minimal incidence angle. This introduces"fake" singularit ies in t he systems, yet t hey can be treated togeth er wit h t he naturalones without any subst ant ial ext ra work.

196 C. Liverani

It tu rns out that the following results can be writ ten in a par ticularly simpleform by int rodu cing the functi on

n(r) = rh'(r) = 2 _ 2 V' (r )r .h(r) 1 - 2V(r)

6.2 Repelling Potential. Sinai [70] and Kubo [381 have given examples ofrepelling pote nt ials, V(r) 2': 0, V'(r) ::; 0, r E (0, R) , for which the system haspositive ent ropy and is ergodic. In their examples, the potenti al was cont inuousbut not C1

. Such results are genera lized by Donnay and Liverani in [20] .The reason why one has not yet been able to make C 1 potentials for which

positive Lyapunov exponents can be proven may be understood by reference toTheorem 6.2. If the potenti al is smooth, then ~e'(O) = 2. For small angles, therepelling nature of the potential causes t ra jectories to rotat e less far around thedisk than they would in the V == 0 case. Hence for small <p, ~e' (<p) < 2. Thusthe values of ~e' fill up some interval [2 - 0, 2] , and the cone-field method alonecan not handle thi s case.

Theorem 6.4 ([20]). If V satisfies conditions (6.3), (6.4) and

1. n(r) is non- in creasing for r E (0, R) .

2. V(R-) = 0

3. V'(R-) < 0

Then~e'(<p) ::; 2 - 0, V O.

Let us call t min the minimal flying t ime between two consecut ive collisions.Using Theorem 6.2, Theorem 6.3, Theorem 4.6, Theorem 5.1, Theorem 5.2,

T heorem 5.3 and Theorem 5.4 we have19

Corollary 6.5 Any repelling potent ial V(r) satisfying the assumptions ofTh eorem 6.4 and for which tmin > O(i R

l_2 will produce a flow (pi, X , /1, ) whichis hyperbolic, ergodic and B ernoulli.

The corollary follows by noting that Theorem 6.2 implies hyperboli city; Theorem 6.3 together with Theorem 4.6 shows th at the possible boundaries betweenergodic component s must be contained in the set of configurations that neverexperience a collision or experience a tangent collision both in the past and inthe fut ure. But such sets do not separate the phase space (as the reader caneasily check)20 thu s the map must be ergodic. Property K follows from Theorem 5.1 and the fact that the above argument applies to all t he powers of t he

19Th e Bernoulli property for the Poincare map was first proved in [75].2oFor th e double tangencies remember condit ion 6 which implies that the images of

the singularity manifold s are transversal.

Interactin g Par ticles 197

Poincare map as well. Finally Bernoulli is a consequence of Theorem 5.2. Thi salso shows that the flow is ergodic. The Bernoulli prop erty follows from the nonjoint integrability of the st rong stable and st rong unstable foliation, as alreadyremarked in sect ion 5.

6.2.1 Soft Billiards. In the previous sit uat ion it is also possible to allowdiscontinuities for V at the boundary: V (R- ) -10. If we choose V (r ) == Vo , r E[0,R-) , we have the case of soft pot entials studied by Baldwin [4], Knauf [361and [20]. For such a V,

[R2 _ [2 ]!

!:::.O' (ep ) = 2 h(R- ) _ [2

When Vo < a then

AO' ( ) 2 2u ep :::; VI - 2Vo < ,

so we get positive Lyapunov exponents provided that tmin > 2R[VI - 2Vo_ 1]-1. The sit uation for Vo E (0,1/ 2) is slight ly different . For (R cos ep)2 = [2 2h(R-) , we have !:::.O(ep ) == O. For [2 < h(R- ), (6.5) gives !:::.O' > 2. Hence softpot ent ials prod uce positive Lyapunov exponents by two different mechanisms.Clearly one can produce examples where V( R-) -I a and V is not constantprovided that 0 has the appropriat e behavior.

6.3 Attracting Potentials. We call a potential with V' (r ) 2 0, r E (0, R),attract ing.

The first examples of at racting potentials that enjoy ergodic behavior wereobtained by Knauf 135J who showed that for potentials with singularities of thetype _ r-2(1- I / n ) , n E Z+ \ {a, I }, th e flow can be regularized, i.e. can be extended to a smooth flow, in an appropriate covering manifold. In addition, sucha manifold turns out to be of negative curvature with respect to the Maupertuismetrics defined by the Hamiltonian [31. The problem is thus solved by using thegenera l Anosov [2] results for geodesic flows on manifolds of negative curvature.

It is easy to understand why the regularization is possible at all in terms ofthe rot at ion funct ion !:::.O.

lim !:::.O(ep ) = 21TI O(0).<p--Hr / 2-

Let 0 E C1([0,R]) and a = 2 - 0(0). T hen V will have a singularity atr = a of the form -l /ro.. It is then possible to define the rot ation function tobe cont inuous and even smooth for ep E [0,1T], provided a = 2(1 - ~) , n E N. Infact , for such a ,

lim !:::.O(ep) = n1T, n E Z.<p-t (~r

(6.6)

198 C. Liverani

T hus we can set b,.()(7f/ 2) = n7f and define

b,.()(<p) = 2n7f - b,.()(7f - <p), <p E (7f/ 2,7f].

Yet , the hyperbolicity does not really depend on the possibility to regularizethe flow. In !201 it is shown that , for any Q E (0,2), one can construct smoot hpotent ials V (r ) with singularity of order - r » for which the flow has posit iveLyapunov exponents almost everywhere and is ergodic.

Theorem 6.6 (120]). Let V be an attracting potential satisfying (6.3), (6.4)and such that n (r) is strictly increasing for r E (0,R) with n(O) > 0 andV (R) = V'(R) = 0 (n (R ) = 2) . Then b,.()'(<p) > 2 for all <p E (0,7f/ 2).

Using Theorem 6.2, Theorem 6.3, Theorem 4.6, Theorem 5.1, Theorem 5.2,Theorem 5.3 and Theorem 5.4 we have

Corollary 6.7 Any attracting potential V(r ) satisfying the assumptions ofTheorem 6.6 will produce a flow (Ipt, M ,J.l ) which is hyperbolic, ergodic andBernoulli.

Remark 6.8 Note that the above conditions cannot be satisfied by a smoothattracting potential with no singularity at the origin and for which the rotationfunction is continuous.

To see this, note that if V is smooth at the boundary then limep->o+ b,.()' (<p) =2 since for ip = 0 the particle is tangent to the disk and hence stays in the V == 0region. And we know that for V == 0, b,.()' (O ) = 2. So we are forced to makeb,.()' > 2 for all <p, since if b,.()' decreases cont inuously from 2 the cone methodbreaks down. But if V is smooth and bound ed, and b,.() E C1 then

b,.() (7f/2 ) = 7f =1'f b,.()'(<p ) dip.

so that b,.()' cannot be always biggcr than 2.

6.4 Smooth or Leinard-Johns Type Potentials. A potenti al V(r) being smooth does not necessarily imply that the rot ati on function b,.()(<p) , ip E[0, 7f / 2), is smooth. It is this observat ion that allows to const ruct smooth,bounded potential s for which the flow has positive Lyapunov exponents andis ergodic. Thi s idea was originally introduced by Donnay in [17] .

One can const ruct a smoot h potential having, on the energy level underconsideration (E = ~) , a closed orbit for some rc < R, hence n (r c) = O. In sucha situation there exists an angle <Pc E (0, 7f/ 2) of entry for which the tra jectorywill become asymptotic to this closed trajectory and never leave the disk. Thusb,. () (<Pc ) will be undefined.

Theorem 6.9 ([20]). There exist smooth potentials, satisfying (6.3), (6.4),with a closed orbit at some rc < R for which the rotation funct ion satisfies

/),.()' (cp) > 2, cp E (0, CPc ),

lim t/),. ()' (cp )I = 00 ,<p-4 O.

(6.7)

By introducing t he closed orb it , we permit th e rotat ion function to start outwith /),. ()' ~ 2 for sma ll angles and then to have /),. ()' ::; 2 - 0 for larger angles.Yet /),.()' never takes values in t he interval (2 - 0,2 ).

T he usual concatenation of t heorems yields the following.

Corollary 6.10 For a potential whose rotation function satisfies (6.7) andfor which tmin > 2R(2 - 0)/0, the flow is hyperbolic, ergodic and Bernoulli.

R emark 6.11 There exist smooth potentials for which (6.7) holds that havea hard core: i.e. for some r* < rc , the potential satisfies V(r) = + 00 for allr E (0, r*] .21 Hence, the particle cannot enter this region.

Not e that the above sit uat ion it is qui te unstable. If the energy is slight lychan ged , the periodic orbit eit her disappears (t hus the rot ation function becom esmooth and nothing can be said with the availab le techniques) or it gives rise toa stable region foliated by invariant tori (condition (6.4) fails) t hus ergodicityis lost. 22

6.5 Non-Ergodicity. After the above discussion it is natural to ask whathappens if the condit ions of T heorem 6.2 are violat ed . T he answer is that thesystem may not be ergodic, as we have just seen at t he end of sect ion 6.4. Letus look at the situation in more detail.

T he first to invest igate this type of sit uat ion has been Baldwin 141 but t hemost complete resu lts are found in the work of Donnay [1 8], [191.

Let us call {lx,l y} t he two sizes of the torus in which t he disk is sit uated.Assume tha t the rotation function is piecewise 0 (4) wit h /)"() (O) = 0, /),.()' (O) = 2,/),.O (k)(O ) = 0, k E {2,3,4}. These cond it ions insure a smooth flow for sp = o.

Definition 6.12 A rotation function is called partially focusing if

a there is an angle .p" for which the rotation function satisfies /)"O' (cp*) = 2,

b in a neighborhood of cp*, /),,0' is continuous and takes value less than 2.

A disk is called partially focusing if its rotation function is partially focusing.

21The potential it is smooth in (r ", 00).22This last case is similar to Donnay's light bulb, see [17] .

200 C. Liverani

Theorem 6.13 ([18]) . Given a partially focusing rotation disk for which

t,.B(<p*) -=1= 2<p* +7r mod (27r) ,

one can place it on a torus in such a way that the resulting billiard system isnot ergodic. Th e time between returns to the disk can be mad e arbitrarily large.

The above theorem immediat ely implies that smooth potentials outside thevery special type discussed in sect ion 6.4 are likely not to give rise to ergodicsystems. Yet , thi s does not rules out the possibili ty of having ergodic systems,it only shows that the event ual ergodicity will depend upon the fine st ructureof the geomet ry of the orbit, so the st udy of such systems stands as a very hardtechnical challenge.v'

A similar situation has been investigat ed also in a rather different context:the approximat ion of billiard table by very steep potentials. Thi s type of problems has been investigated in great detail in [76] , [77], [781 .

Consider a billiard table Be JRn and consider local coordinates (~ , r) near oBsuch that ~ E oB and r is the distance from ~ along the direction norm al to thebound ary. Clearly, such coord inates are well defined only in a neighborhood ofthe smooth pieces of the boundary. Let oeB be the corners of the boundary (thenon-smo oth part) . Then one can consider pote nt ials of the form Vg satisfyingthe following condit ions:

o1. For each compact K cB holds limg--+o IlVgIKll c (T) = 0, for some r 2 2.

2. There exists a funct ion V(r) , V(r) = 0 for all r 2 1, such that for eachJ > 0 there exist s co > 0 for which

Vg(~ , r) = c-1V(c- 1r )

for all e ::; co, d(~,oeB) > J. In addition, V'(r) -=1= 0 for all r E (0,1) ,V(O) = 1, V(I) = 0, V" 2 O.

Then, as e -t 0, the Hamiltonian flow converges, in e(l) , to the billiard flow,away from the singular t ra jectories. In fact , the same holds for more genera lpotenti als, see [761 for details.

Nevertheless, the hyperbolicity (and thus the ergodicity) may not be preserved.

Theorem 6.14 ([76]) . If a billiard has a simple singular (i .e., experiencinga tangency) periodic orbit l , then there exists a one parameter family of Ham iltonians limit ing to the billiard flow as e -t 0 for which there exists a sequenceof in terv als of c converging to zero on which elliptic periodic orbits 19 exists inthe energy level of l . These elliptic periodic orbits lim it to the singular periodicorbit in the limit e -t O.

A similar theo rem holds for singular homoclinic orbits.

23The point is that any periodi c orbit that explores a neighborhood of t he criti calangle ip" is in danger of being ellipt ic. If such a case KAM theory could possiblyapply t hereby producing non-trivial invar iant sets .

(6.8)

6.6 Billiards with Gravity. Up to now we have discussed the motion ofa particle on ']['2 under a potential field. Other, very interesting, possibilities doexist . A first inspiring example can be found in [461 where the authors studynumerically the motion of a par ticle in a wedge under a gravitational potential.

The simplest example of such a system is given by the Hamiltoni an

1 2 2H = 2(PXl +PX2) + X2

on th e configuration space X <p = {Xl 2: 0; Xl cos rp + X2 sin rp 2: O} , for somerp E (0, ~) , where the particle collides elast ically at the boundaries of X <p '

Such a system turns out to be equivalent to two particles of mass m1 +m2 =1, with m1 = sin2 .p,moving on a vert ical line (under the influence of the gravity)and colliding between them and with a floor [87] . In other words we have theHamiltoni an

1 2 2 2

L Pi LH = - - + mi qi2 m

i= l ' i= l

(6.9)

with configura t ion space X = {O < q1 < q2}.The above mentioned numerical simulat ions have shown that (6.8) has dif

ferent ergodic properties depending on rp .24This type of result s has been generalized and rigorously investigated by

Wojtkowski in a series of papers [87], [88], [891 .A more general class of examples (usually called falling balls) is given by the

Hamiltonian

1 n 2 n

H= - ~ Pi +~miV(qi) '2~m c:

i= l ' i= l

(6.10)

The phase space is X = {(q,p) E IR.2n I0 < q1 < . . . < qn} , and the particl escollide elast ically among themselves and with th e floor (that is, the particle q1can collide with the floor). Note th at these systems are flows with collisions andthe Poincare map from collision to collision sat isfies conditions 1 to 3.

To investigate the hyperboli c behavior of such a system it turns out to be convenient to consider the Lagrangian subspaces VI = {(~ , O)} and 112 = {(U~,~)}

where the mat rix U is defined as follows

Since th e vector field, associa ted to the Hamiltoni an (6.10), is (m;lpi , -miV'(qi))E V2 , we are exactly in the setting of section 3. As usual let Q be the associatedquadratic form.

24 If If! > i , then the Lyapunov exponents are almost everywhere different from zero,otherwise there exists a periodic linearly stable orbit .

202 C. Liverani

Theorem 6.15 (Woj tkowski [871 , [88]). If the system determined by theHam iltonian (6.10) has masses that satisf y the in equaliti es m l > m 2 > .. . > 'Inn

and if V' > 0, V" ::; 0, then the flow with collision s is monotone. If V" < °then the flow is eventually strictly monotone.

This shows that a system of n par ticles on a vertical line subject to a Newtonian potential V(q) = -«:' has all the Lyapunov exponents (bu t two) differentfrom zero.25 Thus th e systems, restrict ed to a constant energy sur face, is a completely hyperbolic flow.26

The Theorem 6.15 does not set t le the cases in which the second derivative isnot st rictl y negative or when the masses do not sat isfy the st at ed inequ aliti es.Let us look at these possibilities.

A case that has been deeply investigated is V(q) = -q. In such a caseTheorem 6.15 te lls us only that th e flow is monotone, in fact in [87] Wojtkowskiproved that it has all (but two) the Lyapunov exponents different from zeroon a set of positive measure. Subsequent work of Simanyi 169] complete d thepicture.

Theorem 6.16 (Simanyi [69]). If the system determin ed by the Ham iltonian (6.10) has mas ses that satisf y the inequalities ml > m 2 2': ... 2': m n and ifV (q) = - q, then the flow with collisions is eventually stric tly monotone.

It remains to invest igate the situation with masses that do not satisfy therequired ordering. Fir st of all not e that if the masses are all equal , then there isno real difference, for th e particles, between colliding and pas sing through eachother (provided the par t icles are prop erly renumbered after "collision"). Thusthe syste m with all equa l masses is completely integra ble. The crossing of sucha special case does not improve much the situation as far as complete hyp erbolicity is concern ed: typically stabl e periodic orbits appear. As an example let usquote the following resul t (see [16] for more) .

Theorem 6.17 (Cheng-Wojtkowski [16]). If the sys tem determin ed by theHamiltonian (6.10) , with V(q) = - q, has masses ml < m 2 = m 3 = .. . = m n

or m l = m 2 = '" = m n -l < m n , then there exists a periodi c lin early stableorbit.27

25 What happens is that the Q form increases strictly between collisions if V" > 0while it remains constant if V" = O. At collision it increases only on some vectors,not dissimilarly from the RBS that are discussed in detail elsewhere in this volume.

26Sometime the expression completely hyperbolic is used in the literatur e to meanhyperbolic (in our sense) when one wants to emphasize, like now, that in a systemwith many Lyapunov exponents all are different from zero.

27Note that the authors are not able to establish the applicability of the KAM theorem. Consequently, strictly speaking, t he result does not prevent ergodicity. Nevertheless, it looks extremely unlikely that ergodicity may hold.


(6.11)

Fin ally, it is important to notice th at , if we restri ct th e discussion to linear potentials, th en more genera l problems can be t reated [89J . Consider th eHamiltonian

1 n n

H = 2" L K ijPiPj + L ciqii = 1 /=1

where K is strictly positive definit e and ci > O. The phase space is X = {(p,q) E

jR2n Iqi 2': O}. In addit ion, when a particle reaches the boundary of X (th at is, aqi is equal zero) th e component of the velocity parallel to the face of th e configuration space (the positive cone) is preserved and the component orthogonal tothe face is reversed. Orthogonality is taken with respect to th e scalar productdefined by the matrix K .

Theorem 6.18 (Wojtkowski [89]) . If all the off-diagonal entries of the matrix K - 1 are negative then the Ham iltonian system (6.11) is even tually stric tlymonotone.

The systems (6.11) are nothing else th an a particle in a multidimensionalwedge with an acceleration vector pointing inside it, so that the constant energysurfaces are compact .P" In fact , it t urns out that the falling balls system [82J(with linear potential) is of th e same type (after a symplectic change of variable), with a tridiagonal matrix K and th e accelera tion vector equal to the firstgenera tor of the wedge. This case corresponds to what Wojtkowski calls simple wedges. The idea used in proving Theorem 6.18 is to decompose th e wedgeinto simple wedges. Each one of such simple wedges is equivalent to a systemof falling balls, thus one can use the quadr atic form previously described. Theresult follows th en by checking th at th e form is increased going from one simplewedge to anot her. 29

Other two inte rest ing special systems which turn out to be equivalent toa simple wedge, and hence to th e falling balls system, are the following.

Th e capped system of particles in a line describ ed by the Hamiltoniann 2

H "Pi= L...J 2m ' + m nqn,

i=1 t

with configurat ion space {q E jRn I 0 ::::: ql ::::: q2 . . . ::::: qn} where th e particlescollide among each other and with the floor.

Theorem 6.19 ([89]) . Th e capped system of particles in a line is completelyintegrable if

nfor k = 1, . . . , n - 1,

28The shape of the wedge is determined by the matr ix K that defines the scalarproduct.

29 Note that the quadratic form so defined is only piecewise continuous, since it experience a jump going from one simple wedge to another.

204

and completely hyperbolic if

C. Liveran i

m, > 1M , - 2

.!!!i > m i-l (1 + m i -l )-1 r . 2 2M i - M i -

lM i -l Jor l = , .. . , n -

m n -l > m n - 2 (1 + m n _ 2 )-1M n - l M n - 2 M n - 2 '

where Mi = m i + ...+ m n.

The system of attracting particles in a line describ ed by the Hamil tonian

with configurat ion space {q E jRn+l Iqo ~ q1 ~ ., . ~ qn; mo qo+ · · .+mnqn = O}where the particles can collide elastically with each other and the cente r of massis fixed.

Theorem 6.20 ([89]). Th e sys tems of att racting particles is completely in tegrable if, for some a > n,

m i amo (a - i)(a - i + 1)

and it is completely hyperbolic if the sequence

mo + ...+ mi-1 + ,;ai = ------- •

An inte rest ing system to which Theorem 6.18 applies is the system of noninteracting particles falling onto a moving floor of fin ite mass. Such a system isdefined by the Hamil tonian

n 2 n

H = L ;~ +Lami(qi - qo ),i=l ' i = l

with a > 0 and elastic constraints {q E jRn I qi > qo Vi > O}. It is easy tosee that such a system, via a simple change of coordinates, can be t ransformedin a system to which Th eorem 6.18 applies directly. It is very interesting tonotice that, in this case, no constraint on the masses is needed in order to havecomplete hyperb olicity.

In fact , even more genera l systems can be t reated (with different collisionconst raint among the par ticles) we refer to 189] for a complete discussion.i'?

30 see [90] for a more cond ensed exposit ion.


Up to now we have seen that t he system of falling balls with linear potenti almay be complete ly hyperbolic or non-ergodic according to the mass distribution.Yet , we have not discussed the ergodicity in the hyperbolic case. In fact , it isnot known if, in genera l, such syste ms are ergodic or not .31 The difficulty is t hatall the conditions of Theorem 4.6 are sat isfied except condit ion 6. To be moreprecise the triple collision of three par ticl es is not properly aligned . T his meanst hat T heorem 4.6 applies only to the case of two par ticl es.

Theorem 6.21 (150], 112]). If the system determ ined by the Hamiltonian(6.10) consists of only two particles with masses satisfying the inequalities m1 >m 2 and if V(q) = -q, then the flow with collisions is hyperbolic, ergodic andBernoulli on each constant energy surface.

6.7 Magnetic Billiards. Partially moti vated by physical reasons concerning quantum mesoscopic syste ms a certain amount of atte nt ion has been devotedto a par ticle movin g in a billiard in presence of a magnet ic field 162], [61], 151],[5], [72], [731 , [74] . More precisely, the mod el consists of a plane billiard witha magneti c field perp endicular to the plan e.

In t he case of const ant magneti c field th e particle moves, between collisions ,on arc of circles with t he radius depending on t he st rengt h of t he magnetic field(and the energy ).

While the work on this subject has been mainly numerical, some rigorousresults exists in the case of constant magnetic field.32 Namely, it is possible tot reat th e strong and weak magnetic field case. Clearly, if t he magneti c field issufficient ly st rong, then the par ticle can move in circles that never touch theboundary. In such a case st abl e motion arises (see 173] and 161] for explicitconditions insuring t he exist ence of stable periodi c orbits). The other extremeis the persist ence of hyperbolicity for small magnet ic field in billiards with finitehorizon .P To understand qualit atively t his phenomena it suffices to noti ce th atif a billiard has a finit e hor izorr' " t hen it is typically possible to find a conefamil y such that t he map d¢/ , for t larger than the maxim al flying t ime , isst rict ly mon otone. T his means t hat t he same cone famil y will work for smallper turbations of d¢/ as well. Thus the cone famil y used to prove hyp erbolicityof t he bill iard with zero magneti c field will work for sufficiently small magneticfields as well.35 Not e tha t the above argument does not use t he constan cy of the

31Although they are widely believed to be ergodic.32 Both constant and variable magnetic fields have been investigated numerically, but

most of the results pertain to t he constant field case.33 In the physics literature the positivity of the Lyapunov exponents goes, at times,

under the funny name of hard chaos.34Here by fi nite horizon I mean that there exists a maximal flying time between two

consecut ive collisions with non-flat boundaries.35The reader will cert ainly remember very similar considerat ions at the beginning of

the section regarding an external potential field.

206 C. Liverani

field, yet by restricting the discussion to homogeneous fields Tasnadi [73] findsexplicit est imates of how large the magnet ic field can be taken.

§7. Non-Hamiltonian Systems

In this section we discuss briefly further generalizations. First of all one canconsider , instead of a magnetic field, an elect ric field. If the billiard table issimply connected then we are just in the already studied situ ation of a particlemoving under an external potential. If the domain is not simply connected, itis perfectly possible tha t there is no globally defined potenti al generating thegiven electric field. In particular this is the situation when there exists a closedcurve along which the electric field performs a positive amount of work. In sucha case clearly there cannot be an invariant measure since the par ticle will tendto accelerate indefinitely. Consequently, to have steady states in presence of anelectric field it is necessary, in general , to have some form of dissipation in thesystem.

7.1 External Electric Fields. A form of dissipation widely investigat edin this context are the so called Gaussian Thermostats. For a particle on thetorus this amounts to a motion governed by t he equations

where

{

mq=p

jJ=E(q) -ap

a = (E(q), p)(p, p) .

(7.1)

The motion determined by (7.1) conserves the total kinetic energy and it isreversible but it is not Hamil tonian. i'" The simplest case is the driven Lorentzgas. Thi s corresponds to a Sinai billiard with an external electric field [10] .

Another possible form of dissipation is to introduce non-elastic collisionswith the boundaries. An interesting example can be found in [151 where theauthors study a non-elastic deterministic rule of reflection simulating movingwalls.

This typ e of systems has attracted a great deal of attention lately in connection with the so called Ruelle's chaoti c hypothesis and the Gallavotti-Cohenfluctuation theorem. The description of thi s area of research would lead us quiteafar from the present course, especially given the explosion of results in the subject . Yet, here is a very sketchy and incomplete bibliography for the interest ed

36Yet, it retains a lot of t he sympl ectic st ruct ure of Hamiltonian motions: it is conformally symplectic, see [91] for more det ails on such property and its consequences.

reader. In order to get acquainted with the general ideas [23], [65], [24]. To tack lethe more technical resu lts [11], [25], [66], [67], [26], [29], [68].

7.2 Internal D egrees of Freedom. Another very important fact aboutmolecules is that they have internal degrees of freedom. Not much has been doneto investigate the effect of such internal degrees of freedom on the ergodicity oftwo interacting molecules, yet some results do exists.

The simplest way to introduce internal degrees of freedom is to considerrotating two-dimensional disks of equal mass m and radi us r . To do that in a nontrivial way it is necessary to have a collision rule that couples the translationaland the rotational degree of freedom. Such a collision rule (no-slip collision) hasbeen introduced in [7]. It amounts to the condition that the tangential velocitybetween the disks, at the point of collision, is zero (the two disks are rough andcannot slide one on the ot her). To be more precise it is necessary to introducecoordinates. Let us use the coordinates qi for the center of the disks, Vi for thevelocities of the centers, Wi for the angular velocities. Let I be the momentumof inert ia. Let e be the unit vector parallel to q2 - ql and f the normal vectorsuch that {e,f} form an orthonormal base with the usual orientation. Finallywe denote by A the linear operator that describes the change of the velocitiesat collision. Expanding the velocity vectors Vi in the orthogonal basis {e,f} , wecan consider the vector ((Vl,J), (V2' 1), WI ,W2) E ]R4. The operator A projectedto this four dimensional subspace has the following form:

[ ,L - r e

-'''11+p 1+e 1+e 1+~2

.c. 1 --!:L ~A = 1+e 1+e 1+e 1+e

-1 1 L -1r(1+e) r(1+e) 1+e 1+e

- 1 1 -1 Lr(1+e) r(1+e) 1+e 1+e

where C = J 1 2 < 1. The description of A is completed by its action on the~ rnr-

two dimensional subspace of vectors ((VI ,e), (V2' e)),

A = (~ ~) .

Clearly the dynamics is not Hamiltonian since we have an anho lonomic constraint, yet it is reversible . In [85] Wojtkowski has proven that the system of tworotat ing disk with non-slip collisions on the torus has a linearly stable per iodicorbit provided the disks have a sufficiently large momentum of inertia I andtheir radi us is sufficiently large with respect to the torus size.37 In addition, in[71 it is proven that if one has a ball colliding with two flat parallel walls, thenthere exists a stable periodic orbit (perpendicular collisions with the walls).

37 Again this does not prove non ergodicity since the applicability of KAM theory hasnot been checked, yet it is a pretty conclusive result.

208 C. Liverani

In conclusion, the above analysis points quite convincingly to the conclusionthat internal degrees of freedom can spoil the ergodic properties of the system.j"

7.3 Stochastic Dynamics. A much more drastic way to introduce internaldegrees of freedom, or the influence of an external environment , is to model themvia stochastic interactions among the particles or between the par ticles and theexte rior. Thi s leads us quit e afar from the original model yet it may be relevantto mention such a possibility since, once stochas t icity is introduced, it is possibleto obt ain a very good understanding of the ergodic properties of the syst em. Inpar ticular , it is possible to und erstand the ergodic properties of arbit rarily large(and infinite ) systems.

At this point a word on ergodicity of infinite systems is called for. If oneconsiders an infinite gas of non-interacting particles, or an infinite harmoniccrystal, with respect to a Gibbs measure, then the resulting dynamical systemsturn out to be ergodic, even Bernoulli , [81], [60], [44] . This may seem quiteodd since the finite dimensional systems are completely integrable and thus theexact opposite of ergodic. The explanation of the apparent paradox is that ,for infinite systems, all the invariant measures th at , when restricted to finitesystems, would correspond to the different ergodic components become mutuallysingular. Such a phenomenon is clearly due to the presence of infinitely manyparticles: by looking at a finite region one sees cont inuously new particles comingfrom out side, thus the behavior appears completely random. In essence, the classof measures th at are absolute ly cont inuous with respect to a given one is a rathersmall set in infinite dimensions and ergodicity in such a class it is not a veryrelevant concept.i'?

A different context in which such type of problems is central is the deri vat ionof hydrodynamic laws from microscopic dynamics. In such a contex t a modelsimilar to the previous examples has been introduced by Olla, Varadhan andYau [56] . The model consist s of infinitely many ident ical particles (we call n thephase space oflocally finite"? configura t ions w = (qi ,Pi) , qi ,Pi E JR3 ) und ergoinga motion determined by an Hamiltonian

¢ being a stri ctly convex function with at most linear growth. Let L be theassociated Liouville operator. 41 In addit ion, there is a long range stochast ic

38See [90] for similar considerations in the case of Hamiltonian systems with linearpotential and elastic constraints.

39This was already recognized in [30] .

40That is {q;} has no accumulations points in ]R3 .41 The Liouville operator is nothing else than the generator of the flow associated to

the Hamiltonian vector field .

Interacting Particles

interaction among par ticles. The equations defining the mot ion read as

209

{

dq., = 1J'(po) dt

dp., = - L {3# o V' (qo - q{3 )dt + cbo(w) dt + eL~= l L {3# o a: ,{3 (w)dW: ,{3 '(7.2)

where w~ ,{3 is a family of independent one-dimensional Wiener processes for

() = 1,2 , ..., d and 0' -I {3 such that w: ,{3 = -w~ ,o ; E E IR+; 1J' and V' denotethe gradient of 1J and V , respectively. The coefficient s bo , a~ ,{3 : n H 1R3 areappropriate smooth local functions, chosen in such a way th at total energy andmomentum are both preserved by th e randomized evolut ion (7.2).42 Any Gibbsstate P with energy H will be a reversib le measure for the stochast ic pa rt ofth e evolut ion:

Jcp(w)L1jJ(w) P(dw) = J1jJ (w)Lcp(w) P(dw)

for all smooth local funct ions cp, 1jJ : n H IR. The operator L is the generator ofth e stochast ic part of the dynamics and reads

d

L1jJ = L(bo, : 1jJ ) + ~ L L L (a~ ,{3 ' (D~ ,{3 1jJ)a~ , {3 ) 'o El Po O=loEl {3#o

where D~ ,{3 1jJ is the matrix of second derivati ves obtained by applying Do,{3 =a/apo - a/ap13 twice to 1jJ . Since the Liouville operator is ant isymmetric with

respect to Gibbs distributions, the full generator , L = L +cLalso satisfies thestationa ry Kolmogorov equation, JL1jJ dP = 0, for a wide class of test functions1jJ and any Gibbs state P .

Without entering in furth er technical details it suffices to say that thestochastic interaction is a diffusion conserving th e total energy and momentum, thu s it can really be thought of as th e result of some complex internalmotion of the constituent of th e particles.

In this situation the concept of ergodicity t hat is employed (and that provesthe most fruitful) is the classification of the invariant measures in the class ofmeasures of finite specific ent ropy. To be more precise let :FA be th e space ofsmooth functions localized in th e region A C ]R3. Given two prob ability measuresP, p Ion n let PA,P~ be th eir restrict ion to the region A. T he relative entropyof P~ with respect to PA is define by43

HA(PIIP ) = sup {IBf (F) -log EP (exp(F ))} .F EFA

42rn the following we assume also that they sat isfy some extra technical conditions,see [221 for details.

43By lEP we mean the expectation with respect to the measure P .

210 C. Liverani

A measure pi has finite specific entropy with respect to P if

HA(PIIP)sup IAI < 00 .

AC IR3 1 +All the above suggests to call an infinite system ergodic if the only space

time invariant measures , in the class of the measures that have finite specificent ropy with respect to a given Gibbs measure, is just the closure of the convexcombinations of Gibbs measures.

The st ronger result in the above direction can be found in 122]' 147] and refersto a generalization of the model discussed above: the stochastic interaction canbe arbitrarily weak (e arbitrarily small) and of short range .v' Let us call p(w) =limlA I---+ oo IAI -1IwA I the density of particles in the configurat ion w = (q,p).

Theorem 7.1 ([47], [22]) . Suppose that P is a translational invariant stationary measure with finit e specific entropy, and let Pc := 3/(47rRY) · If P( {wip(w) > pi}) = 1 for som e pi > Pc, then P is a convex combination of Gibbsmeasures.

To conclude this brief comments it is worth mention that , in general, nosystem is really isolat ed. As already mentioned one can wonder about the influence of the external world on the system und er considerat ion. A typical way todo this is by introducing stochas tic boundary condit ions. This is a ra ther vastsubject that it is not reasonable to review here, let us just mention [27] where itis shown that a gas of intera cting particles with sto chast ic boundary condit ionshas indeed a stationar y measure, [48] where the Lorent z gas with stochast icboundary conditions is investigat ed and [21] where a stationary measure is obtained for an anhar monic chain connected to two different stochast ic boundarycondit ions that simulate two heat bath at different temperatures. T his shouldgive a feeling for the fact that , once stochasticity is introduced in the syst ema priori , much stronger result s can be obtained .

§8. Final Considerations

From the above review we have seen that, in general, as soon as we try to considermore realisti c det erministic models of a gas of molecules ergodicity tends to fail.Th e few successes are limited to two particles in two dimensions, no resultsare available in three dimension or for more t han two particles. To underst andthe typ e of difficulties involved it may be useful to consider the vaguely relat edproblem of construct ing higher dimensions convex billiards. T his can be done

44The range Rl of the stochastic interaction must be larger than the range Ro of thedeterministic interaction, also, for simplicity, the deterministic potential is assumedto be repelling.


[91 but it is a very delicate business since stable orbits arise very easily [861 (seethe related article in the present volume for a discussion). As a consequence oneneeds some understanding of the geometry of the trajectories which is typicallyhard to achieve.

On the ot her hand , stochasti c models that seem to be rath er reasonableyield very good ergodic properties. In fact , prob ably more than is really neededto just ify the successes of statistical mechanics. Nonetheless, if one wants touse them as a justification of the validity of statistical mechanics it remains tounderst and in which sense a dynamical system behaves as a stochastic one evenwhen the dynamical systems is not ergodic.

Th e point could be th at the typical system exhibits both integrab le andstochastic behavior and the stochast ic behavior may be prevalent in some appropriate sense. Unfortunately, none of the techniques discussed in th is reviewseem to have any hope to be applicable in such a situat ion. The study of systems with both positive measure entropy and a positive measure of invarianttori st ands as a challenge.

Finally, it should be emphasized th at weaker properties than ergodicity maysuffice to obtain the right thermodynamical behavior for all the reasonable measura ble quantities. In fact , the physically relevant thermodynamics quanti ties(the ones needed to define a macroscopic st ate) are rather few (enormously lessthan the space of £1 observables, with respect to which ergodicity is defined)and they are often almost constant on phase space, so their average values mayvery well be independent on the initial condition even if the system it is notergodic (see [431for genera l comments and references and [28] for a case st udyalong these lines).

Although there seems to be a rather large consensus in the statist ical physicalcommunity on the above point of view, no subst antial progress seems to beavailable in the direction of a rigorous implementations of such a weakenedtheory of ergodicity.t''

In conclusion, much work is st ill needed to clarify the above issues.

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Scaling Dynamics of a Massive Pistonin an Ideal Gas l

J. L. Lebowitz , J. Piasecki and Va. Sinai

Contents

§l. Introdu ction .§2. Formulation of the Problem . . . . . . .§3. Time Evolut ion of the Piston 's Velocity§4. Derivat ion of Equation (8)§5. Concluding RemarksAppend ix .References . . . . . . . . .

218218220222224225226

Abstract. We study the dynamical system consisting of N non-interact ingpoint par ticles of mass m, in a cubical domain !"h of sides L, separated intotwo regions by an idealized movable wall: a massive parti cle (piston), of crosssectional area L2 and mass M L rv L2 . The piston is constrained to move alongthe z-axis and undergo es elastic collisions with the gas par ticles. We find that ,under suitable initi al conditions, there is, in the limit L -+ 00 , a scaling regimewith t ime and space scaled by L , in which th e motion of the piston and the oneparticle distribution of the gas sat isfy autonomous coupled equations.

1 The research of JL L was supported by NSF Grant NSF DMR-9S1326S, and AFOSRGrant F49620-9S-1-0207. Th e work of JP was supported by KBN (Committee forScientific Research, Poland) Grant 2 P03B 127 16. JP also acknowledges the hospitality at the Department of Mathemat ics of the Princeton University. The researchof VaS was supported by NSF Grant DMS-9706794, and RFFI Grant 99-01-00314.We thank C. Gruber for many useful comments .

218 J . L. Lebowitz, J . Piasecki and Va . Sina i

§1. Introduction

The stat istical descrip tion of the time evolut ion of a Hamil tonian many body system of N interacting particles in a domain A C ~d is convenient ly given by thewell known BBGKY hierarchy for the correlat ion funct ions ! j (r l , VI , ..., r j , Vj, t),j = 1, ..., N [I). Thi s descrip tion remains meaningful (formally at least ) in thelimit N -t 00 . It focuses attention on the lower order correlat ion functions, sayh and 12 , where the relevant physical information is contained.

In the case of non-interactin g par ticles, the hierar chy decouples and the evolut ion of fr depends only on the exte rnal potentials, including elast ic reflection sat the bound aries of A. One may then st ill have, when the one particle dynamics is sufficiently chaot ic, a relaxation to "equilibrium" of moment s of h ,as in the periodic Lorentz gas, with fixed convex scat terers, where the densityn(r, t) = Jfr (r , V , t)dv sat isfies, in suitable scaling limits, the diffusion equation[21 . In the absence of such external chaot ic dynamics any relaxation must cornefrom the interaction between the particles or, less sati sfactorily, from some formof "phase mixing". The former case is notoriously difficult leadin g to the prob lem of "t runcat ing" the BBGKY hierarchy. This can be done rigorously only insome special limiting situations as in the derivations of the Boltzmann and ofthe Vlasov equati ons [31.

In the present note we consider the case where the part icles of the systeminteract with each other indirect ly through elast ic collisions with a moving massive part icle (the equal mass case has been ana lyzed in 14]) . Thi s one dimensionalproblem which is a variat ion of the model introduced in [5], is a caricat ure of thenotorious "piston" problem which has recently attracted much attent ion [6]-[10) .We shall not deal with that problem here. Instead we outline, using a dynamicalsyst ems approach, the derivat ion of a nonlinear differential equat ion for the motion of the massive piston coupled to linear part ial differential equations for theevolut ion of the one par ticle distribution of the light particles. This amountsto an effect ive truncati on of the BBGKY hierarchy for this simple case (seeAppendix).

§2. Formulation of the Problem

Consider a cubical domain [2L of sides L separated into two parts by a wall whichcan move freely without friction inside [2L. Each part is filled by a noninteractinggas with a fixed number of particles. The wall moves along the x-ax is under theact ion of elast ic collisions with the particles and has no other intrinsic degreesof freedom. We assume that the mass m of each particle of the gases is fixedwhile the mass of the piston ML grows as L2 . We express this by requiring thata = 2mL 2j(Nh + m) remains constant when L -t 00 .

T he posit ion of the piston is characterized by a single coordinate XL, 0 <X L < L and the piston itself can move only in the X-direction. Its velocity is

Scaling Dyn ami cs of a Massive Pi ston in an Ideal Gas 219

denot ed by VL . We shall consider the situat ions where VL as well as the velocit ies of the particles are of 0(1) . Since the components of the particle velocitiesperpendicular to the X -axis play no role in the dynamics of the piston we mayassume that each par t icle has only one component of velocity directed alongthe X -ax is. We also assume that the particles have a smooth four dimensionaldistribution so that at any inst ant of tim e there is, with probabili ty one, at mostone particle colliding with the piston.

If V , v are the velocities of the piston and of a part icle before the ir collisionand V' , v' are th ese velocities after the collision, then the laws of elast ic collisionsgive

V' = (1 - E)V + w (1)

v' = -(1- E)V + (2 - E)V (2)

where E = EL = 2m /(ML +m) = a1L2.

Each collision changes the velocity of the piston very little and the situat ionis close to the one considered in [11]. It is reasonable to expect th at for L -+ 00

the velocity of the piston can be decomposed into a sum of two terms

(3)

where W(T) , T = ifL, is a deterministic function of T and ~iI )(t) is an errorterm which tends to zero as L -+ 00, at least in probabili ty. In this paper wedo not estimate all appea ring remainder terms . This will be done in a futurepublication . Our purpose here is to derive an equat ion for w(T). It is natural tocall w(T) the velocity of the piston in th e Eulerian regime.

It is clear that w(T) depends on the statist ics of the collisions which in turndepend on stat ist ical properties of the gases. We assume th at their stat isticsat t = 0 is fully determined by their first correlat ion funct ions which we writeas P"L(X,v;0), p!(X,v;0), corresponding respectively to the gas in the left andright part of the volume specified by XL(O) E (0, L) , the position of the pistonat t = O. Thi s means th at for any domain C c [0, XL(O)] X R I the numb erof particles (X ,v) E C has a Poisson distribution with parameter L2 fe dXdv·P"L (X ,v;0). T he analogous statement holds for particl es with in [Xd O), L]x R I .

We will assume th at P- and p+ are actually L-ind ependent functions of therescaled coordinate x = XIL,

p!(X,v;O) = Jr'f(x ,v;0). (4)

Another simplifying assumption which we make to avoid (over some initi al period of time) recollisions of a gas particle with the piston before the particle hit sthe walls of n at X = 0 or X = L is that

Jr'f(X,ViO) =0, if Ivl :::;vo or Ivl ~ VI (5)

where Vo and VI are some positive constants independent of L. Note th ere is nodispersion in the position XL (or velocity VL ) of the piston at t = O.

220 J . L. Lebowitz, J. Piasecki and Va. Sinai

§3. Time Evolution of the Piston's Velocity

We shall now derive equations for the deterministic component of the velocityof the piston. Denot e by Pt (X,v; t) the first correlat ion functions of the gasesat t ime t. They give the expected densit y of the particles at (X ,v). If C is againa domain in [0, Xdt)] x IR I (or in [X d t ), L] x IR I

) , and M(C) is the numb er ofpar ticles (x,v) E C then

(6)

where the first term is the expected numb er of par ticles in C and viI )(C) is a remainder term . While the distribution of the particles will not remain Poissonianduring the evolut ion, we expect that in typical situations vP\ C ) behaves as thesquar e root of the first term in (6). We shall also assume that Pt are definedat X = XL where they represent the density of par ticles in an immediate onesided neighborhood of the piston . These particles consist of two groups-thoseabout to collide with the piston and those which have just collided. When thevelocity of the piston is Vdt) , the density of the first group is given by

( ){

PL (X d t ),v; t) i f v > VL(t)PL v· t =

1 p!(XL(t) ,v;t) i f v < VL(t).(7)

It is clear that for any fixed L the Pt(x,v; t) as well as th e dist ribu tion ofthe piston 's position and velocity at tim e t will be connected via the BBGKYhierarchy to the higher order corre lat ion functions- correlations indu ced by thecollisions between th e gas particles and th e piston. We expect however thatunder the assumpt ions (4) and (5) equation (3) will be valid in the limit L -+ 00.

We shall now writ e down equat ions for a quanti ty WL(t) which is, for largeL , the leading order of the piston 's velocity VL(t) . In the limit L -+ 00 , WL(t) -+W(T) , and W(T) will sat isfy the limit ing form of the equat ions for WL(t) ,

Here a = 2mL2 j (ML + m) is independent of L, and we have defined

Qo(t) = Jsgn(v - Wdt))qdv; t)dv,

QI(t) = Jvsgn(v - Wdt))qL(v;t)dv,

Q2(t) = Jv2sgn(v - WL(t))qL(v ;t )dv.

(8)

(9a)

(9b)

(9c)

Scaling Dyn amics of a Massive Piston in an Ideal Gas 221

where qdv; t) is formally the same as pdv;t) defined in (7) with Vdt) replacedby Wdt)

( ) {PI,(X d t ),v;t ) i f v> Wd t )ai. v· t =

, p!(Xdt) ,v;t) if v < Wdt) ,

and Pt (X,v; t) are evolved according to the following equations: Inside the volume !h , away from the boundaries, the correlation functions Pt sat isfy simpledifferential equations of free dynamics

({) e ) '1' ( .) _{)t + v {)X PL X, v , t - O.

The obvious boundary conditions at the boundaries X = 0, X = L , are

pI,(0, Vi t) = pI,(0, - v i t) ,

p! (L ,v;t) = p! (L, - v;t)

The boundary condit ions at X = XL look similar,

pt(Xdt) ,v;t) = pt(Xdt) , 2WL(t) - v;t) ,

(10)

(lIa)

(lIb)

(12)

for v < Wdt) in pI, and v > Wdt) in Pi- In (12) we have used the approximation V i = - v + 2V for the velocity of a gas par ticle afte r a collision with thepiston which is valid to 0(1 / L 2 ) , see (2).

Since we shall discuss the dynamics of th e piston on the intervals of tim e tproportional to L the neglect of the terms proportional to L2 in (12) does notlead to significant distortions of the evolution. (N.B. the number of collisionsany particle has with the piston in the time interval (0, t) is fixed when L --t 00

while, during the same time, the piston has 0(L2t ) collisions.)To get the limit ing form of eqs. (8)-(12) , we assume that , if the initial con

ditions sati sfy (4), e.g. Pt(X,v ; 0) are independent of X , then also the functionsPt (X ,v;t) will be slowly changing funct ions of X and t. In other words weassume th at keeping t ]L = r fixed while letting L --t 00

X tPt(X, v;t) = 7rt(L ' v;I) --t 7r1 (x, v; r ). (13)

If this is so then Qo(t), Ql(t) , Q2(t), are also slowly changing functions of tso th at dQ;/dt = 0(1/ L), i = 1,2 ,3. We can then writ e Wdt) = W (oJ(t) +W2 )(t)/L , where W (O)(t) is the root of the quadrati c equation

(14)

For w2 J we then have the differenti al equation:

dW~:J (t) = -2a[Ql (t) _ Qo(t)W(oJ(t)]W2 J(t) + Qo(t)(:2J(t))2 - L:tW (oJ(t) .

(15)

222 J. L. Lebowitz, J. Pi asecki and Va. Sinai

The solut ion of (15) is bounded as long as Qo, Q1,Q2 and their first derivativesremain bounded. For this reason the scaling limit of Wdt) is W (O)(t) which canbe written in the rescaled t ime as W (O)(TL) = W(T) . Set ting

1 fTZXd t ) = Y(T) = y(O) + Jo w(s)ds (16)

(17)

and rewriting Eqs. (10)-(12) in terms of Jr and the rescaled coordina tes we obtaina set of coupled equations for Y(T) and Jr'f(X, V,T) which can be considered asthe Euler equations for our model.

§4. Derivation of Equation (8)

Take a t ime step ~ = ~L , ~ -+ 0, as L -+ 00 , and put t« = n~ . Denote byXn,Vn the coordina te and the velocity of the piston at time tn. We have from(1) and (2)

k"Vn +1 = (1 - tl"Vn + f L(l - f)k ,,-j . Vj

j =l

Here kn is the tota l numb er of collisions during the tim e interval (n~ , (n+ 1)~) ,

Vj is the velocity of the colliding particle before the j-th collision. Then

where

k"

Vn+1 = Vn(1 - f kn) + eL Vj +X~1) +X~2 )j =l

(18)

(19)

(20)k"

X};) = f L vj [(1 - f) k n-j - 1] .

j= l

Introduce the random variable N(tn , v , v + dv,~) equal to the numb er ofcollisions of the piston during the t ime interval (tn,tn + ~) = (tn, tn+l) withparticles having before t he collision the velocity belonging to the interval (V,v +dv). Then

N(tn , v , v+dv,~) = EN(tn ,v , v+dv,~)

= L2n(v, tn)~dv(1 + 0(1)).(21)

We shall find the expression for n(v, t) through th e first correlation functionsPL(X,v;t ). Consider a left particle with position X and velocity v. The conditions for the occurr ence of the above mention ed collision take the form

(22)

Scaling Dynami cs of a Massive Piston in an Ideal Gas 223

If we neglect the fluctuations of the velocity VL during the tim e inte rval (tn ,tn+l)and assume that the deterministic component Wdt) is constant dur ing (tn, tn+dwe can simplify (22) to get

(23)

The expected numb er of particles sati sfying (23) can be expressed through thefirst corre lat ion function and written as

(24)

Thus, for (v - WL(t)) > 0, n-(v, t) = P"L(XL,Vi t)(v - Wdt)) . In the same way

(25)

for (v - WL(t)) < O.Let Pi be the rectangle n!::l. :::; t :::; (n + 1)!::l., L~1 :::; v :::; t-!;~ on the (t ,v)

plane, / 1 is a constant, 1 < / 1 < 2. Then

(26)

where X~3) is a remainder. Further,

(27)

Here t5~1) is another remainde r. Thus

(28)

with another remainde r X~4). Here and below we do not est imate all appea ringremainders, correct ion terms, etc. We can also write

",,_i N(t .i: i + l !::l.) <~ v <",,(i+l)N(t .i. i + l !::l. ) (29)L..J L» n, L / l ' t.».: - L..J J - L..J L» n , i - . : t.».: .

i j =1 i

Replacing N by its mathematical expectation we can writ e

. . 1 k«

L2 "" _z_ n(_z_ t )_ .!::l. + X( 5) < "" vL..J L/1 L / 1' n L/1 n - L..J J

i j=1

2 L (i + 1) i 1 (6)< L --n(- t )·_ ·!::l.+X .- L/ 1 L/1 ' n L/1 ni

(30)

224 J. L. Leb owitz, J . Piasecki a nd Va. Sina i

We denote by X~5 ) ,X~6) other remainders. From the last inequalit ies we get

(31)

Therefore

W ((n + 1 )~ ) - W(n~) = -a Ro(tn)~ · W(n~) + a R1(tn) . ~ + X~8) (32)

with another remainder X~8) and Ro(t ) = J n( v , t )dv , R1(t ) = J vn(v, t )dv .Using the equat ions (24), (25), where n (v , t ) has been expressed th rough thefirst correlat ion functions we have

Ro(tn) = Jvsgn(v - W d tn ))qdv ;tn)dv-Wdtn)Jsgn(v - W d tn ))·qd v; tn)dv

=Ql(tn) - Wdtn) . QO(tn) ,

R 1(tn) = Jv-Wd tn» OV2p[; (X L , v ;tn)dv - Jv- Wd tn)<ov2pt (X L , v; tn)dv

- Wdtn) J v sgn(v - Wdtn))qd v ;tn)dt= Q2(tn) - W d tn)Ql (tn).

Finally we get

Thus we see that the last expression is the difference approximation of (8).


As was stated above th e est imation of all appearing remainders will be donein anot her publication. Here we would like to mention several problems arisingfrom this paper.

1. Does the piston under the Euler dynamics converge to a stationary state?Our results do not indicat e immediat ely the presence of such convergence. Itis not difficult to give examples where the piston makes several oscillations.In some sense the dynamics in these cases is close to the dynamics of massivepar ticles in the field of finite ly many light particles (see [111) .

2. It would be interestin g to replace the ideal gas by the gas of interactin gparticles like the Knudsen gas or by Lorentz models with the length of free pathproportional to L.

Scaling Dyn ami cs of a Massive Piston in an Ideal Gas

Appendix

225

(A I )

(A3)

At the end of the Introdu ction we made a remark that the derivation of theautonomous coupled equations (8)-(12) amounted to an effective t runcation ofthe BBGKY hierarchy. Here we show that these equat ions follow indeed from thehierarchy if one assumes that the piston moves along a deterministic t rajectory.

The following notati on will be used:h (XL,VL; t) = probability density for finding the piston at point XL with

velocity VL at tim e t .

f~ 'f ) (XL ,VL,X ,v; t) = joint density at tim e t of two-particle states : thepiston at (XL ,VL), the gas par ticle to the left (to the right ) of the piston at(X,v).

The state of the piston h (XL, VL; t) evolves in the course of time owing tofree motion and to binary collisions with the gas particles. These two mechanisms are rigorously taken into account by the first equation of the BBGKYhierarchy

(:t + VL 8~L) h (X L,VL;t) = JdvlVL - vi

x {B(v - VL)f~+) (XL ' VI ,X L,V'; t) - B(VL - v)f~+)(XL ' VL,X L,V;t )

+B(VL - v) f~- )(XL ' VI ,XL,V'; t) - B(v - VL )f~- ) (XL ' VL ,XL ,V;tn

Here the primed velocities VI, v' are related to VL,V by the elast ic collisionlaws (1) and (2).

Let 1li (VL) be a funct ion of t he piston velocity. From (AI) we find that therate of change of the mean value

is given by

:t < Ili >t= L2JdVJdX JdvlV - vl[lli (ev + (1 - E)V ) - 1li(V )]

x [B(V - v) f~+)(X, V,X ,Vit) +B(v - V)f~-)(X, V,X ,v; t)] (A2)

In part icular, choosing 1li (V ) = V we find

ddt < V » i =

= L2EJdVJdX Jdv(V - v)2[B(v - V) fJ-)( X ,V, X ,V; t) - B(V - v)

x f~+ ) (X,v,X, v ;t ) ]

226 J . L. Lebowitz, J . Piasecki and Ya. Sinai

In order to derive the equa t ion describing t he det erminist ic component inthe motion of t he piston we assume the probability density h (X ,V i t ) to becentered on a single traj ectory

h (X,V ;t ) = 8[X - X dt)]8[V - Wdt)]

Here 8( .) is the Dirac dist rib ution. Consequently, we also pu t

(A4)

JJ'f \ X ,V,X, Vi t) = 8[X - X d t )]8[V - Wdt)]p('f )(X L,v; t ), (A5)

where p('f )(X L, Vi t) represent the densities of the gas par ti cles on both sides oft he sur face of the pist on . Then , inserting (A4) and (A5) into (A3) we find theevolut ion equa t ion for the deterministic velocity Wdt) of the piston. It reads

(A6)

It can be readily checked that equat ion (A6) is identi cal with equation (8)of Sect ion 3. It has been obtained here from the hierarchy equation (AI) byassuming the deterministic motion of the piston along a given trajectory.

The st atist ical fluctuations around t his trajecto ry have been neglected . Theconsistency of t his motio n wit h the changes in the states of the sur rounding twovolum es of t he idea l gas requires the equations (10)-( 12) to be satisfied , leadingto a closed syste m of coupled equations.

References

[1] H. Spohn, Large Scale Dynamics ofI nteracting Part icles (Springer, I3erlin, 1991).[2] L.Bunimovich and Y. Sinai, Commun. Math. Phys. 78 , 479 (1981) ; L. A. Buni

movich, Y. Sinai and N. Chernov, Russ. Math . Surveys 45 , 105 (1990); J. L.Lebowitz and H. Spohn, J . Stat. Phys. 28 , 539 (1982); 29 , 39 (1982).

[3) O. Lanford, Time Evolut ion of Large Classical Systems, J . Moser, ed., Lectur eNotes in Physics, Vol. 38 (Springer, Berlin, 1975) pp. 1-111 ; Physica A 106, 70(1981); W. Brown and K. Hepp, Comm. Math. Phys. 56, 101 (1977).

14] J. Piasecki, Ya.G. Sinai, A Model of Non-Equilibrium Stat ist ical Mechanics, Proceedings from NASI Dynamics: Models and Kinetic Methods for NonequilibriumMany-Body Systems, Leiden, 1998, Kluwer (2000).

15] J . L. Lebowitz, Stationary Nonequilibrium Gibbsian Ensembles, Physical Reviews, 114 , 1192 (1959).

[61 E. Lieb, Some Problems in Stat ist ical Mechanics that I Would Like to See Solved,Physica A 263 , 491 (1999).

[7] Ch. Gruber, Thermodynamics of Systems with Internal Adiabatic Const raints:Time Evolution of the Adiabat ic Piston, Eur.J .Phys. 20, 259 (1999).

Scaling Dyn am ics of a Massive Piston in an Ideal Gas 227

[81 J . Piasecki, Ch. Gru ber , From t he Adiabat ic P iston to Macroscopic Motion Induced by Flu ctuat ions, Physica A 265 , 463 (1999); Ch. Gruber, J . Piasecki,Stationary Motion of th e Adiabati c P iston , Physica A 268, 412 (1999).

[91 Ch. Gruber, L. Frachebo urg, On t he Adiabat ic Properties of a Sto chastic Adiabatic Wall: Evolut ion , Stationar y Non-Equilibr ium, and Equ ilibri um St at es,Physica A 272, 392 (1999).

[10) E. Kestemont , C. Van den Broeck, M. Malek Mansour , T he "Adiabat ic" P iston :And Yet It Moves, Euro phys. Let t. 49 , 143 (2000).

[l l ] Ya.G. Sina i, Dynami cs of Massive Par ticl e Surround ed by Light Par ticles, T heoretical and Math ematical P hysics (in Russian), 121 , Nl , I lO (1999).

II. Physics

Kinetic Theory Estimates for theKolmogorov-Sinai Entropy, and the

Largest Lyapunov Exponents for Dilute,Hard Ball Gases and for Dilute, Random

Lorentz Gases

R. van Zan, H. van Beijeren and J. R. Dorfman

Conte nts

§l. Introduction . . . . . . . . . . . . . . . . . . .. . . . . . 233§2. The Dynamics of Hard Ball Systems . . . . . . . 237§3. Est imates of the Kolmogorov-Sinai Entropy

for a Dilute Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 2403.1 The KS Entropy as an Ensemble Average . . . . . . . . . . 2413.2 Informal Evaluati on of the KS Entropy at Low Densities . 2433.3 Toward the Formal Evaluation

of the KS Entropy at Low Densities . . . . . . . . . 247§4. The Largest Lyapunov Exponent

for the Hard Ball Syst em at Low Density . . . . . 2514.1 Kinet ic T heory Approach . . . . . . . . . . . ... . . 2514.2 Compa rison with Simulatio ns . . . . . . . . . . 260

§5. T he Dilute, Random Hard Ball Lorentz Gas .. . .. . . . . . . . . . 2645.1 Informal Calcu lat ion of the KS Entropy

and Lyapu nov Exponents for the Dilute, Random Lorentz Gas . 2675.2 Formal Kinet ic Theory for the Low Density Lorentz Gas 271

§6. Conclusions and Open Problems. . 274References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

232 R. van Zon , H. van Beijeren and J. R. Dorfm an

A bstract. The kinetic theory of gases provides methods for calculat ing Lyapunov exponents and ot her quantities, such as Kolmogorov-Sinai ent ropies, thatcharacterize the chaotic behavior of hard ball gases . Here we illust rate the useof these methods for calculati ng the Kolmogorov-Sinai entropy, and the largestpositive Lyapunov exponent , for dilute hard ball gases in equilibrium. T he calculation of the largest Lyapunov exponent makes interesting connections withthe theory of prop agati on of hydrodynamic fronts. Calculat ions are also presented for the Lyapunov spectrum of dilute, random Lorentz gases in two andthree dimensions, which are considerably simpler than the corresponding calculat ions for hard ball gases. The art icle concludes with a brief discussion of someinteresting open prob lems.

Kinetic Theory Estimates

§1. Introduction

233

The purpose of this art icle is to demonstrat e that familiar methods from thekinet ic theory of gases can be extended in order to provide good est imates forthe chaot ic prop erties of dilute, hard ball gases 1. The kinetic theory of gaseshas a long histo ry, extending over a period of a cent ury and a half, and is responsible for many central insights into , and results for the prop erties of gases,both in and out of thermodynamic equilibrium [11 . Strictly speaking, there aretwo familiar versions of kinetic theory, an informal version and a formal version.The informal version is based upon very elementary considera t ions of the collisions suffered by molecules in a gas, and upon elementary prob abilist ic noti onsregardi ng th e velocity and free path distributions of the molecules. In the hand sof Maxwell, Boltzmann, and others, the informal version of kinet ic theory led tosuch important predictions as the independence of the viscosity of a gas on itsdensity at low densities, and to qualit at ive result s for the equilibrium thermodynamic prop erti es, the transpor t coefficients, and the structure of microscopicboundary layers in a dilute gas. The more formal theory is also due to Maxwelland Boltzmann , and may be said to have had its beginning with the development of th e Boltzman n transpor t equation in 1872 [2] . At that t ime Boltzmannobtained, by heuristic arguments, an equat ion for the time dependence of th espatial and velocity distribution function for particles in the gas. This equat ionprovided a formal foundation for the informal methods of kinet ic theory. It leadsdirectly to th e Maxwell-Boltzmann velocity distribution for the gas in equilibrium . For non-equilibrium systems, the Boltzmann equation leads to a versionof the Second Law of Thermodynamics (the Boltzmann H-theorem) , as well asto the Navier-Stokes equat ions of fluid dynamics, with explicit expressions forthe transport coefficients in te rms of the intermolecular potentials governingthe interactions between the par ticles in the gas [31 . It is not an exaggerat ionto state that the kinet ic theory of gases was one of the grea t successes of nineteenth cent ury physics. Even now, the Boltzmann equation remains one of themain cornerstones of our understandi ng of nonequilibrium processes in fluid aswell as solid systems, both classical and quantum mechanical. It continues tobe a subject of investigat ion in both the mathematical and physical literature,and its predictions often serve as a way of distin guishing different molecularmodels employed to calculate gas properties. However, there is still not a rigorous derivation of the Boltzmann equation that demonstrat es its validity overthe long tim e scales typically used in applicat ions. Nevertheless, the Boltzmannequati on has been generalized to higher density, and in so far as they arc available, the predict ions of the genera lized Boltzmann equation are in accord withexperiments and with numerical simulations of the prop ert ies of moderat elydense gases [4] .

1 We will use the term hard ball to denote hard core systems in any number ofdimensions, rather than using the term hard disk for two dimensional, hard coresystems, etc.

234 R. van Zan, H. van Beijeren and J . R. Dorfman

In spite of the many successes of the Boltzmann equat ion it cannot be a stri ctconsequence of mechanics, since it is not time reversal invariant , as are the equations of mechanics, and it does not exhibit other mechanical phenomena, suchas Poincare recurrences [51. Boltzmann realized that the equat ion has to beunderstood as representing the typical behavior of a gas, as sampled from anensemble of similarly prepared gases, rather than the exact behavior of a particular laboratory gas. He also understood that the fluctuations about the typ icalbehavior should be very small, and not important for laboratory experiments.To support his arguments , he developed the foundations of statist ical mechanics, introducing what we now call the micro-canonical ensemble. This ensembleis described by giving all systems in it a fixed total energy, and then assumingthat the probability of finding a system in a certain small region on the constantenergy surface is proportional to the dynamically invariant measure of the smallregion, given by the Lebesgue measure of the region divided by the magnitudeof the grad ient of the Hamiltonian function at the point of interest on the surface. As we know, this micro-canonic al ensemble forms the start ing point forall statist ical calculations of the thermodynamic prop erti es of fluid and manyother systems.

In his attempt to provide a mechanical argument for the effectiveness of themicro-canonic al ensemble for calculat ing the thermodynamic prop erties of fluids, Boltzmann formulat ed the ergodic hypothesis, which, in its modern form,st ates that the time average of any Lebesgue-integrable dynamic quantity of anisolat ed, many particle system approaches the ensemble average of the quantity,taken with respect to the micro-canonic al ensemble [61. Thi s hypothesis is thesubj ect of several ar ticles in this Volume, but its value for the foundations ofstati sti cal thermodynamics is often question ed 17] . T he question able points canbe summarized in a few items: (1) The ergodic hypothesis applies to classicalsystems while nature is fundamentally quantum mechanical. (2) Even grant ingthe approximat e validity of classical mechani cs for many purposes, no laborator y system is truly isolat ed from the rest of the universe . Instead , laboratorysyst ems are const antly perturbed by out side influences, and these sources of randomness, together with the simple laws of large numb ers for systems with manydegrees of freedom, may be responsible for the ut ility of the micro-canoni calensemble for the calculation of equilibrium prop erties. (3) Even granting theability to isolat e a laboratory system from the rest of the universe, the t imeit would take for a system's phase space trajectory to sample all of the available phase space on the constant energy surface is just too long for ergodicbehavior to be a physically reasonable explanat ion for the effectiveness of themicro-canoni cal ensemble. It seems more likely that th e equilibrium behaviorof a system of many particles depends on a numb er of these factors , includingperturbations from the environment , the fact that thermodynamic syst ems havea large numb er of degrees of freedom , and the fact that th e physically relevantquantities are projections of phase space quantities onto a subspace of a fewdimensions. Thus even though the time scale for ergodic behavior on the full

Kinetic Theory Estimates 235

phase space may be unreasonably long, the pro jected behavior on relevant subspaces may not take very much time for the establishment of equilibrium valuesof th e measurable quantit ies, such as pressure, temperature , etc . Pendin g thefurther clarification of these and similar issues, it seems fair to say that thecomplete underst anding of the reasons for the validity of the micro-canonicalensemble as the basic understructure for statist ical thermodynamics has yet tobe achieved.

In addit ion to resolving the various issues needed to complete our understanding of the equilibrium behavior of fluids, we would also like to underst andthe dynamics of the appro ach to thermodynamic equilibrium on as deep a levelas possible. Such an understanding would enable us to provide a just ificati on ofthe Boltzmann equation and its many generalizat ions, as well as of the successfuluse of hydrodynamic or stochast ic equati ons in nonequilibrium situations. Thecounterpart of Boltzmann 's ergodic hypothesis for nonequilibrium phenomenais the assumpt ion that a dynamical system should be mixing, in the sense ofGibbs. Th at is, given some initial distr ibution of points on the constant energysurface in phase space, in a region of positive measure, a syst em is mixing ifthe distribution of the points eventually becomes uniform over the energy surface, with respect to the invariant measure 181 . If one can prove that an isolat eddynamical system with a large number of degrees of freedom is mixing, thenone can show th at the phase space distr ibution funct ion for the system will approach an equilibrium distribution at long t imes, and consequent ly, quanti tiesaveraged with respect to th is distribution function will approach their equilibrium values. Needless to say, the same concerns listed above suggest that theapproach to equilibrium may be a very complicate d affair with a number ofpossible factors act ing in concert or individually as circumstances require. Onemight also ask why a phase space distribution funct ion is needed at all, sincea laboratory system corresponds to a point in phase space at any given tim e,and not to a distribution of points in phase space. T he usual argument given instatistical mechanics texts is that it is easier to describe the average behavior ofan ensemble of points than to solve the complete set of the equat ions of motionof a single system, and to draw conclusions from such a solut ion. Of course,computer simulat ions of fluid systems are often attempts to solve the equationsof motion for an individual system, but they too are influenced by noise in theform of round-off errors, and so do not really describe an isolated dynamicalsystem, unless one resort s to certain lat tice-type models that can be treatedby integer arithmet ic on the computer. Our hope, often unst at ed, is that theproperties we explore using the methods of st at ist ical mechanics are somehowtypical of the behavior of an individual system in the laborato ry, even if weknow th at this canno t be st rictly t rue. We hope, and occasionally can prove,that the deviations from typical behavior are small.

In any case, it is clearly important to know what role the dynamics of thefluid system might play in the approach to equilibrium. Certainly the equilibrium and nonequilibrium prop erties of the syste m are sensitive to the underlying

236 R. van Zan , H. van Beijeren and J . R. Dorfman

molecular structure of the fluid and to the interactions between the par t icles ofwhich it is composed [9, 101. Our experi ence with the Boltzmann equation alsoassures us that the role of molecular collisions cannot be underestimat ed, evenif we are not ent irely certain why the Boltzmann equation works for an individual laboratory system. Therefore, when faced with a plausible model of a fluid,one would like to know if the dynamics of the model is ergodic, mixing, K, orBernoulli. It is of considerable interest to establish th e dynamical properti es ofa large isolat ed system of par ticles as the st arting point for our invest igation ofthe foundations of nonequilibrium statistical mechanics, and then to look at theconsequences of: (a) exte rnal noise on the system, and (b) the restri ction of ourinterest to only a small class of functions of the dynamical variables needed forphysical applicat ions.

As a step in this direction we show here that kinetic theory can be usedto demonstrate (not prove) that isolated hard ball systems are chaotic dynamical systems [111 . We will show, in fact , that , at low densities at least , hardball systems have a positive Kolmogorov-Sinai ent ropy, which we can est imate,and that the largest positive Lyapunov exponent can also be est imated. Theseestimates, in fact , are in good agreement with the results of numerical simulations. What these est imates are unable to tell us is whether or not the systemsare indeed ergodic , mixing, K, or Bernoulli , since we cannot use these kinetictheory techniques to show that the phase space consists of only one invariantregion and not a countable number of them. In fact there is some evidence th atif the intermolecular potential is not discontinuous but smoot her than a hardsphere potential, then there may be some elliptic islands of positive measure inthe phase space [121 . Strictly speaking, the ergodic hypothesis is not valid forsuch potenti als. It remains to be seen whether or not thi s phenomenon is ultimat ely of some importance for stat ist ical mechanics, especially for systems withlarge numbers of particles, and at temperatures where quantum effects may beneglected.

As a by-product of the analysis given here we will also be able to calculate thelargest Lyapunov exponents and Kolmogorov-Sinai Entropy for a dilute Lorentzgas with one moving moving particle in an array of randoml y placed (but nonoverlapping) fixed hard ball scat terers [13, 14]. This system is much easier toanalyze than a gas where all the particles are in mot ion, and was the first typeof hard ball syst em whose chaot ic dynamics were studied in detail , either byrigorous methods or by kinetic theory.

The plan of thi s art icle is as follows: In Section II we will set up the equationsof motion for hard ball systems that will enable us to analyze the separat ionof initially close (infinitesimally close, actually) trajectories in phase space. Thedynamics of th e separation of trajectories in phase space is, of course, an essential ingredient in analyzing Lyapunov exponents and Kolmogorov-Sinai (KS)entropies. In Section III we will apply these results to a calculat ion of the KSentropy of a hard ball gas using informal kineti c theory rather than a formalBoltzmann equation approach. Thi s informal method gives the leading density


behavior of the KS ent ropy, but more formal methods are needed to go further.We will outli ne the more formal method based upon an extension of the Boltzmann equation, but we will not go too deeply into its solution, since it rapidlybecomes very technical. In Section IV we outline a method for est imating thelargest Lyapunov exponent for a dilute hard ball gas using a mean field theorybased upon the Bolt zmann equat ion [15, 16]. In Section V we apply these methods, both informal and formal, to the calculat ion of the KS entropy and largestLyapunov exponents for a dilute Lorentz gas with fixed hard ball scatterers.We conclude in Section VI with remarks and a discussion of outst anding openproblems.

§2. The Dynamics of Hard Ball Systems

In this sect ion we will present a method due to Dellago, Posch and Hoover [17] fordescribing the dynamical behavior of infinitesimally close trajectories in phasespace for hard ball systems. We begin with a consideration of a system of Nidenti cal hard balls in d dimensions , each of mass m , and diameter a. Their posit ions and velocities are denoted by Ti and Vi , respectively, where i = 1,2 , ... Nlab els the part icles. For simplicity we can imagine th at the particles are allplaced in some cubical volume V = Ld and that periodic boundary conditionsare applied at the faces of th e cube. The dynamics consists of periods of freemotion of the par ticles separat ed by instantaneous binary collisions betweensome pair of particles. During free motion, the equations of the system are

(1)

At the inst ant of collision between particles i and i .say, there is an instantaneouschange in the velocities. It is convenient to write the dynamics in terms of thecenter of mass motion (Rij ,V;j ) and relative motion (fij , Vij),

Rij = (fi + Tj)/2, V; j = (Vi + vj) /2,

fij = fi - Tj, Vij = Vi - 0 ·

Th e change can be described by the equations

where the matrix M, describes a specular reflection on a plane with normal e ,i.e., the unit vector in the direct ion from par ticle j to i at the inst ant of the(i, j) collision,

Ma- == 1 - 2;:';:' . (2)


Nondotted products of vectors are dyadi c products and 1 is the identi ty matri x.In te rms of the individual particle velocit ies, the dynamics is describ ed by

(3)

These equat ions (plus the dynamics at the boundaries) are all th at one needsin order to determine the trajectory of the system in phase space. However ,in order to determin e Lyapunov exponents and other chaot ic prop ert ies of thesystem, we need to examine two infinit esimally close t rajectories, and obtainthe equat ions that govern their rat e of separa t ion.

Equations for the rate of separation of two phase space t ra jectories of hardball systems have been developed by Sinai using differenti al geomet ry 19, 181.This leads to an expression for the rate of separation in terms of an operatorthat is expressed as a cont inued fraction. Here we adopt a somewhat differentbut equivalent approach that is nicely suited to kineti c theory calculations.To obt ain the equations we need, we consider two infinitesimally close phasespace points, r (the reference point) and r + I5r (the adjacent point) , given by(i1 ,iit,is ,V2, .. . , rN,VN) and by (i1 + 15r1 ' V1 + 15v1 ,is + l5iS ,V2+ 15v2 , . .. , rN+I5rN' VN + I5vN), respectively. The 2N infinitesim al deviation vectors 15ri ' I5videscribe the displacement in phase space between th e two trajectories. Thevelocity deviation vectors are not all independent since we will restrict theirvalues by requiring th at the total momentum and tot al kinetic energy of thetwo tr ajectori es be the same. That is,

N

LI5Vi = 0,i = 1

N

L Vi ' MJi = O.i=1

(4)

where, in the energy equat ion, we have neglected second order terms in thevelocity deviations. We will not use these equat ions in any serious way sincewe will be interested in the limit of large N , in which case they are not veryimportant. In the case of the Lorent z gas, to be discussed in a lat er section, theconservat ion of momentum equation is not relevant , but we will requir e thatboth of the two trajectories have the same energy, so that the velocity deviationvector is orthogonal to the velocity itself.

Between collisions on th e displaced trajectory, the deviations sat isfy

(5)

The treatment of the effect of the collisions on the deviation vectors is morecomplicated. One assumes that the two t rajectories are so close together that thesame sequence of binary collisions takes place on each t raj ectory over arbitrarilylong times. Let us suppose th at we consider a collision between par ticle i and jon each trajectory. Now, since the trajectories are slightly displaced, the (i ,j)collision will take place at slightl y different times on each t rajectory. Thi s slight

Kinetic Theory Est imates 239

(in fact infinitesimal) time displacement must be included in the analysis of th edynam ical behavior of the deviation vectors at a collision.

The dynamics of the deviations can be considered separately for the cente rof-mass coordinates and the relat ive coordinates. The center-of mass coordinatesbehave as in a free flight , so that

OR;j = stu; OV;j = oV; j.

For the relative coordin at es, we consider the relative coordinates of two infinitesimally close trajectories, fij ,Vij and T;j ' Vi{

reference t rajectory(collision at t = 0)

t < 0:Vij(t) constantTij(t) = Vij t + aa

t > 0:Vij(t) = M , . Vijfij(t) = M a- . Vijt + aa

adjacent tra jectory(collision at t = M)

t < M:Vij (t) constantT;j( t) = Vij(t - M) + ao"

t > M:Vij (t) = M a- - . VijT;j( t) = M a-- . Vij (t - ot) + air"

Th e transformations at a collision in terms of the deviation vectors are foundwith the aid of

ofij = T'ij(O-) - fij (O- ),

8~j = T'ij (ot+) - Tij (Ot+),

OVij = Vij(O-) - Vij (O- ),

oV:j = V;j (ot+) - vij( M+ ),

where the superscripts + and - indicate immediat ely after and immediatelybefore the collision, respectively. We should use Eq. (5) in between collisions,i.e., from the last collision up to t = 0-. Then we use the collision rule, tobe derived in this sect ion, that links or: and o~ to ofi and OVi' Eq. (5) isused again from t = 8t+, start ing with or: and o~ . It is also possible to uscEq. (5) from t = 0+, as if the values of or:j and O~j in the above equat ionwere valid at t = 0+. T he difference is an erroneous addit ional MO~j for ofij ,when we apply Eq. (5) from t = 0+, but this addit ional term is quadratic in thedeviations and therefore negligible. In this way, the collision may also be viewedas instantaneous for the deviati on vectors.

We can now write

where Oa = ir" - a , and in the last equality we neglect terms quadratic in thedeviations. Because /0"1= la*1 = 1, we have Oa .a = O. Taking the inner productwith aof the above formula gives

a ·orijM = - -.- _- ,

(J" Vij


Substitution of these result s into 8r:j = ir:« - ira - M, . Vij 8t and 8V:jMa- ' . Vij - M, . Vij, yields

8r';j = M; . 8fij,

811ij = -2Qa-(i ,j) ·8fi j + M, · 8Vij ,

where Qa-(i , j) is the matrix

Q. ( . .) _ [(0-. vij )l + o-Vijj· [(0-. vij )l- Vijo-]o Z, J - ( . ~) .

a o . Vij

In terms of the individual part icle deviat ions , the collision dynamics read s

8r'; = 8ri - (M'ij . 0-)0- ,

8fj = 8fj + (8rij . 0- )0-,

81! = 8v· - (8v · .. 0-)0- - Q . (i J') . 8r· ., " J (j , 'J'81Jj = 8vj + (8Vij .0-)0- + Q a-(i , j) · 8fij .

(6)

(7)

Eqs. (5) , (6) and (7) are the dynamic al equ ations that govern the t ime dependence of the deviation vectors, {8ri ,8v;} . T hey have to be solved togetherwith th e equa t ions for the {fi ,Vi } in order to have a complete ly determinedsyste m. That is, in order to follow the deviation vectors in t ime, one needs toknow when, where, and with what velocit ies the various collisions take place inthe gas.

In the next sect ion, we will use th ese equa t ions to provide a first est ima te ofth e Kolmogorov-Sinai ent ropy for a dilute gas of hard balls in d dim ensions.

§3. Estimates of the Kolmogorov-Sinai Entropyfor a Dilute Gas

We consider th e hard ball syste m from the previous section when th e gas isdilute, i.e. nad « 1, with n t he density NjV , and it is in equilibr ium withno external forces act ing on it. Since the hard ball system is a conservative,Hamiltonian syste m, one can easily show that all nonzero Lyapunov exponentsare paired with a corresponding exponent of identical magnitude but of oppositesign [9, 10, 19]. Obviously there are an equal number of posi tive and negativeLyapunov exponents , and the sum of all t he Lyapunov exponents must be equalto zero. For the system we consider here, the Kolmogorov-Sinai (KS) ent ropy isequa l to th e sum of the positive Lyapunov exponents , by Pesin's theorem [20] .We will compute the KS ent ropy per par ticle in th e thermodyna mic limit , fora dilute hard ball gas, using methods of kinet ic theory [11 , 21].

To carry out this calcul ation we will use the fact th at when an infinitesimally small 2Nd-dimensional volume in phase phase is proj ected onto the N ddimensional subspace corresponding to the velocity directions, the volume of


(8)

this projection must grow exponent ially with time t , in the long t ime asymptotic limit , with an exponent that is the sum of all of the positive Lyapunovexponents. That is, when we denote this proj ected volume element by JVv(t) ,as t -t 00 ,

;~:~~~ = exp {t A~O Ai} '

The same result also holds for another small volume element , denoted by JVr(t) ,that is the projection onto the N d-dimensional subspace corresponding to theposition direct ions'', The advantage of using an element in velocity space residesin the fact that the velocity deviat ion vectors do not change during the tim eintervals between collisions in the gas, but only at the inst ant s of collisions. WewiII make use of this fact short ly.

Our first step will be to obtain a genera l formula for the KS ent ropy ofa hard ball system, which, in principle, should describe the complete densitydependence of this quantity. Then we wiII apply this result to the low densitycase, using both informal and formal kinetic theory methods.

3.1 The KS Entropy as an Ensemble Average. Our object here isto express the KS ent ropy for a hard ball system as an equilibrium ensembleaverage of an appropriate microscopic quant ity, which in turn can be evaluatedby standard methods of statistical mechanics. To do this we rewrite Eq. (8) asa tim e average of a dynamical quanti ty as

. I JVv(t)"" Ai = lim - In J:V ( )'~ t->oo t u v 0Ai >O. Ilt

d JVv(r )= lim - dr -d In J:V ( )'

t->oo t o r u v 0

= / !£ In JVv(r) )\ dr JVv(O) ,

(9)

where the angular brackets denote an average over an appropriate equilibriumensemble to be specified further on. Here we have assumed that the hard ballsyst em under considera t ion is ergodic so that long time averages may be replacedby ensemble averages . Now we can use elementary kinetic theory arguments togive a somewhat more explicit form to the ensemble average appearing in Eq.(9). Since the volume element in velocity space does not change during the

2 It is wort h point ing out that while bot h §Vv an d JVr grow exponent ially in time,t heir combined volume, i.e, t he origina l 2Nd-d imensional volume, stays constant .This seemingly paradoxical statement can be und erstood by realizing that almostall projections of t he 2Nd-dimensiona l volume onto N d-dimensional subpaces willgrow exponent ially in t ime with an exponent given by the sum of t he largest N dLyapunov exponents.

242 R. van Zan , H. van Beijeren and J . R. Dorfm an

t ime between any two binary collisions, and since the binary collisions in a hardball gas are instantaneous, the ensemble average of the time derivative may bewritten as

Here the step function, 8 (x ), is equal to unity for x > 0, and zero otherwise.The prim e on the velocity space volume denotes its value immediately afterthe collision between particles i and i . while the unprimed quanti ty is its valueimmediat ely before collision. In the derivation of Eq. (10) we consider the rateat which binary collisions take place in the gas , and then calculate the changein velocity space volumes at each collision. Thus, we have integrated over allallowed values of the collision vector a for a collision between particles i andj with relative velocity Vi j, and included, by means of a delta function, thecondition that the two particles must be separated by a dist ance a at collision.The other factors in Eq. (10) take into account the rate at which collisionsbetween two particles take place in the gas. A more formal derivation in termsof binary collision operators is easy to construct, but not necessary for ourpurpose here [21] .

Suppose, for the moment , th at the N d-dimensional velocity deviation vectorimmediat ely after the (i, j ) collision, (8vl ,8v2, .. . ,817; , ... ,8Vj,... ,8vn ) is related to the velocity deviation vector immediately before collision through thematrix equation

so, 8Vl8v2 8v2

817; 8Vi= A i j · (11)

8iT'· se,J

se; se;It then would follow that due to the occurrence of the (i ,j) collision

8V'8V: = Idet A i j l ,

and the sum of the positive Lyapunov exponents would be given by

(12)

L Ai = N(~ - 1) (ad-

1Jda8(- V12 . a) lv12 . a 18(f'12 - aa) In Idet Ad) .Ai >O

(13)

Here we have assumed th at the ensemble average is symmet ric in the particleindices, and chosen a parti cular pair of particles, denoted by particle indices1,2. We now must argue for the validity of Eq. (11) and then calculate thedetermin ant of the matrix A 12 .

If we examine Eqs. (7) , we see th at we can obt ain an equation of the form ofEq. (11) if we can relat e the spatial deviation vectors <S f; and <Sfj immediatelybefore an (i ,j) collision to the velocity devia tion vectors immediat ely beforecollision. Such a relation must be linear since we are keeping te rms only tofirst order in the deviations. We therefore make the Ansatz that , in general, thespatial deviation vectors are relat ed to the velocity deviation vectors througha set of d x d mat rices, called radius of curvature (ROC) matrices (even thoughthey have the dimension of time), Pkl' via

<Srk(t) = L Pkl . Ml1.I

Then some simple algebra shows that

(14)

det A12 = det [Ma-(1 ,2) - Qa-(1 ,2) . (Pll + P22 - P12 - P21)] ' (15)

We can carry the calculation of the KS entropy one step further now by insertingEq. (15) into the expression for the KS entropy, Eq. (13), to obtain

L ).i=~ad-1JdX1 dX2dpll dp12dp21 dp22da8( -V12 . a) IV12 . al<S(r12 - aa)>. , >0

XIn 1 det [Ma-(1 ,2) - Qa-(1 , 2) . (Pll + P22 - P12 - P21)] I

(16)

where Xi = (f; ,Vi) , dXi = df; dVi , and we have defined a new pair distributionfunction, .r2(X1 , X2,Pll ' P1 2' P21 ' P22) by

.r2(X1 , X2 , Pll ' P12' P21 ' P22)

= N(N -1) JdX3···dx N II'dP i j F N(x1, x2, ..., xN,Pll "" ,PN N)' (17)

Here :FN is the normalized ensemble distribution function for the posit ions andvelocities of all of the particles and for all of the ROC matrices. The primeon the product means that integrat ions are not carried out over the four ROCmatri ces whose indices are 11,12,21 and 22.

To proceed further , we will need to det ermine the pair distribution function,F2 , and then evaluate the averages needed for the KS ent ropy. We will do thisusing both informal and formal kinetic theory arguments . We mention here thatso far we have made no low density approximat ions, so that Eq. (16) is a generalexpression for the KS entropy of a hard ball system, given the validity of theAnsatz, Eq. (14).

244 R. van Zon, H.van Beijeren and J . R. Dorfman

3.2 Informal Evaluation of the KS Entropy at Low D ensities. A verysimp le and useful approximation to the KS entropy [n ] for a dilut e hard ba llgas can be obtained by noticing that the spatial dev iation vectors for particles1 and 2, immediately before their mutua l collision can be written in the form

8f;. = ti8vi + 8ri(7i,0) for i = 1, 2. (18)

Here 7i,0 is the t ime of the prev ious collision of par ticle i with some ot her par ticlein the gas , and t, is the time interval between that previo us collision and th e(1,2) collision. The quant ity, 8f;. (7i,0) is the spatial deviation vector for particlei immediately afte r the previous collision.

The following argument is correct for dispersing billiard systems such asthe Lorentz gas, and gives the leadin g order in th e density for the hard ballgas, system which is only semi-dispersing 3 . Comparing the two te rms on theright-hand side of Eq. (18), one finds th at their rati o scales, on the average , asthe mean free time which is proport ional to the inverse first power of the gasdens ity. T herefore it seems reasonable that for low densities, the terms in Eq.(18) proport ional to the t, should be dominant and that the terms 8ri can beneglected . If we do this, we can use the approximations

P 12 = P 21 = 0

Pu = til for i = 1, 2.

If we insert this approximation in Eq, (15), we find that

(19)

(20)

T he determinant can be evaluated using the exp licit expressions for M and Qin Eqs. (2) and (6) . For two dimensional systems we find

Idet [M a- (I, 2) - Qa- (I , 2)(t1 + t2)]I= 1 + jV!2l(t l ~ t2), (21)a cos

where cos¢ = 10-· v!2l/ lv!2l, with - 7f/2 :::; ¢:::; 7f /2 . For the three dimensionalcase we find

Idet [Ma- (I ,2) - Qa- (I, 2)(t1 + t2)]I

= 1 + Iv!2l(t1 + t2) (1 + cos2 ¢) + ( IV1 2!(t1 + t2)) 2acos ¢ a

(22)

3 A dispersing billiard takes any infinitesimal, parallel pencil of trajectories in configuration space before a collision into a defocusing pencil in all directions. A semidispersing billiard leaves rays st ill parallel after collision in at least one planethrough the pencil. A simple dispersing billiard is the outer surface of a sphere,and a simple semi-dispersing billiard is the outer surface of a circular cylinder.

We can now inser t these expressions into the right-hand side of Eq. (16) toobtain explicit expressions for the KS entropy per particle of dilute ha rd ballgases, provided we can specify the pair distribution funct ion , :F2 . With theapproximations made above in Eq. (19), a consistent equilibrium approximationfor the pair function at low densities is provided by the form"

:F2(Xl , X2, Pu , P12' P21, P22)

= n2<po (iTt )<Po(V2 )v(e,)v(v2)e-( I1(ih)t l +1I(V2)t2 )O( P12)O(P21) ' (23)

Here <PO(Vi) are the normalized Maxwellian velocity distribut ions, V(Vi) is t hecollision frequency for a part icle with velocity Vi ,

and we have used the low density expression for th e dist ribution of free flighttim es, t i between collisions. Eq. (23) is equivalent to the approximation

(25)

where

(26)

It is now a simple matter to calculate hKS/ N . To lowest order in density wecan expand the logari thm of the determinan t about the term with the highestpower of the t ime, which is linear in time for d = 2, and quadratic in time ford = 3. By the usual scaling arguments , the addit iona l terms in the expansionof the logarithm appear to be at least one ord er higher in the density than theterms we keep. Thus, for two dimensional syste ms

T he integral may now be performed , in part , numerically, by using the equilibrium values for the velocity dependent collision frequencies. A similar calculationmay be done for the three dimensional case as well. In each case we find that [111

(28)

4 We note that some delta functions are needed to convert Pii from a 2 x 2 matrixto a scalar quantity t i , but this is simple to fix , and we do not provide t he det ailshere.

246 R. van Zon, H. van Beijeren and J . R. Dorfman

Here lid is the average collision frequency per particle at equilibrium ,

where (3 = (kBT) -l , T is the temperat ure of the gas , and kB is Boltzman n'sconstant . In addition, nd is a red uced dens ity given by

The quantities Ad,Bd are numerical factors given by

A2 = 1/ 2, and B2 = 0.209,

A3 = 1, and B3 = 0.562.

(30)

(31)

(32)

T he values for Ad are in excellent agreement with comp uter simulations byDellago and Posch , but the values for the Bd are too small by factors of 3 or

10 r-----,--....------,--...,.-----,r----,----,

8

6

4

2d=2

theoryfit, N =36fit, N =64

MD, N = 36DSMC, N = 36

MD,N= 64DSMC, N = 64

+x)I(

o

O'-------'---.L..---'----'----'-------I.---'10- 7

v

Figure 1: A plot of t he results for the KS ent ropy per particle for a dilut e gas oftwo-dimension al hard balls, as a function of the collision frequency II in units of.,jkBT/(ma). T he solid line is the result given in Eq. (31), and the data points arethe numerical results of Dellago and Posch. Here N is the number of particles used inth e simulations. The data points are lab eled according to the computat ional method,molecular dynamics (MD) or direct-simulation Monte-Carlo (DSMC) . Also plotted , asthe dashed curves, are fits to th e data points, to functions of the form (28) , wit h Adand Ed as fitting parameters .

Kinetic Theory Estimates

20 ~---,---~--~---,---~--~----.

247

16

12

8

4d=3

theoryfit, N = 32

fit, N =108MD, N= 32

DSMC, N = 32MD,N = 108

DSMC,N= 108

+x)I(

o

v

Figure 2: A plot of the results for the KS entropy per particle for a dilute gas ofthree-dimensional hard balls, as a function of the collision frequency t/ in units of

/~kBT/(ma) . The solid line is the result given in Eq. (32), the data points are resultsof Dellago and Posch and the dashed curves are fi ts. The notation is the same as inthe previous figure.

so. These results are illustrated in Figs . (1) and (2) together with the results ofnumerical simulations by Dellago, and Posch.

The source of th e discrep ancy between the values of Ed can be attributedto our neglect of the spa t ial deviation vectors in Eq. (18) . For semi-dispersingbilliards , the spatial deviations in certain directions remain comparab le to th ecorresponding component of tir5vi. Therefore, they may contribute to the firstcorrection to the density logarithm in Eq. (28). In th e next subsection we willpursue this point a bit further.

3.3 Toward the Formal Evaluation of the KS Entropy a t Low D ensities . As we have seen above, the KS entropy of an equilibrium hard ball systemmay be calcu lated if one knows th e pair distribut ion function ;:2' The simpleapproximation to thi s functio n, Eq. (23), correctly gives the leading order termat low density, but not the next term in a density expansion. At low densities ,one might expect that the two colliding particles, 1, 2 in the above expressionsare uncorrelated just before their collision, so that a useful approximation to

248 R. van Zan, H. van Beijeren and J. R. Dorfman

the pair function would st ill be of the form

but we should use a better approximat ion to the one particle distribution function, :Fl , than that used above. More specifically, we consider the set of reduceddistribution functions, :Fl , :F2 , .. . , which satisfy a set of BBGKY hierarchy equations, and then use this set to find a good, low density approximation to theequation for :Fl , in much the same way as the BBGKY hierarchy is used toderive the standard Boltzmann transport equation and its higher density corrections [21]. However as we now consider a set of extended distribut ion funct ionswhich include the P matri ces as variables in addit ion to the positio ns and velocit ies of the par ticles, the hierarchy equat ions must be extended to include thesenew variables as well. The full hierarchy technique requires the use of somewhatcumbersome clust er expansion methods , and has been out lined elsewhere [211 .Nevert heless, the low density approximation to the equat ion for :Fl is physicallyplausibl e since it is a direct extension of the Boltzmann equation to includethe additional ROC matri ces as variables. Thi s exte nded Boltzmann equationis given by

[~ + £ 0(1)] :Fl( Xl ,Pll ,t)

= Jdx2dp12dp2ldp22T_(1, 2):Fl (Xl ,Pll ' t):Fl (X2 ' P22' t)J(P12)J(P2l) ' (34)

Here we have included a possible t ime dependence of the dist ribution functions,the operator £0(1) is th e free st reaming operator which accounts for the changein :Fl due to th e free motion of the par ticle. It is given by

(35)

Here we assume that there are no external forces acti ng on the particles, so thatin the course of free motion for a t ime interval, (7, t + 7), the position of thepar ticle changes according to i l (t + 7) = i l (7) + tVl (7) , and the ROC matrix,according to PI (t +7) = PI (7) +t1. The derivatives with respect to the positionvari able and with respect to the diagonal components of Pll reflect thi s freeflight motion. The right-hand side of th e extended Boltzmann equation, Eq. (34),expresses the change in :Fl due to binary collisions, and we have supposed thatthe two colliding particles are uncorrelated before their collision, as is norm allydone in the derivations of the Boltzmann equat ion. In this te rm we have defined

Kinet ic T heory Est imates

a binary collision operat or by

7-(1 ,2) = ad- 1Jdo- lih z . 0-18(ih z . 0- ) [J(1'lZ - ao-) Jdp~ l dp~zdp~l dp~z

x II J(Pij - Pij (P~ 1, · · ·, P~z ) )P~ ( 1 , 2)i,j=l,Z

249

(36)

Here P~ is a substi tut ion operator that replaces ROC matrices to its right bythe corresponding primed matrices, and velocities to its right by resti tuting velocities, namely those which for a given 0- produce ih, iJz after a binary collision.The delta functions in the ROC matri ces require that the primed ROC matricesbe rest ituting ones, namely, those which produ ce the unprimed matrices afterthe 1,2 binary collision.

The restitutin g values of the ROC matrices can be found by means of Eqs.(6) and (7), and the Ansatz, Eq. (14). To do this we consider only the ROCmatri ces involving the two colliding particles, and ignore the other ROC matricesinvolving other par ticles, such as P13' etc . In the equations below we will denotewith primes the values of resti tuting quanti ties, i.e., the quanti ties before collisionwill be denoted with prim es, and those aft er collisions without prim es. T henbefore collision

and it follows th at

where

Jr{ = P~l . JiJ{ + p~z . JiJ~ ,

J~ = P;z . JiJ~ + P;l . JiJ{ ,

JR~z = p~ . JV{z + p~ . JiJ{z,- I I - I , -,

Jr1Z= Pc . Jv1z + Pd . JV1Z'

1P« = "2 (Pll + P1Z + PZ1 + pzz) ,

1Pb = 4; (Pll - P1Z + PZ1 - pzz) ,

1Pc = "2 (Pll - P1Z - PZl + pzz) ,

Pd = (Pll + P1Z- PZ1 - pzz) .

(37)

(38)

(39)

(40)

Now one writes the collision equations for the spatial deviations of the centerof mass and relat ive coordinates in te rms of the varia bles before (prim ed) andafter collision (unprimed) ,

Pa . JV1Z+ Pb . JiJ1Z= p~ . JV{z + p~ . J~z ,

Pc . JiJ1Z+ Pd . JV1Z= M &. [p~ . J~z + p~ . JV{z ],


as well as the collision equation for the deviation of the relative and cente r ofmass velocities,

Mh2 = M, . Jv;.2 - 2Q~ . [p~ . Jv;.2+ p~ . JV{2],

JV12 = JV{2' (41)

(43)

Then we use the fact th at th e deviations of the relative and cent er of massvelocities before collision are independent of each other , and th at there areno orthogonalization or normali zation constraints on the components of thesevectors that would make some of th em dependent on the others". Therefore,upon inser ting Eq. (41) into Eq. (40), and comparing coefficients of JV{2and ofJv;.2 ' we obtain

Pa - 2Pb . Q~ . p~ = p~ ,

Pb . M o- - 2Pb . Q~ . p~ = p~,

Pc . Mo- - 2pc . Q~ . p~ = Mo- . p~ ,

Pd - 2pc . Q~ . P~ = M, . P~ ' (42)

In order to complete the specification of the restituting ROC matrices, weuse the fact that the particles are taken to be dynamically uncorrelated beforecollision so that P~2 = P~l = 0, and we can also expr ess the matrix Q~ in termsof the unprimed relative velocity after collision: Q~ = Qa(Vi2) = - QnV12).Thus, the primed variables before collision can be expressed in te rms of theunprimed variables after collision, provided the Eqs. (42) for the p' matri cescan be solved. It is important to point out th at , as we will see in Section V,the solution of these equations is very different for dispersing billiards such asencountered in the Lorent z gas , and semi-dispersing billiard s as occur in thehard ball gas where all of the particles move. The difference resides in the factthat for dispersing billiards the elements of ROC matri ces after collision are ofthe order of a/v and therefore much smaller than the typical matrix elementbefore collision, which is on the order of a the mean free time. For semi-dispersingbilliards thi s is no longer true, and some of the matrix elements remain large aftercollision as well. Thi s is equivalent to the st atement that the spatial deviationvectors immediat ely after collision are not negligible for semi-disper sing billiardsystems.

It is an elementary matter now to write expressions for the low density KS entropy in terms of the distribution functions and the elements of the ROC mat rices.One need only replace F2 in Eq. (16) by F l (Xl , Pll)Fl (X 2 ' P22)J(P12)J(P2l) ' anduse the leading order term in the expression for the determinants, namely

21vdPLlIdet [M o- (l, 2) - Qo-(l, 2)(Pll +pdl l ~ ¢ ,a cos

5 The constraints mentioned earlier, Eq. (4), on the deviations ofthe total momentumand energy of the N -body system do not have any significant effect on the dynamicsof two particles if N » 2.

Kinet ic Theory Estimates

for two dimensional systems. Here the matrix element p1.1. is defined by

251

(44)

Here the subscript ...L on the two dimensiona l unit vector Vl21. denot es a unitvector in a direction perpendicular to V12 . For the three dimensional syst em, weobtain

I [ ( ) ( )( )]1 (2IV121) 2[ 1l 22 12 21 ]det M, 1, 2 - Qa 1, 2 Pll + P22 ~ -a- P1.1.P1. 1. - P1.1.P1.1. .

(45)

Here

ij _ 1 ( ' i ( ) . j )P1.1. - 2 vl21. · Pll + P22 . "ia.i , (46)

where the unit vectors V12, Vi21. ' v~21. form an orthonormal coordinat e basis inthree dimensions.

It now remains to solve the extended Boltzmann equat ion for :Fl . Th is appears to be somewhat difficult due to the complicatio ns inherent in the resti tut ing, or gain term . If one ignores this term entirely, for whatever reason, thesolut ion becomes simple. In fact one recovers th e approximate form for :Fl givenby Eq. (26), and therefore one recovers the same expressions for the KS entropyas given by Eqs. (28- 32) [211 . To do better , one has to include the rest ituting term in the equation for :Fl . How to (approximately) solve the extendedBoltzm ann equat ion then, is st ill under investigat ion.

§4. The Largest Lyapu nov Exponentfor the Hard Ball System at Low Density

In this sect ion, we will obt ain an estim at e for the largest Lyapunov exponentAm a r . This exponent determines the asymptotic growth of all the deviationvectors: 15vi(t) ,15f'; (t) '" eA m a x t . We will calculat e it in the low density regime,nd « 1, when a Boltzmann equation gives an appropriate description of th ebehavior of the one-part icle distribution function . The result s will be checkedagainst simulat ions.

4.1 Kinetic Theory Approach. At low density, two colliding particles,i and j, have spent a long t ime in free flight before the collision. Therefore, inEq. (18), just before collision we keep only the dominant term involving thefree flight tim e t i. We saw in the previous section that for the KS entropy thisapproximat ion turned out to be unsatisfactory for describing the behavior of

252 R. van Zan , H. van Beijeren and J. R. Dorfm an

the ROC matrices. For the leading behavior of the velocity deviations, however ,it will be a valid approximat ion.

We insert this in Eq. (7) and keep only the leading terms in the free flightt imes,

(47)

Let us denote the typical velocity of a part icle in the gas by Vo = J kBTlm.One sees that Jil;. l vo and J1";1a are of the same order order of magnitude, whichjustifies the approximat ion tha t Jr'; « tiJVi' Notice we have eliminated theposition deviation vectors, but now we need to know the free flight t imes betweencollisions. To compare different cont ribut ions in Eq. (47), we want to know theorder of magnit ude of the velocity deviat ion vectors. We define clock values k;

to represent their order of magnitude measured in inverse powers of Tid :,

(1 )ki

JVi == Jvo nd ei , (48)

where ei is a unit vecto r and Jvo is a fixed infinitesimal velocity. To obtain thelargest Lyapunov exponent, it is enough to know the tim e evolut ion of theseclock values. They will grow linearly with tim e,

The average collision frequency Vd sets a time scale propor t ional to the density.It is extracte d so that the clock speed w can be interpreted as the averageincrease of a clock value in a collision. We see from Eq. (48) th at the clockspeed is related to th e largest Lyapunov exponent via Am a x = -WVd In Tid .

After the collision, the two par ticles have equal clock value, given by

k' = k' = In IQa(i , j)(tiJvi - tj Jvj )/ Jvol' J - In nd

= max (ki , kj ) + 1+ 0 (II In nd).

(49)

(50)

To see how Eq. (50) follows from Eq. (49) consider the following. If one of theclock values, say k i , is larger than the other one, the corresponding velocitydeviation JVi rv n;/i is very much larger than JVj rv n~kj . We will considerthe case that the two clock values differ by less than one in a moment . We saypar ticle i is the dominant par ticle. Only the dominant term inside the logari thmin the denominat or of Eq. (49) has to be taken into account , as th e other oneis much smaller. The free flight tim es scale, for low densities, inversely withthe density Tid , whereas the terms Qa(i , j ) and Vo do not contain any densitydependence. To see the density dependence more explicit ly, we scale the freeflight time as s, = ndti, so th at Si is density independent . The denominator

Kinet ic T heory Estimates 253

in Eq. (49) is then seen to be the sum of a term proportional to - In nd anda density independent ter m which fluctuates from collision to collision:

In IQa- (i, j )tiov;jovol = In IQa- (i, j )si (l / nd)ki +leil= (ki + 1)(- In nd)+ In IQa- (i , j )Siei l,

which leads to Eq. (50). The fluctuating correct ions to the addit ion of 1 to thedominant clock value scale with density as 1/ In nd. Finally, let us consider whathapp ens when Iki - kj l < 1. In that case, it is not necessari ly t rue that one ofthe two terms in the denominator in Eq. (49) dominates. To show that Eq. (50)then st ill holds, i.e. k~ = kj = max(ki , kj ) + 1 plus correct ions which scale withthe density as 1/ In nd, we consider the ext reme case th at k, = kj exactly:

In IQa-(i,j)(Mvi - tjOVj) /ovo l = In IQa-(i,j)(1 /nd)ki+l(Siei - sjej )1

= (ki + 1)(- In nd) + In IQa-(i,j)(S iei - sjej )l,

where Sj = ndtj . Again , the second fluctu ating term is density independent , andEq. (50) follows.

The clock values have a well-defined dynamics in the limit nd -t 0, whichgives the leading behavior of w. For this leading behavior, we can neglect the0 (1 / In nd) ter ms in Eq. (50):

(51)

In some collisions the neglected terms may be large, but the lower the density,the rarer such collisions become. If these ter ms were kept , they would give riseto terms of the order of In- 1(nd) and higher in w, so we would get an expansionof the form:

T he coefficients wo, W I , etc., in fact are st ill density dependent due to the neglected terms in Eq. (47), and to correlat ions between collisions. The former willgive rise to powers of the density, the latter may also give rise to a nonanalyt icdependence on the density [231 . Such terms however are at least accompanied byone factor of the density. Th at means that for nd -t 0, the limiting values of thecoefficients give the asymptotic behavior of w. Corresponding to the expansionof w, the Lyapunov exponent has an expansion of the form

(52)

We will calculate the first term in the density expansion of Am ax (which givesthe leading behavior) by calculat ing the leading clock speed Wo o As we alreadyargued, it is enough to use the dynamics in Eq. (51), for which we can restrictourse lves to integer clock values. The calculat ion will be based on an exte nded

254 R. van Zan, H.van Beijeren and J . R. Dorfman

Boltzmann equat ion for th e one particle distribution function f(k ,V, t) of clockvalues and velocities . Colliding particles are assumed to be uncorrelated beforethe collision, i.e., the probability that a particle with clock value k, and velocityVi and a particle with clock value kj and velocity Vj collide is prop ortionalto f (ki,Vi, t) f (kj ,Vj, t) ; thi s is the Stosszahlansatz. For th at collision we stillneed to specify er and this vector will, in a collision with given Vij, be drawnfrom a prob ability distribution proportional to ler .Vi j I. We demand that er .Vijbe negative, so that the particles are approaching each oth er. The gas will beuniform in equilibrium, therefore we neglect any spatial dependence. Consideringgains and losses, we can construct the following extended Boltzmann equationfor the tim e evolut ion of 1:

8!(k ,vd ( ~)f(k~) jd~ jd'8(' ~) d-II ' ~ Iat = -v VI ,V + V2 (J - (J . VI 2 na (J • VI2

[

k -2 k-2

X f(k - 1,v;.) 1];00 f(l ,~) + f(k - 1 , ~) 1];00 f(l , v;.)

+f(k - 1, v;.)f(k - 1 , ~)] , (53)

where in the integral , primed and unprimed quant ities denote values before ,respectively after collision. v(VI) is the velocity dependent collision frequency,given by Eq. (24), where we take the velocities to have their equilibrium distribution CPO. In principle, we could also allow for an initially nonequilibrium ornonst ationary velocity dist ribution, but thi s will not influence the asymptot icclock speed, as the velocity distribution funct ion will be stationary in due course,whereas the clock values keep on growing indefinitely. The extended Boltzmannequat ion can be simplified using cumulati ves, defined by

k

C(k,v) == 1(~) L f(l ,v).CPo V 1=-00

Because of this definition , C(k ,v) will tend to 1 as k tends to infinity for all V.In terms of the cumulat ives, the Boltzmann equation reads

oC(k,vd (~)C(k~) jd~ jd'8(' ~) d 11 ' ~ I (~)at = -v VI ,VI + V2 (J - (J . VI 2 na - (J . VI 2 CPo V2

xC(k - 1, v;.)C(k - 1,~) . (54)

This is th e appropriate equation to calculate Wo from. The assumptions mad e inits derivation were that colliding particles are uncorrelated, that the clock valuesincrease according to (51), that spatial fluctuations do not play any significantrole and that the velocity distribution has reached its equilibrium form . Theequat ion is similar to that found in earlier work [15, 16], where we had not

Kinetic T heory Estimates 255

accounted for the velocity dependence. We will follow the same approach usedthere, to find a better estimate for Wo, and thus for the leading behavior of thelargest Lyapunov exponent for low density.

The Boltzm ann equation is of the same type as equations encountered infront propagation [25]. In this analogy, C = 1 is an unst able phase, that lives atthe higher clock values, and C = 0 is a stable phase, th at propagat es from thelower clock values to the higher ones. Because the JVi grow exponent ially, thisprop agation means th at the clock values are increasing linearly in time:

C(k,Vi t) = F(k - WOl/d t , v). (55)

We assume that the equat ion is of the pulled front type, meaning th at theequation lineari zed around the leading edge (F = 1) sets a crit ical velocityof the front . Fronts exist for any velocity above this crit ical one. If the init ialcondit ions of C are sufficient ly steep, then a front will develop with this criticalvelocity. In our case, as an initi al condition, almost any (single) Jf will do tosee an exponent ial growth with the largest Lyapunov exponent, as almost anyJf will have some component along the fast est expanding direction in phasespace . Because of the finite numb er of particles, the init ial dist ribu tion of clockvalues corresponding to a single Jf, will have a finite support . Thi s is as steepan initial condit ion as one can get, so the crit ical velocity is the clock speed Wothat determines the Lyapunov exponent .

We will now show how the linearized equation sets a critical velocity, andthen we will determine it for the hard ball gas in two dimensions. We insert theAnsatz (55) into the Boltzmann equati on and concentrate on the leading edge.That is, we write F(k ,v) = 1 - b..(k ,v) and get , to linear order in b.. ,

ob..(k,VI) (_ )"'( _)-WO//d ok = - 1/ VI u k,VI

+ JdV2Jdo- 8(0- . vI2)nad-

I I0- . v12 lifJo(V2 )

x [b..(k - 1 ,~) +b..(k - 1,V';)] .

(56)

We now take a linear sup erposition of exponentially decreasing functions in k:

(57)

For this to be a solut ion of Eq. (56), only certain characteristi c values "Ii and corresponding characteristic functions Ai(v) are allowed, determined by the charact eristic equat ion:

WO"ll/d A(VI )= - I/(vd A(vd + e"Y JdV2 Jd0-8(0- ,v12)nad-

II0- 'VI2!

XifJO(V2 ) [A(~) + A(V';)].

256 R. van Zan , H. van Beijeren and J. R. Dorfman

(58)

which is found by inserting A(v)exp( - ,k) into the linear ized equation, Eq. (56).The characterist ic equat ion is non-linear in f. To deal with it , we rewrit e thisequation first as a generalized eigenvalue problem:

A WW01' A = LA,

where A = e-1' is the eigenvalue and the operat ors are defined by

(Wwo1'A) (Vl)=(vd- 1v(vd + wo,)A(vd ,

(L A) (vl )=vi1JdV2Jda 8 (a· v12)nad-1Ia'Vd'PO(V2) [A(~ ) + A(V;)].

For solving Eq. (56), first the whole spectrum {Ak (w,)} of the generalized eigenvalue problem (58) should be determined, with the corresponding eigenfunct ions{Ak(v;wo')}' The eigenvalues should also equal e1' . So next , for every Ak, thesolutions for , of

for fixed Wo , have to be found, and they are characteristi c values. For eachterm in Eq. (57), the ,i is one of these solut ions, with characteristic funct ionAi (v) = Ak(v;WO'i) . The superposit ion (57), with arbit rary coefficients, then isthe genera l solut ion of the linearized equation (56), for a given clock speed Wo 0

However , we don 't need the full solut ion of Eq. (56). For the leading edge,k -t 00 , only the term with the slowest decay in Eq. (57) surv ives. The characteristic value with the smallest real part , we call this , 0, corresponds to theslowest decay. As the distribution function f should be positive, F should bean increasing funct ion and ~ should be monotoni cally decreasing, meaning th at,0 cannot have an imaginary par t . The requirement that 1'0 be real is essent ial:it will be the determining factor for the crit ical velocity Wo, as follows. Thesmallest characterist ic value 1'0 is a function of the clock speed W00 The inverseof this function, woho) , has a minimum. Th at minimum is the critical value,because, for clock speeds Wo below thi s minimum, there are no real 1'0.

The characterist ic value with the smallest real part , was requir ed to be real ,so we only need to consider the eigenvalue problem (58) for real Wo,. In th atcase, the operators are self-adjoint with respect to the inner product

(A IB ) =Jdv'Po (v) A(v) B(v). (59)

The self-adjointness means tha t all the eigenvalues Ak are real. As we want thesmallest real " we are interested in the largest eigenvalue, which we will callAo. The self-adjointness of the operators, together with the fact tha t W Wo 1' isa positive operator, can be used to derive a maximum principle for this eigenvalue,

(60)

Kineti c Theory Estimates 257

The corresponding eigenvector Ao(v) is t he A(v) for which th e expression takeson its maximum value. On ce we have Ao(wo'Y) , 'Yo is the sma llest real solutionof

(61)

Notice that 'Yo then indeed still depends on the value of Wo in the Ansatz (55).As in previous work [15, 16], the crit ical velocity is det ermined as th e smallest

Wo possible when 'Yo is real and posi tive, i.e., the minimum of woho). Thisminimum is obtained from Eq . (61) taking the derivative with respect to 'Yo andusing wbho) = O. This gives

This condit ion together with Eq. (61) can be captured concisely by saying thatthe critical velocity is the minimum of the following function wover real positivevalu es of Wo'Yo:

_ Wo 'Yow(wo'Yo) == I A ( )- n 0 Wo'Yo

(62)

Once the crit ical clock speed Wo has been obtained , the cha rac te rist ic funct ion Ao(v) , determined by Eq . (58) , gives the velocity distribution in the headof th e front . More precisely,

(63)

is the velocity distribution of the par ticl es with clock values k larger than someko, for ko tending to infini ty.

We have no exact express ion for t he eigenvalue Ao, but th e variat iona l principle will allow us to ca lculate approxima tions to it , as follows. Ao was given inEq . (60) as th e maximum over all functions A(v). So inserting any function A(v)would give a lower bound on Ao. Using this lower bound in Eq. (62) will thengive a lower bound on W Oo We take a fixed form of A(v; bo,bl , •• . ) , depending onvari ational parameters {b;}, and det ermine the maximum in Eq . (60) over thesepar am et ers . This gives an approxim ation to Ao and Ao(v), which can be systemat ically improved, and checked for convergence, by including more variationalparam eters.

We will now apply this pro cedure explicit ly to the two dim ensional case. Theextrac t ion of the factor I/ d in the definition of L and W wo'Y removes the densityfrom the equat ions . We can get rid of the temperature dependence by rescalingthe velocities by the typi cal velocity vo: v ---+ vi vo. The opera to r W wo'Y thenbecomes

258

and L is

R. van Zon , H. van Beijeren and J. R. Dorfman

1 j' - HV21

2 J(LA)(ih) = r:;; dih e dala ·1)211(a · ih2) [A(V;) + A(V;)] .2y 1r 21r

As our variational eigenfunction we choose: A(ii) = a+boliil 2+bll iii4 +b2vx +b3vy.a is not a variational parameter, but is set by the normalization requirement ,(A 11) = 1. The operator L contains no preferred direction, so the eigenfunctionsare expected not to contain any either. Indeed, it turns out that the maximumin Eq. (60) is assumed for (b2 = 0,b3 = 0). Therefore we will only work withthe nontrivial variational parameters bo and h. Because the variational eigenfunction is linear in the parameters, there is an equivalent method that we willuse. We construct an orthonormal basis out of 1, liil2 and liil4, and determinethe matrix elements of the operators Land W WOI" The largest eigenvalue ofthe truncat ed matrix (in which all other matrix elements are set to zero) is anapproximation and lower bound to the real eigenvalue Ao.

The basis vectors are

11) = 1, 12) = ~ liil2 -1, 13) = il iil4 -liil2+ 1,

which are orthonormal with respect to the inner product defined in Eq . (59).A tedious but straightforward calculation gives the truncated matrices on thisbasis,

I:4231651128

I )-32

R and L 3 x 3 =12818091024

I21182764

1 )-16

2764 .

641512

For many values of WO I th e eigenvalue problem

AW3 x 3 A = L 3 x 3 AWo'Y '

was solved numerically using "Maple V", and the largest eigenvalue, Ao(wol)was taken to det ermine w, according to Eq. (62). The minimum Wo is assumedat about WOIO = 3.5, therefore, in Fig. 3, the values of w were plotted for2 < WO I < 4.

The minimum of wcan be determined numerically. It is given by

Wo ~ 4.735 ,

wo,o ~ 3.497 ,

and the normalized eigenfunction is approximately

Ao(ii) = 0.63943 + 0.162891ii12 + 0.004351IiiI 4.

(64)

(65)

3.63.553.53.453.4............................................. . .............................................4.5


6

4.736

5.54.734

4.732W 5

43.532.54L------....L..--------'---------'------------'

2

Wo')'

Figure 3: Several approximations for the function w. A 1 x 1 truncation of the matr icesyields the dotted curve. The solid curveresults fromthe 3x3 matrices. As the differenceis so small, only the blow-up in the inset can show the 2 x 2 results separately as thecurve of points.

In Fig. 3, also the result is plotted when the matric es are truncated further.Keeping only 11) means no variation al par ameters and no velocity dependence.Hence we recover the result Wo ~ 4.31107 .. . of previous work [15, 16], in whichthe velocity dependence of the collision frequency was neglected. The enhancement of taking two basis vectors, 11) and 12), is ra ther large, but taking onemore gives only a small difference, of about 0.1%, as the inset of Fig. 3 shows.Therefore we assume th at these values have converged up to at most 0.1%.

The results can be qualitatively understood as follows. In the first place,par ticles with a higher velocity have a higher collision frequency, and so theirclock values increase faster than average . Thus, one expects higher velocities tobe more prominent in the head than in the rest of the distribution. Indeed, thisis seen in Eq. (65), where according to formu la (63), the positive value of thecoefficient 0.16289 signifies a shift to higher velocities in Phead as compared to'Po. Secondly, collisions of other particles with the head just synchronize thispart icle to the one in the head. So the collision frequency of the part icle in thebulk is irrelevant and unable to compensate the increase in collision frequency inthe head. Th is increase is what really causes the increase of w over the velocityindependent est imate of 4.31107 .. . in Refs. [15, 16], as can be seen from the


relative increase in this frequency,

which mat ches closely the relat ive increase in clock speed, wo/4.3 1107= 1.0983 . . .

4.2 Comparison with Simulations. The result s for the largest Lyapunovexponent of a system of N hard balls will now be checked against simulations.We want to check the following points: the validity of the clock model, therelati on between the Lyapunov exponent and the clock speed, and the velocitydistribution in the head.

The simulat ion method we shall use is a variant of the Direct SimulationMonte Carlo Method (DSMC), tha t was also used by Dellago and Posch 126]for the calculat ion of Lyapunov exponents in a "spatially homogeneous system".Th ese authors also checked the equivalence for low densities of DSMC and molecular dynamics simulat ions. The low densities that we are interested in, are notaccessible to the molecular dynamics simulat ions. The DSMC method simulates the Boltzmann equation, and consists of the following. In each t ime ste p,a pair of par ticles is picked at random to collide. The probability of accepti nga pair is prop ortional to the relative velocity so that the collision frequencyof two particles with given velocity is also linear with the relative velocity. Inthe original met hod due to Bird [28], the system is divided into cells, and onlypar ticles within one cell can collide, but as we are simulat ing a homogeneoussystem , the whole system is in one cell. Once a pair is picked, a collision norm al;, is drawn from a distribution proportional to IV21 . ;' 1, as in the Boltzm annequation. The velocit ies of the pair are transformed according to Eq. (3) andtheir deviations are subjected to the tra nsformations in Eq. (18), to accountfor free flight , and, subsequently Eq. (7). Even though in the DSMC methodthe positions themselves are discarded, we st ill keep track of the deviations inposition by integrating the deviations in velocity by means of Eq. (18).

In a parallel simulati on, for the same collision sequence, the particles aregiven separate clock values, which are updated in a collision according to Eq. (51).Thi s enables us to check how well the clock values from the dynamics of Eq. (51)represent the act ual dynamics of the velocity deviati ons.

Initially, the velocities of the particles are picked from a Maxwellian distribution, scaled by vo, i.e., with kBT/m set to one. The initial velocity andposition deviations are unit vectors drawn from an isotropic distribution, oVo isset to one (this can be done because the deviations follow a linear equat ion) ,and correspondingly all clock values are zero.

The first check is to see whether the clock values are accurate in describingthe behavior of the velocity deviat ions. As mentioned above, in the simulation the particles have their real deviations in position and velocity as well as

Kinet ic Theory Estimates 261

a clock value, evolving independentl y of the deviations, according to Eq. (51).If that equation were exact , for each particle the clock value k, would equalIn 18v;j8vol/l ln ii i for all time , if it did so initi ally. But Eq. (50) shows thatthere are density correct ions of order II In ii . If the difference between ki andIn 18v;j8vo l/lln iii became very big too often , the maximum of the clock valuesmight not correspond to the maximum of the velocity deviations, and the dynamics from Eq. (51) would not even be approximately correct . If things go aswe expected, the clock values will give the right w in the limit of zero density,i.e., wo , with deviations that scale as 1I In ii .

Two simulations for N = 128 particles, and d = 2 dimensions, were performed for, in tot al, 3,887,196 collisions, one simulation at a density ii2 = 10- 4

and one at a density ii2 = 10- 12 . In Fig. 4 results are plotted for both simulations. For each particle a symbo l is plotted , a plus for the simulat ion atdensity ii 2 = 10-12 and a diamond for the simulation at density ii2 = 10- 4 .

The horizontal position of each symbol is given by the final clock value k i ofparticle i (of which 226885 is subtracted) , the vertical position by the final valueof In 18v;j8vo l/ lln ii2 1. The scale of the velocity deviations is a bit different forthe two simulations, therefore different scales were used on the verti cal axis, asindicated in the figure.

From the plot , we can see the following. There is a linear relation betweenIn 18v;j8vol /l ln ii2 1and k i , around which there are fluctuations. The fluctuationsget smaller, and the slope gets closer to one, as the density decreases. Smallfluctuations indicat e that the maximum in Eq. (51) is correct , i.e., the particlewith the largest clock value has the largest velocity deviation. The value ofIn 18v;j8vol / lln n21 is not exactl y equal to the clock value ki . Eq. (50) showsthat the correction in the increase of the clock value in a collision is of the orderof II In n2. Because such a correct ion is present in each collision, the correctionto the relation In 18v;j8vol/ llnii2 1 = k; also grows. The relat ive correct ion is,for density n2 = 10- 4 of the order of 24%, while for the density n2 = 10- 12

it is about 8%. The difference by a factor of three shows that the correction isof the order of 111 In n21, as In 10- 12I In 10-4 = 3 also. We conclude that thedynamics of the clock value in Eq. (51) indeed describes the dynamics of theIn 18vi I accurat ely for low densities.

The second check is on the asymptotic form (52) of the Lyapunov expon ent,and its relation to the leading clock speed. To check the leading behavior of thelargest Lyapunov exponent as a function of density, we need to perform a numberof simulations for different densities, all very low, because the coefficient in thenext term in Eq. (52), although it could be calculated in principle, is not yetknown and th us we cannot assume that we can neglect it. The leading clockspeed wo , which can be determined from the parallel calculat ion of the clockvalues, Eq. (51), does not depend on density and can be determined from justone simulation at an arbitrary density.

In this case, simulations are done with N = 64 particles, again in d = 2dimensions . The Lyapunov exponent divided by the collision frequency is plotted

R. van Zan, H. van Beijeren and J . R Dorfman

302515 20

k, - 226885

+ +++++

OOO~~

10

++=1=

00 °~O

262

30

~C'l25..,.

~

I I00

20,...., ,....,

II IIC'l C'l

'': '': 15OM0)ll':J"' ,...., 10""" 00r-o,...., N

I I5

C'l C'l

' ': ' ':..s..s 0---- <,1;5' 1;5' -5"0 "0

6 6-10.. ..

+ 0 5

Figure 4: Check on t he clock model, for N = 128 and d = 2. For each particle,IOvil/lln n2 1is plot ted against ki . The diamonds are from a simulation with n2 = 10- 4

,

after 3,887,196 collisions. T he plus signs denote results are from a simulat ion wit hn2 = 10- 12

. Also plot ted is a line wit h slope 1.

as a function of the density n2 in Fig. 5. To justi fy the form of the densityexpa nsion given in (52), we also plotted the function - Wo In n2 + W I , whereWo is the clock speed found from the parallel simulation of the clock values,Wo = 3.4479 ± 0.0016 and W I is obtained from fitting (using only points fromdensities n2 smaller or equal to 10- 5 ) . T his function describes the simulationpoints very well, i.e., the slope is indeed given by the clock speed. The deviationsat higher density may be accounted for by higher powers of 1/ In n2 .

Surprisingly, the value of Wo is far from the predicted value in Eq . (4.1). Thisis due to finite size effects, which have been well st udied by now [16, 29, 30].Corrections due to the finite ness of N are , for large N , of the order of In- 2 N ,but for N = 64 we are not yet in the asymptotic regime for which this scalingholds. The investigati on for larger numbers of particles and the calculation ofthe finite size corrections will be done in future work, but we remark that fora simplified clock model, in which the collision frequency is ta ken to be velocityindependent, this was already car ried out in [161 .

T hirdly, we will check the velocity distribution P head (v) in the head of theclock distribut ion. T his dist ribution is determined as follows. After the simulat ion has run for some tim e, we determine the average of the clock valuesand shift all the clock values by that amount , so we can access the stationary

Kinetic Theory Estimat es 263

0.010.0010.00011e-051e-061e-071e-08

50

60

20

1O~.L..---'-----'--'-----'----'------.J.-'-----'------'-----'--'----'----'--'-'-----'----'-----'..J'-----'-----'--'--"----'-_1..-L..J

1e-09

30

Figure 5: Density dep endence of the largest Lyapunov exponent Am a x for N = 64,d = 2. The plus signs denote DSMC results. The linear curve is a function of th eform (52) , where the first two terms are included . For Wo , the clock speed from thesimulation is taken , WI is a fit parameter.

F(k ,v) rather than the shifting C(k , ii, t). We then pick some lowest clock valueko; all particles with a lower clock value are discarded. Of the remaining particles , which are those in the head of the distribution, the velocity distributionis constructed. We adjust ko such that k > ko can be identified with the tail ofthe distribution. To get good statistics (which is difficult because there are notmany particles in the head) , we measure this distribution at several times , andaverage the result .

As the velocity distribution Phead (v) only involves clock values, there is nodensity dependence and a simulation at one density is enough. We ran a simulation with N = 128 particles, and d = 2. The distribution of the clock valueswas obtained, and a value of ko = 7 seemed a reasonable start of the tail ofthe distribution. We have checked that the results do not change much wheninstead we take ko = 6 or ko = 8.

Combining Eqs. (65) and (63) gives the explicit prediction for the velocitydistribution in the head :

This prediction has been plotted in Fig. 6 together with the distribution foundfrom the simulations, and the velocity distribution in the whole gas. Even though

264

0.7

0.6

0.5

"tl 0.4e"(

0.3

0.2

0.1

R.van Zon, H. van Beijeren and J . R. Dorfman

v

Figure 6: Comparison of the velocity distribution in the leading edge. The histogramresults from the simulations for N = 128, in d = 2 dimensions. The solid curve is a plotof the prediction (66). The dotted curve is the two-dimensional Maxwell distribution,vexp(-v2/2).

the statistics isn't perfect, there is a very good agreement between the two. Itshould be noted that leaving out the v4 term in Eq. (66) gives only a smalldifference. It is also evident that the velocity dist ribut ion in the head of theclock dist ribution has higher mean velocity than the velocity distribution characterizing the full gas, which is also plotted in figure 6.

§5. The Dilute, Random Hard Ball Lorentz Gas

A very useful and simple example of a dynamical system which is a dispersingbilliard, rather than a semi-dispers ing billiard is provided by the Lorentz gasmodel. In this model one places fixed, hard balls of radius a in ad-dimensionalspace , and then considers the motio n of a point particle of mass m and withinit ial velocity if, in the free space between the scatterers [3]. The particle movesfreely between collisions and makes specular, energy conserv ing collisions withthe scatterers. We consider here only the case where the scatterers are notallowed to overlap each other. One can imagine that the scatterers are placedwith thei r centers on the sites of a regular latt ice, or that the scatterers areplaced randomly in space . Here we consider the case where N scatterers are

Kine tic Th eory Estimat es 265

placed randomly in space in a volume V in such a way that the average distancebetween scatterers is much great er than their radi i, a. That is, the scatterersform a quenched, dilute gas with rui" « 1.

In this section we show that the simplifications in the dynamics of this modelproduce simplifications in the calculat ions of the Lyapunov exponents and KSentropies of the moving particle, due primarily to the dispersing nature of thecollisions of the moving particle with the scatterers. A Lorentz gas in d dimensions may have at most d - 1 positive Lyapunov exponents . That is, the phas espace dimension for the moving particle is 2d, the requirement of constant energyremoves one degree of freedom , and the flow direction in phase space has a zeroLyapunov exponent associated with it, since two points on the same trajectorywill not separate or approach each other in the course of time . Thus there can beat most 2d - 2 nonzero Lyapunov exponents, of which only d - 1 can be posit ive,since the exponents come in plus-minus pairs for this Hamiltonian system.

To analyze th e Lyapunov spectrum we consider two infinitesimally closetrajectori es in phase space and follow the spatial and velocity deviation vectorstSr,tSv separat ing the two trajectories, in tim e. By arguments almost identical to(but simpler than) those present ed in Section II for the case where all particlesmove, we have the following equations of motion for the position, r, velocity,v spatial deviation vector, tSr, and velocity deviation vector , tSv, of the movingparticle, between collisions

r=v,

v= 0,

tSf = tSv,

tSiJ = o. (67)

At a collision with a scatterer , th ese dynamical quantities change according to-of _

r = r ,-of - 2(- A) AV = V - v · 0' 0' ,

tSr' = M", . tSr,tSV' = M ", . tSv - 2Q", . tSr. (68)

(69)

Again, the primed variables denote values immediately after a collision. whilethe unprimed ones denote values immediately before the collision. The distancefrom the center of the scatterer to the moving particle, at collision, is oir , andthe tensor Q", is given by

Q . _ [(cT · v)l + cTV] . [(cT· v)l - veT]17- a(cT'v) .

It should be noted that if the velocity deviation vector and the velocity vectorare orthogonal before collision, then they will be orthogonal after collision as

266 R. van Zon, H. van Beijeren and J. R. Dorfman

well. That is, if v·Ov = 0, then VI . 01J' = 0, also. Thus we can take Ov to beperpendicular to v for all tim e, and without loss of generalit y, we can take Or tobe perpendicular to v as well. Of course, the condition that v·ov= 0 is simplythe statement that the two trajectories are on the same constant energy surface.We will denote the spat ial and velocity deviat ion vectors for the Lorentz gas witha subscript 1.. to indicat e th at they are defined in a plan e perp endicular" to v.It follows that we may replace the d x d ROC matrix defined by

Or = P ' ov

by a (d - 1) x (d - 1) matrix P1- defined by

(70)

(71)

For the two dimensional case P1- is a simple scalar which we denote by p. Oneeasily finds that between collisions p grows with t ime as

p(t) = p(O) + t ,

The change in p at collision sat isfies the "mirror" equation [181

1 1 2v- =-+--.pI P a cos 1>

(72)

(73)

Here the primes denote values immediately after a collision, and v is the magnitude of the velocity of the particle. If the radius of curvature p is positive,initially, it will always remain positive, and it also follows from Eq. (73) thatthe value of vp after collision is less than half of the radius of the scatterers.Consequentl y the radius of curvat ure typically grows to be of the order of a meanfree time, and it becomes much smaller immediately after a collision with a scatterer . For three dimensional systems a similar situat ion results. Then P1- IS a2 x 2 matrix which sat isfies the free motion equation

(74)

and changes at collision according to

(75)

Here th e inverse radius of curvat ure matrices [p~] - l and [p1-]-l are definedin plan es perpendicular to if' and to ii, respectively. In the hybrid not ation ofEq. (75), in which both d x d matrices and d - 1 x d - 1 matri ces figure, the

6 Of course , t he component s of 85,8r in th e dir ection of 5 are not related to t henon- zero Lyapunov exponents or the KS ent ropy, since these components do notgrow exponent ially.


inverse matrices in the directions along V' and ii, respectively, may be defined by[p~l - l .V' = 0 and [p. r ' .iJ= O. Th e final matrix in the right-hand side of Eq.(75) can then be restri cted to the plane perpendicular to V' straightforwardly.

It is worth pointing out some important differences between the ROC matrices defined here for the Lorentz gas and those defined earlier for the regular gasof moving particles. Here the ROC matrices are defined in a subspace orthogonal to the velocity of the moving particle. Further the change in the mat rixelements at collision is from a typically large value on the order of a mean freetime to an always small value, on the order of the t ime it takes to move a distance equal to half the radius of a scatterer . T his lat ter property is a propertyof dispersing billiards. For the regular gas case, only a few of the elements ofthe ROC matrices become small after a collision, which means that one cannotfind an accurate approximation to the ROC matrices by considering only onecollision. T his latter prop erty is associated with semi-dispersing billiards wherea reflection from a scatterer does not change the diagona l components of theROC matrix that correspond to the flat direct ions of the scatterer , at all.

5.1 Informal Calculation of the KS Entropy and Lyapunov Exponents for the Dilute , R andom Lorentz Gas. Here we show that simplekinetic theory methods allow us to compute the Lyapunov exponents and KSent ropies of Lorentz gases in two and three dimensions [14] . To do that weuse methods similar to those in Sect ion III. That is, we consider the equat ionsfor the deviation vectors, Eqs. (67 - 69) above. The velocity deviation vectorchanges only upon collision with a scatterer. We will base our calculat ion on theexponential growth rate of the magnitude of the velocity deviation vector, andfor three dimensional systems, on the exponent ial growth rate of the volumeelement in velocity space.

We begin by writing the spat ial deviation vector jus t before collision as

Ji = tJiJ+ Jr(O), (76)

where Ji(O) is the spati al deviati on vector just after the previous collision witha scatterer. T his equat ion is essent ially the same as Eq. (18), but now it isa good approx imation to neglect the spat ial deviation vector vector Ji(O) sinceJr(O) is of relative order a/ vt compared to the term tJiJ, in all directions of Jr,perpendicular to iJ. Thus we neglect this ter m and insert Eq. (76) into the lastequality of Eq. (68) to obtain"

JiJ' = M ; . JiJ - 2tQa . JiJ == a . JiJ, (77)

where we have defined a matrix a that gives the change in the velocity deviationvector at collision. T hen we can express the velocity deviati on vector at some

7 In prin ciple , t he term M a . ov in Eq. (77) can be neglected also, but only fordirection s perp endicular to the velocity. If one is careful to consider only deviationsov in the subspace perp endi cula r to t he velocity of t he particl e, it is possib le tocarry out the calculat ion with this ter m neglected .

268 R. van Zon , H. van Beijeren and J . R. Dorfm an

time t in terms of its initial value as

MJ(t) = aN · a N - I · · · al . ov(O) , (78)

where we have labeled the successive collisions by the subscripts 1,2 , ...,N. Wecan determine the largest Lyapunov exponent by examining the growth of themagnitude of the velocity deviation vector with t ime, and the KS entropy as thegrowth of the volume element with time. Therefore, with the approximat ionsmentioned above,

(79)

where oif is the velocity deviat ion vector immediately after the collision labeledby the subscript i. Similarly, the sum of the positive Lyapunov exponents is givenby

N I N'" Ai = hK S = lim - N '" In Idet a, I·~ t-so: t Z::Ai >O 1

(80)

To evaluate the sums appea ring in Eqs. (79,80) , we note th at to leading orderin the density none of the collisions are correlated with any previous collision,that is, the leading contribut ion to the Lyapunov exponents comes from collisionsequences where the moving particle does not encounter the same scatterer morethan once in the sequence 8 . Therefore we can treat each term in the sums in Eqs.(79, 80) as being independent of the other terms in the sum. We have expressedAm a x and the sum of the positive Lyapunov exponents as ari thmetic averages,but for long tim es and with independently distributed terms in the average,we can replace the arithmet ic averages by ensemble averages over a suitableequilibrium ensemble. That is

and

L x, = v (lnldetal )Ai >O

(81)

(82)

8 In two dimensions the particle will hit the same scat te rer an infinit e numb er of ti mes.However t he effects of such processes are of higher density, and can be neglected heresince the number of collisions between successive collisions with the same scattererbecome typically very large as t he density of scatte rers approaches zero.


where se: and bij+ are the velocity deviation vectors before and after collision,respect ively, v is the (low density)value of the collision frequency, N [t as tbecomes large, and the angular brackets denote an equilibrium average.

We now consider a typical collision of the moving particle with one of thescatterers . T he free t ime between one collision and th e next is sampled from thenormalized equilibrium distribu tion of free times [3], P(T) given at low densitiesby

(83)

The construct ion of th e matri x a requires some geomet ry and depends on thenumb er of dimensions of the system. In any case we take the velocity vectorbefore collision, ij to be directed along th e z-axis, and take a·ij = -v cos ¢ , where- 7r / 2 :S ¢ S; 7r/ 2. The velocity deviati on before collision se: is perp endicular tothe z-axis. Then it is a simple matter to compute Ibij+I /lbij-1 and [det a]. Fortwo-dimensional systems bij and the matrix a are given in this representati onby9

(1 ) ( (1 + A) cos 2¢ sin 2¢ )

bv = 0 IJ'v I; a = (1 + A) sin 2¢ - cos 2¢ , (84)

where we introduced A = (2VT)/( a cos ¢ ). To leading order in ur]a we find that

Idet a ] = A. (85)

For three dimensional systems the unit vecto r a can be represented as a =- cos ¢ Z+sin ¢ cos a x+sin ¢ sin a y. Now the ranges of the angles ¢ and a areoS; ¢ S; 7r/ 2 and 0 S; a S; 27r. Th ere is an additional angle 't/J in the x, y planesuch tha t the velocity deviat ion before collision se: = Ibij - I[xcos 't/J + Ysin 't/J] .It is somewhat more convenient to use a symmetric matri x, a = (1 - 20-0-) . a ,given by

1 + A(cos2 ¢ + sin2 ¢ cos2 a)Asin2 ¢ cos a sin a

o

A sin2 ¢ cos a sin a1 + A(cos2 ¢ + sin2 ¢ sin2 a)

o n(86)

One easily finds

Ibij+l _ 2TV [ cos2(a - 't/J) . 2( _ 0/') 2-t.] 1/2

I, ~ I - 2 -t. + sin a 'f/ cos 'f/ ,uV - a cos 'f/

(87)

9 In contrast to the ROC matrices p, a is a d x d matrix. If one chooses one ofthe basis vectors of a perpendicular to ii, the remaining ones are the basis of thecorresponding d - 1 x d - 1 matrix, from which one can also obtain Eq. (85), and,in the three dimensional case, Eqs. (87) and (88) .

270

and

R. van Zon, H. van Beijcren and J. R. Dorfman

[det a] = [det a] = (2:rr (88)

to leading order in irr / a.To complete the calculat ion we must evaluate the averages appearing in Eqs.

(79,80) . That is we average over the distribution of free times and over the rateat which scattering events are taking place with the var ious scat tering angles.Additionally in 3 dimensions an average over a stat ionary distribution of angles'Ij; has to be performed in general. Due to the isotropy of the scat tering geomet ry'Ij; can here be absorbed in a redefinition a' = a - 'Ij; of th e azimuthal angle a .T his will not be t rue any more if the isotropy of velocity space is broken (e.g.by an external field). The appropriate average of a quantity F takes the simpleform

(F) = ~100

dr Jdir cos <jJP (r )F, (89)

where P(r) is the free time distribution given by Eq. (83) and J is a norm alization factor obtained by set ting F = 1 in the numerat or. The integration over theunit vector o; i.e., over the appropriate solid angle, ranges over - 7r / 2 ::; <jJ::; 7r/2in two dimensions and over 0 ::; <jJ ::; 7r / 2 and 0 ::; a ::; 27r in three dimensions.After carrying out the required integrations we find th at

>.+ = >'max = 2nav[-ln(2na2) + 1- Cj + .. . , (90)

for two dimensions. Here C is Euler's constant, and the terms not given explicitlyin Eq. (90) are higher order in the density. Similarly, for the three dimensionalLorentz gas we obt ain

>.t ax = na2v7r[-ln(ii /2) + In2 - ~ - C] + ... ,

>.t ax+ >'~in = 2na2v7r[-ln(ii /2) - C] + ... ,

from which it follows that

>'~in = na2 v7r[- In(ii/ 2) - In2 + ~ - C] + ." ,

(91)

(92)

(93)

where ii = na37r. We have therefore determined the Lyapunov spectrum for theequilibrium Lorent z gas at low densities in both two and three dimensions [13,14]. There is good agreement with simulat ions, as shown in figure 7. We noteth at the two positive Lyapunov exponents for three dimensions differ slight ly,and th at we were able to get individual values because we could calculate the

Kineti c T heory Estimat es 271

largest exponent and the sum of the two exponents. We could not determineall of the Lyapunov exponents for ad > 3 dimensional Lorent z gas this way.Moreover, for a spati ally inhomogeneous system, such as those considered in theapp lication of escape-rate methods, the simple kinetic arguments used here arenot sufficient and Boltzmann-type methods are essential for the det erminationof the Lyapunov exponents and KS entropies.

In Fig. (7) we illustrate the results obtained above for the Lyapunov exponents of the dilute Lorent z gas in both two and three dimensions, as functionsof the reduced density of the scat terers, and compare them with the numerical simulations of Dellago and Posch [31, 321. As one can see the agreement isexcellent.

5.2 Formal Kinetic Theory for the Low D ensity Lorentz Gas. T heformal theory for the KS entropy of the regular gas is easily app lied to theLorent z gas , which is, of course, considerably simpler. Thus by following thearguments leading to Eq. (13) for the sum of the positive Lyapunov exponentsfor the regular gas , we find that the KS entropy for the equilibrium Lorent z gasis given by

L x, = ad-

1JdxdpdRdCr8(- e-fi)lv· 8"!t5(f - R - afi).\ i>O

x In Idet [M, +2Qa . p] 1.1'2 (X, R,p). (94)

Here R denotes the location of the scatterer with which the moving particle iscolliding, and .1'2 is the pair distribution function for the moving particle to havecoordina te, r, velocity, v, ROC matrix, p, while the center of the scatterer islocated at R. At low densities we may assume that the moving particle and thescatterer are uncorrelated , so that the density expansion for .1'2, immediatelybefore collision would have the form

(95)

where n is the number density of the scatterers and .1'1(X, P) is the equilibriumsingle particle distribution function for the moving particle.

We may easily construct an extended Lorentz -Boltzmann equat ion (ELBE)for .1'1 along the lines used previously for the extended Boltzmann equation inSection III. The ELBE is given by [13, 14J

[:t+ .co] .1'1(x,p) = ad-

1JdRdfi8(-e. fi)l(v , fi) 1

x [<5(f - R- afi )Jdpl<5(P - P(pl))P~-<5(f-R+afi)] .1'1(X,P) ' (96)

Here the operator P~ is a substitution operator that replaces velocities, vandROC matrices, p, by their restitution values, i.e., the values needed before collision to produce the values v,and p after collision with a scatterer with collision

272 R. van Zon, H. van Beijeren and ,J. R. Dorfman

0.100

0.010

0.0010.0001

0.100

0.010

0.0010.0001

0.001

0.001

n

n

0.01

0.01

0.1

0.1

Figure 7: A plot of t he Lyap unov exponents, in units of v / a, for the moving particle ina random, dilute Lorentz gas in two dimensions (top) and three dimensions (bottom) ,as functions of t he density n , in units of a- d . T he solid lines are the resu lts given bykinetic theory, Eq . (90), respectively, Eqs . (91) an d (93) , and the data points are thenumerical results of Dellago and Posch.

Kinet ic T heory Estimates 273

vector a. Also, the free par ticle streaming operator on the left-h and side of Eq.(96) is given by

(97)

since between collisions, r varies as r(O ) + tV and p varies as p(O ) + tl.Returning for the moment to Eq. (94) for the KS entropy, we see by evaluat

ing the determinant in the integrand on the right side, that not all componentsof p contribute. In fact , only the components of p in the plane perpendicularto v contribute to hK S ' Here we will work out the details of the calculat ion ofhK S for the two dimensional case, leaving the details of the th ree dimensionalcase to the litera ture [1 41 . For d = 2 we can easily evaluate the determinant inEq. (94), and find th at it is

(2vp )Idet [M& + 2Q&. p] I = 1 + --A-. .

a cos 'f'(98)

Here the scalar p is the component of the p matri x given by p = Vi- . P . vs..where Vi- is a unit vector orthogonal to V. Moreover , for low densities we canuse the approximation nF l (x, p) for F2 . Th en the expression for hK S becomes,at low densities

where x = r ,v , and we have to determine F1(x ,p) as the solut ion of th e ELBEwhere we integrate over all components of p , except the one diagonal componentp. The ELBE then becomes, in the spatially homogeneous, equilibrium case

f) l 1r/

2

aF1(r,v ,p) = nav d¢ cos ¢p - 1r/2

X [('0dplr5(P- aC~)Fl (r, i! ,P' ) - F l (r ,V, p)] .in 2v + a coso vp'

(100)

The argument of the r5 function is simply obtained by using th e mirror formulagiven by Eq. (73), but now the unprimed variable is the value after collision,and the primed variable is the value of p before collision. A furth er, and usefulsimplificat ion result s from the observation that p' is typically of the order of themean free time between collisions, which, for low density, is much larger thano.l». Therefore th e delta funct ion can be replaced by

r5 (p _ a cos ¢ ) ~ r5 (p _a cos ¢ ) .2v + acos rf> 2v

v p'

(101)

274 R. van Zan, H. van Beijeren and J . R. Dor fman

In a spatially homogeneous and isotropic equilibrium state, F 1 has to be a function of the norm of the velocity vector Ivl and the radius of curvature p only.Using that the magnitude of the velocity, lvi, always stays the same , we knowthat there is a solution of the form

(102)

with <p(v) = (21rVvO) -lJ(lvl - vo) is the normalized equilibr ium spat ial andvelocity distribution function for the moving particle, Vo is its constant speed,and V is the volume of the system. Now all we have to do is to determine 'I/J (p).An inspection of Eq. (100), with the approximation, Eq. (101) shows that 'I/J (p)satisfies the equation

8'I/J(p)8P + 2nav'I/J(p) = 0,

for p ;::: a/(2v) , with solut ion

'I/J (p) = (l /to)e- t/ to for o>a/(2v).

(103)

(104)

Here to is the mean free time given by to = (2nav)-1 . In the case th at p <a/(2v) , one can easily solve the full equation with the delta function to find

Combining these results with Eq. (102) and inserting them in Eq. (99) we recover the result , Eq. (90) for the KS entropy, equivalentl y the positive Lyapunovexponent, for the low density, equilibrium, random Lorentz gas. A similar, butsomewhat more elaborate calculation can be carri ed out, for the three dimensional case, to obtain exactly the same result as obt ained in the inform al theoryfor the KS ent ropy. To obt ain the largest Lyapunov exponent by more formalmethods, one has to resort to methods for dete rmini ng the largest eigenvaluefor products of random matrices. This is well describ ed in the literature [141 andwe will not pur sue thi s issue further here.

§6. Conclusions and Open Problems

In thi s ar ticle we have given a survey of the applicat ions of the kineti c theoryof dilute, hard ball gases to the calculat ion of quantities that characterize thechaotic beh avior of such systems. Result s have been obt ained for the KS ent ropyper particle and for the largest Lyapunov exponent of dilu te hard ball systems,as well as for the Lyapunov spect rum for the moving particl e in the dilute, ran dom Lorent z gas, with nonoverlapping, fixed, hard ball scatterers. All of these


results are in good to excellent agreement with the results of compute r simulations. In the st udy of the largest Lyapunov exponent , we have developed a veryinteresting clock model which seems to explain many feat ures of the behaviorof this exponent . Moreover, the method for treating the clock model revealsa deep and, perhaps unexpected , connectio n between the theory of Lyapunovexponents and the theory of hydr odynamic fronts.

We emphasize that the results given here apply to a dilute gas in equilibrium , but nonequilibrium situations have been t reated by these methods as well.For example, it is possible to calculate the Lyapunov spect rum for a dilu te, random Lorent z gas in a nonequilib rium steady state produced by a thermostattedelectric field, at least for small fields, and to obt ain result s that are in excellentagreement with compute r simulat ions [33, 34, 32]. Calculat ions are current lyunderway for the largest Lyapun ov exponent for a hard ball gas, when the gasis subjected to a thermostatted, external force that maintains a steady shearflow in the gas. Furthermore, one can use kinetic th eory to calculate the Lyapunov spectrum for tr ajectories on th e fractal repeller for a Lorent z gas withopen, absorbing bound aries [1 3, 35]. Such results are useful for underst andingescape-rate methods for relating chaotic quantities for tra jectories on a fractalrepeller of an open system to the transport prop erties of the syste m, as describedby Gaspard and Nicolis [9, 10, 36]. These results will event ually be exte nded tohard ball gases as well.

Of course , many problems remain to be solved. Here we ment ion some of themost immediate probl ems:

1. All of the results described here apply to hard core syste ms. Th at is thepar ticles interact with a potential energy that is either zero, beyond a givenseparation, or infinite, below that separation. It is worth st udying theproperties of dilu te syste ms with smoother potential energies for a numberof reasons: (a) The results of Rom-Kedar and Turaev 11 2] suggest that thedynamics of par ticles with short ranged but smoot h pote nt ials may exhibitregions of nonhyperboli c behavior. It is important to know more aboutthese regions and to assess their effect on the overall chaotic behavior ofgases interacting with such potentials. (b) We know very lit tle about thechaot ic prop erties of dilute gases interacting with long range forces, suchas Maxwell molecules and Coulomb gases.

2. An open problem, even for dilute gases, is to obt ain the complete spectrum of Lyapunov exponents for a hard ball gas. Thi s is a very challengingproblem in mathemati cal physics, and no easy approach is in sight . Thereis a very tantalizing set of numerical results (see the paper by Posch andHirschi in thi s volume 1371) showing that the smallest nonzero Lyapunovexponents have a hydrodynamic st ructure , in that the exponents themselves seem to scale as the inverse of the linear size of the system, andthat the spatial deviat ion vectors seem to form collect ive modes of bothtransverse and longitudinal types. Eckmann and Gat [38] have proposed

276 R. van Zan , H. van Beijeren and J. R. Dorfm an

an explanation of this hydrodynamic-like behavior of the lowest Lyapunovmodes using techniques from the theory of random mat rices. It would beuseful to understand and to extend their results using kinetic theory methods.

3. The extension of these results to gases at higher densities remains anopen, challenging problem. The kinetic t heory of dense gases has exposeda numb er of effects caused by long range dynamical correlat ions betweenthe particles. Such effects include nonanalytic terms in the density expans ion of transp ort coefficients and long tim e tail phenomena in t imecorre lat ion functions, among others [4] . It is of some interest to see the effect of these dynamical correlat ions on the chaot ic prop erties of the gas, aswell. Furthermore, a high density hard ball system may form a glass, andthere would be useful information obt ained about the glassy state if onecould study the chaotic behavior of the gas through the glass transition.

4. The extension of the clock model to higher density systems can also be expected to reveal new and interesting phenomena connected to the effects ofdensity and other fluctuations in the gas upon the clock speed and relatedquantities . At present there are some weak numerical indicat ions [391 thatthe largest Lyapunov exponent might diverge in the limit of large numb ersof part icles as In N , where N is the numb er of particles in the system. Itmight be possible to confirm or to rule out this possibility by extendingthe clock model so as to include the effects of fluctuat ions in the fluid.

5. Lyapunov exponents and KS entropies are not the only prop erties th atcharacte rize chaotic systems. There are many more quanti ties, such astopological pressures, fractal dimensions, etc., that remain to be exploredby the methods out lined here.

6. A subject of considerable interest and act ivity is the physics of gases thatmake inelast ic collisions, i.e. the physics of granular materials [401 . Onewould like to know what the chaot ic prop erties of such syste ms might be,or , more generally, how to define such properties in nonstationary systems.

In conclusion, we have described in this art icle only the first steps, taken overthe last few years, to develop useful methods for the calculations of Lyapunovexponents and KS entropies for the systems of particles th at can be treat ed bykinet ic theory. We are particularly delighted that kinetic theory has somet hingto cont ribute to the field of the chaotic behavior of large systems of particles,and th at the ideas of Maxwell and Boltzmann are st ill having new and fruitfulappli cat ions.

Acknowledgements: The authors would like to th ank Professor Harald A.Posch, Professor Christoph Dellago, and Dr. Arnulf Latz for many helpful conversat ions, and for collaborat ions on much of the research describ ed here. here;We are also grateful to Professor Christoph Dellago and Professor Harald A.

Kinet ic Theory Estimates 277

Posch for supplying Figs. 1, 2 and 7. H. v. B. and R. v. Z. are support ed byFOM , and by th e NWO Priority Program Non-Linear Systems, which are financially supported by the "Nederlandse Organ isatie voor Wetenschapp elijk Onderzoek (NWO)". J . R. D. would like to thank the Nat ional Science Found ationfor support under grant PHY-9600428.

References

[I] S. G. Brush , The Kind of Motion We Ca ll Heat : A history of the kinetic theory ofgases in the 19th cent ury, Studies in statistical mechan ics vo!. 6, Nort h-Holland,Amsterd am (1976).

[2] L. Boltzmann, Lectures on Gas T heory , S. G. Brus h, t rans!., Dover Publications,New York, (1995).

[31 S. Chapman and T. Cowling, The Mat hematical Theory of Non-uniform Gases,third editi on, Cambridge University Press, Ca mbridge (1970).

[4] ,J. R. Dorfman , and H. van Beijeren, in Stat istical Physics, Part B, B. ,J. Berne,ed. , Plenum Publishing Co., New York , (1977) ,p.65.

[51 P. Ehrenfest and T . Ehrenfest, The Conceptual Foundations of t he Stat ist icalApproach in Mechanics, Cornell University Press, Ithaca, (1959).

[6] G. E. Uhlenbeck and G. W. Ford , Lectures in Statistical Mechani cs, AmericanMathemat ical Society, Providence, (1963).

[7] ,I. L. Lebowitz, Physica A, 194, 1, (1993).[81 V. I. Arno ld and A. Avez, Ergodic Problems of Classical Mechanics, W. A. Ben

jamin, Inc, New York , (1968) .[91 P. Gas pard,C haos , Scat terin g Theory and Statist ical Mechanics, Cambridge Uni

versity Press, Cam bridge (1998).[10] J . R. Dorfman , An Introdu ct ion to Chaos in Non-equilibrium Stat ist ical Mechan

ics, Cambridge University Press, Cam bridge (1999).[11] H. van Beijeren, .1 . R. Dorfman, Ch. Dellago and H. A. Posch, Phys. Rev. E 56 ,

5272 (1997).[12] V. Rom-Kedar and D. Turaev, Ph ysica D 130, 187(1999).[1 3] H. van Beijeren and J. R. Dorfman, Phys. Rev. Lett . 74 , 4412 (1995) ; Erra t um

Phys. Rev. Let t . 76 , 3238 (1995).[1 4] H. van Beijeren , A. Lat z and J . R. Dorfman , Phys. Rev. E 57, 4077 (1998).[15) R. van Zon, H. van Beijeren and Ch. Dellago, Phys. Rev. Let t. 80, 2035(1998).[16] R. van Zon, H. van Beijeren, J . R. Dorfman , Kinetic Theory of Dynam ical Sys-

te ms, in "Proceedings of t he NAT O-ASI on Dynam ics: Models and Kine tic Methods for Non-equilibrium Many Body Systems, J . Karkheck (ed .) Kluver, 2000".

[171 Ch. Dellago, H. A. Posch and W. G. Hoover , Lyapunov instability in a syste mof hard disks in equilibrium and nonequilibrium ste ady states, Phys, Rev. E. 53 ,1485-1501 (1996).

[18] Ya. G. Sinai, Russian Math. Surveys 25 ,137, (1970) ; Ya. G. Sinai (ed.) , Dynamical Systems, World Scientific Publishing (1991).

[191 E. Ott , Chaos in Dynamical Systems, Cambridge University Press, Cambridge(1993).


1201 Ja. B. Pesin , Sov. Math . Doklady 17 (1976) , 196; Reprinted in R. S. MacKayand J . D. Meiss, Hamiltonian Dynamical System s (Adam Hilger , Bristol , 1987).

[21] J . R. Dorfm an, A. Lat z, and H. van Beijeren, Chaos 8 , Issue 2 (1998) 444[221 J . R. Dorfm an , (to be publi shed) .[231 H. van Beijer en, I-I. Kruis, D. Panja and J. R. Dorfman , to be published .[241 C. Cercigna ni, T heory and Application of the Boltzmann Equation, Scot tish Aca

demic Press, Ed inburgh/London (1975).[251 W. van Saarloos, Phys. Rev. A 37, 211(1988) ; U. Ebert and W. van Saarloos,

Ph ys. Rev . Lett . 80, 1650 (1998); U. Ebert and W. van Saarloos, Front propagat ion into unstable states : Universa l algebraic convergence towards un iformlytranslating pulled fronts, Physica D (to appear).

[26] Ch. Dellago and H. A. Posch, in [271 , pp . 68~83 (1997) .[27] Special issue on th e Proceedin gs of the Euroconference on Th e Microscopic Ap

proach to Complexity in Non-Equilibri um Molecular Simulations. CECAM atENS-Lyon, France,1996, edited by M. Mareschal, Ph ysica (Amsterdam ) 240A(1997) .

[28] G. A. Bird, Molecular Gas Dynamic s, Clare ndon, Oxford (1976) .[29] E. Brunet and B. Derrid a, Phys. Rev. E 56 , 2597 (1997) .[30] A. Lernarchand and B. Nawakowski, J . Chern. Phys. 109, 7028(1998) .[31] Ch . Dellago and H. A. Posch, Phys. Rev. E, 52 , 2401, (1995).132) Ch . Dellago and H. A. Posch, Phys. Rev. Lett. 78 , 211 (1997).[ 3:~1 H. van Beije ren, J. R. Dorfman , E. G. D. Cohen , Ch. Dellago and H. A. Posch,

Phys. Rev. Lett . 77 , 1974 (1996).[341 A. Lat z and H. van Beijeren and J . R. Dorfman , Phys. Rev . Let t . 78 , 207 (1997) .[351 H. van Beijeren, A. Lat z, and J . R. Dorfma n (to be publi shed) .[361 P. Gaspard and G. Nicolis, Phys. Rev. Lett . 65 , 1693, (1990).[37] Lj. Milanovic, H. A. Posch and Wm. G. Hoover , Mol. Phys. 95 , 281 (1998); H. A.

Posch and R. HirschI, Simulat ion of billiards and hard bod y fluids, t his Volume[38] J .-P. Eckmann and O. Gat , J . St rat . Phys. 98 , 115, (2000) .[39] D. J . Searles, D. J . Evans and D. J . Isbister, in [27], pp. 96~104 (1997) .[40] M. H. Ern st , in "Proceedings of th e NATO-ASI on Dynamics: Mod els and Kineti c

Methods for Non-equilibr ium Many Body Systems", J . Karkheck (ed.), Kluver ,2000.

Simulation of Billiardsand of Hard Body Fluids

H.A. Posch and R. HirschI

Contents

§l. Int roduct ion .§2. Dynam ics in Phase and Tangent Space§3. Billiards in Two and Three Dimensions

3.1 The "Diamond" and t he "Stadium"3.2 The Cylindrical Billiard .

§4. T he Lorentz Gas . . . . . . . . . . . . . .4.1 Th e Galton Board or t he Periodic Loren tz Gas4.2 The Externally-Driven, T her mostated Lorentz Gas4.3 The Random Loren t z Gas . . . . . . . . . . . . . .

§5. Lyapunov Spect ra for Hard Diskand Hard Sphere Fluids in Equilibrium . . . . . . .

§6. "Lyapunov Modes" for Hard Ball Syst ems . . . . . .§7. Hard Ball Systems in Nonequilibrium Stead y Stat es§8. Conclusions and OutlookReferences . . . . . . . . . . . . . . . . . . . . . . . . . .

280281283286287288288294295

298301310311313

Abstract. Recent computer simulations have contributed significant ly to ourund erstanding of the Lyapunov instability of hard par ticl e systems in equilibrium and in nonequil ibrium steady states. We discuss a very genera l method forthe computation of the full Lyapunov spect ra and apply it to billiards and tomany-body hard disk and hard sphere systems. The velocity corre lat ion function of billiard flows is also discussed . For hard disk and hard sphere systemst he perturbed states associated with the sma llest Lyapunov exponents (in absolute magnitude) are shown to reveal collect ive dynamic mod es. We st udy theproperties of the se mod es and prov ide exa mples for hard disk systems in twodimensions. It is suggested that there is a connection with the dyna mic mod esfamiliar from fluctuating hydrodynamics. Th e largest Lyapunov exponent, however, is associated with localized perturbations in th e fluid.

280 H.A. Posch and R. Hirschi

§1. Introduction

Computer simulat ions have been, and will continue to be, an essent ial drivingforce in th e development of statis tical mechani cs. They have frequently led toqualit atively new and unexpected insights, which are now gradually being castinto theorems by mathematical physicists. One example is the existence of multi fractal measures in phase space for thermost ated nonequilibrium systems insteady states, anot her the appearance of collective hydrodynamic modes in thetangent-space dynamic s of hard body fluids in equilibrium. This book providesmore examples of a similar nature. The st udy of billiards and of hard ball systemshas also benefitted enormously from this exchange between simulat ion and theo ry.

In this ar ticle we review recent work on billiards and hard ball systems intwo and th ree dimensions, where th e emphasis is on the computation of theLyapunov spectra of such systems. The interest in billiards and hard par ticlesyste ms has many roots. Firstly, theoretical treatm ent is comparatively simple,and secondly and even more importantly from the physical point of view, theirergodic and dynamical properties are thought to be typical of realistic physicalsyst ems. Billiard s are models for many problems in classical mechanics, stat ist ical physics, optics and acoust ics. Many-bod y hard sphere syste ms are paradigmsfor fluids and serve as reference systems for sophist icated perturbation theoriesfor the st udy of dense gases and liquids. The investigation of their Lyapunovspectra has uncovered links to the theory of transport processes in nonequilibrium fluids, and the recent ly-discovered mode-like st ructures associated withinfinitesimally perturbed states , which are discussed in Paragraph 6, promise toestablish links with fluctuating hydrodynamics.

Our method of computi ng the Lyapunov spectra for systems wit h hard elast ic interactions is based on a general scheme developed recently [1]. It has beenapplied to systems both in equilibrium and in nonequilibrium steady states andmay involve many degrees of freedom. Thi s algorithm is outlined in Paragraph 2.The models st udied in Paragraph 3 include various types of billiards such as thediamond and the stadium [21 . T he Lorentz gas in two and three dimensions is ofparticular importance. The externally-driven version of this model is, arguably,the simplest system which displays the relevant prop erties, such as ergodicityand the existence of multifractal attractors and repellors in phase space, thoughtto be typical of many-body systems. Our recent work [2-6] on this model is summarized in Paragraph 4. In the following Paragraph 5 we discuss the applicat ionof our methods to the study of hard ball syste ms in two [1 ,7] and three dimensions [8,91. Our recent discovery of long-wavelength pat terns in connect ion withthe study of t he small (in absolute value) Lyapunov exponents for hard disk andhard sphere fluids is the topic of Pa ragraph 6. These results suggest a new linkwith classical fluid dynamics. Since nonequilibrium steady st ates are believedto be understood reasonably well, only a very short account is given in Paragra ph 7. We conclude with a discussion and some remarks for future studies inParagraph 8.

Simul ation of Billiards and of Hard Body Fluids

§2. Dynamics in Phase and Tangent Space

281

The phase space dynamics of a particle system with elast ic hard collisions ischaracterized by a cont inuous flow or streaming, which is interrupted by discretecollisional events . Recently, a very general scheme for the computation of theLyapunov spect rum for such systems has been developed [1 ,2 , 51 . In the followingwe outline this scheme, which is a generalizat ion of the classical algorithm byBenettin et al. [10,11] and Shimada et al. [12] .

We denot e by I'{r) an L-dimensional state vector at t ime t , and writ e theequations of motion for the continuous streaming in the form

with the formal solution

r = F (r) ,

f(t) = <1>t(f(O)) .

(1)

(2)

The system is assumed to be bounded. Collisions occur at times {Tl ' TZ , T3 , ... } ,

and in each encounter an initial state I', is mapp ed into a final st at e I'f by thecollision map

ff = M(fi ) . (3)

M (I') is assumed to be reversible and differentiable with respect to the phasespace variables. Then the combined t ime evolut ion in phase space , the so-calledreference trajec tory, becomes

r (t) = <1>t-1"n 0 M 0 <1> 1"n - 1"n - 1 0· ·· 0 <1>1"2-1"1 0 M 0 <1>1"1 (f (0)) . (4)

The stability of thi s trajectory with respect to small perturbat ions of the initial condit ions is obt ained by introducing (infinitesimally displaced) perturbedtrajectories I', (t) , connected to I' (t) by a parameterized pa th with parameters such that lims-+o I' , (t) = I' (t) . The vectors 5f (t) = lims-+o (Fs (t) - f (t)) / s.are tangent vectors and evolve cont inuously during the st reaming inte rvals according to the linearized equations of motion

(5)

Here, D (f) = 8F/ 8f is the Jacobian mat rix. The formal solut ion of this linearized equation is written

5f(t) = L t · 5f (0), (6)

(7)

where the prop agator matrix L t is a time-ordered exponent ial of th e integratedJacobian [131 . Dur ing a collisional event a tangent vector is instantaneouslymapped, according to (3), from 5fi to 5ff by

8M5ff = S · 5f i == 8f

i. bTi ,

282

l'

H.A. Posch and H.. Hirschi

'1

Figure 1: Schemat ic represe ntation of a collision for a reference trajectory (smooth)and an offset t ra jectory (das hed) in phase space.

where S is the linearized collision map . In Fig. 1 the collisions of a reference trajectory (smooth) and of an offset trajectory (dashed) are schemat ically depicted.

By strict ly keeping only terms linear in of i , the linearized map (7) becomes

8M [8M ]Ofj =8fi · ofi+ 8fi ·F(fi) -F(M(fi)) OTc · (8)

In this map , OTc is the delay time , by which the collision of the offset t raj ectoryis delayed with respect to that of the reference t ra jectory. It may be positiveor negati ve, and it depends on I', and - linearly - on of i (see Fig. 1). As willbe shown for the applications below, OTc is determined from the spatial offsetcomponent ort hogonal to the collision plan e and the spatial normal velocity ofthe reference traj ectory immediately prior to collision. Combining the soluti on(6) for the streaming intervals with the lineari zed collision map (7) applied atthe collision times, the trajectories for t he offset vectors in tangent space maybe constructed according to

Thi s is the linearized flow corresponding to the phase flow in (4).For chaot ic systems the norm of this infinites imal perturbation grows or

shr inks exponentially depending on the direction of the initi al displacement

Simulation of Billiards and of Hard Body Fluids 283

(10)

vector 15f(O). Thi s motivates the definition of the Lyapunov exponents accordingto

. 1 l15f (t )1A= t~ t log l15f (0)1 '

The multiplicati ve ergodic theorem by Oseledec 114,15] states th at for ergodicsyste ms, for which the ergodic domain is the full L-dimensional phase space,there are L orthonormal initial vectors 15ft (0) , which yield a set of L exponents{At} . The ordered set {AI 2 A2 2 . . . 2 h} is commonly referred to as theLyapunov spectrum of the system. The At are indep endent of the metric and ofthe initial condit ions of the reference trajectory.

To be precise, our simulat ion takes place in an extended phase space , and thedynamics may st ill have to fulfill a number of const raints , such as the conservation of energy and momentum which will confine the flow to a manifold of lowerdimension. As a consequence, a number of Lyapunov exponents vanish , one foreach dynamical constraint, plus an add it ional exponent for the non-singular expansion of perturbations along the trajectory. Thus, a two-dimensional billiardwith a four-dimensional extended phase space and a consta nt kinetic energywill have two vanishing exponents. For a hard sphere gas in three dimensionseight exponents have to vanish, three each for the conserved momentum and forthe center of mass, one for the conserved energy, and one for the perturbat ionin flow direction. From the numerical point of view, these vanishing exponentsare useful to assess the accuracy of the simulat ion. For symplectic systems theLyapunov spectrum has also a very pronounced symmet ry. We will come backto this point below.

For the numerical implement ation the reference tra jectory (4) and a complete set of tangent-vector trajectori es (9) are computed simultaneously. Thedifficulti es associat ed with the choice of the unknown initial vectors {15fl(O)} ,taken to be orthonormal and with arbitrary orientation , and with the roundingerrors of the computer are overcome by periodic reorthonormalization of the offset vectors during the simulat ion [11 ,1 6]. The Lyapunov exponents are obt ainedfrom the t ime-averaged logari thms of the corresponding normalizing factors.Since the number of equat ions required for a full Lyapunov spectrum increaseswith the square of the numb er of particles, the numb er of degrees of freedom islimited to a few thousand with the power of a present-day workstation.

§3. Billiards in Two and Three Dimensions

Billiards are flows on a manifold with a closed bound ary, where a point par ticlemoves with consta nt speed v within the domain and is elastically reflected fromthe bound ary. An in-depth treatment of various types of billiards may be foundelsewhere in this book.

So-called everywhere- dispersi ng billiards were st udied by Sinai [171 and Bunimovich 118] . The boundary is convex (as seen from inside of the domain) such

284 H.A. Posch and R. HirschI

2K

Figure 2: The "diamond" and t he "stadion" in two dimensions.

that a narrow local bundle of traj ectories is dispersed by the reflections everywhere on the boundary due to the negative curvat ure. These systems wereshown to be chaot ic and K-flows with at least one positive Lyapunov exponent .An example is the "diamond" shown in Fig. 2.

Anot her class involves only f ocusing collisions on a concave boundary (asseen from inside of the domain ). T he most famous example is the "stadion"in Fig. 2, for which the boundary consists of two semicircles joined smoothlytogether by straight lines. T he presence of these st raight segments is essent ialfor the sta dion to be ergodic and to have the K property, as was shown byBunimovich [19, 201.

The boundary of semi-dispersi ve billiard s cont ains also a neut ral , non-dispersive component. A three-d imensional example of such a system is the "cylindri cal billiard", shown in Fig. 3 below, which is non dispersive for the directionsparallel to the axes of th e cylindrical components forming the boundary.

In the following the position and momentum component s of the state vectorI' = (q , p ) and of the offset vectors 8f = (8q , 8p ) are treated separa tely. For th ebilliards th e streaming between successive collisions is a free motion for which(1) simply is q = pj m, p = O. Eqs. (2) and (5) become

and

q(t) = q (O) + (p(O) jm)t , p (t ) = p (O)

8q(t) = 8q (0) + (8p(O) j m)t, 8p (t) = 8p (O),

(11)

(12)

respectively. The collision map (3) for specular reflect ion , for which only thesign of the velocity component perp endicular to the sur face is reversed , is

q f = qi , Pf = Pi - 2 (p i·n) n , (13)

where n is the unit vector perpendicular to the surface at the collision point .

-1

1

Simu lation of Billiards and of Hard Body Fluids

1r-

1

Figure 3: T he "cylindrica l" billia rd in three dim ensions.

285

The linearized collision map follows from (8) and yields [1]

bql = bqi - 2 (bqi . n ) n ,

bPI = bPi - 2 (bPi ' n ) n - 2 ((Pi ' bn) n + (Pi ' n ) bn ).(14)

Here, the vector bn == Du]{)qi . bqc is tangent to the bound ary and denotes thevariat ion of n due to the spatial displacement between the collision points forthe offset and reference trajectory, bqc == bqi + (p;/m) . bTe (see Fig. 1). Thedelay time bTc is most conveniently obtained from

(bqi . n )bTc = - ,

(p;/m ' n )(15)


the trajectory displacement perpendicular to the collision surface divided bythe norm al velocity. The terms containing bn in (14) are a consequence of thecurvature", of the boundary and vanish for flat surfaces. If the surface is partof a circle or a sphere, with center Xo and radius R, the norm al vector is simplyn = (q, - xo) / R, and points into a direct ion outside of the domain for concave boundaries, and inside for convex (where "concave" and "convex" refer toa viewing point inside of the domain) . It follows that bn = "'bqc, where x = 1/ Ris th e curvature. For a cylindrical surface, the component of bn parallel to theaxis of the cylinder vanishes.

We have discussed this algorithm in some detail since it is readily applied tothe billiards t reat ed below, and, with lit tle adaptation, also to hard ball systemsin two or three dimensions [1 ,8] . The observables usually st udied for billiards [211include the Lyapunov spect rum, the Kolmogorov-Sin ai ent ropy h tcs , the velocitycorre lat ion function of the flow, C(t) = (v (O) . v(t) )/ v2 , and the collisionalvelocity correlation decay of the Poincare sect ion, Cm - n = (v n . v m ) /v2 , whereV n is the velocity vector immediately after the n-th collision, and v = [v] . Forbilliard s such as the Lorent z gas, for which the basic domain is unfolded intoan infinite lat tice and permits the point par ticle to move through the lattice,the diffusion coeffic ient, related to the integral of the velocity autocorrelat ionfunction by the Green-Kubo relat ion, and the mean-square displacement arecommonly studied. Although our main concern is the Lyapunov spectrum, wepresent a few examples for some of these properties below. The applicat ionof return-tim e statist ics has been advocated by Artuso for systems exhibit ingintermittency and a power-law decay of corre lations [22].

3.1 The "D ia m on d" and the "Stadium". The diamond and the stadiumbilliards in Fig. 2 are examples for everywhere-dispersive and nowhere-dispersivesystems, respectively. The diamond consists of the arcs of four circles, withcurvature ", = 1/ R, which cross at the vertices of a square with diagonals oflength two (Fig. 2). The stadion is bounded by two semicircles of unit radius ,joined smoothly by two st raight lines of length 2",. The velocity of the pointpar ticle is uni ty, and the particle collides elastically with the bound ary. Bothbilliards have been invest igat ed extensively by Benettin [23] . For x = 0 they areintegrable, whereas for x > 0 they are known to be K systems. In the transitionregion for small '" the maximum Lyapunov exponent scales according to

Al '" ",f3 , (3 = 1/2. (16)

In Fig. 4 thi s is demonstrated for th e diamond (label D) and for the stadion(label S). The same power law is found for other dispersive billiards , and evenfor sequences of products of randomly perturbed conservat ive matri ces, if theyare hyperboli c on the average. Thus, (3 = 1/2 seems to be a universal exponentfor the onset of stochastic behavior in such Hamiltoni an syste ms. There exists a formal analogy with the mean-field app roximat ion of second-order phasetransitions [231.

Simulat ion of Billiards and of Hard Body Fluids

10

.....

287

0.1

0.01

0.0010.0001 0.001 0.01

K

0.1 10

Figure 4: Log-log plots of Al as a function of '" for the diamond (D) and t he stadium (5) . The fit of Eq. (16) to th e asy mptotic da ta gives for the diamond Al =(1.64 ± 0.03) exp{(0.501± O .OO:~)", } , and for t he stadion Al = (1.26± 0.03) exp{(0.496 ±0.003)K} .

The absolute value IC(t)1 of the velocity correlation function for the diamond is shown in Fig. 5 for Ii = 0.25 and Ii = 1/ 24 , respectively. The t ime isgiven in units of the Lyapun ov time. The correlat ion functions oscillat e, and theasymptotic decay of the amplitudes is exponent ial. It should be noted that forvery small Ii the syst em, although ergodic in principle, does not trace out thefull phase space in the t ime available to the simulat ion (about 108 collisions).Therefore an ensemble average over 105 trajectories of 1000 time unit s each,with initial conditions chosen randomly from the Lebesgue measure in Birkhoffcoordinat es, was used to generate these correlation funct ions and the Lyapunovexponents in Fig. 4.

Also, diam ond billiard s in three dimensions have been st udied recent ly [241 .Th e preliminary results confirm the conclusions reached with the two-dimensional case. There are 6 Lyapunov exponents in the extended phase space, ofwhich two vanish as explained before. As required for symplectic systems, thenonvanishing exponents are paired such that the Lyapunov spectrum becomes{>'1 ' >'2, 0, 0, - >'2, ->.t} . Both positive exponents are found to scale according to(16), with exponents very close to f3 = 1/2 [24].

3.2 The Cylindrical Billiard. Anot her three-dimensional billiard currentlyunder investigation is depicted in Fig. 3. It consists of 6 intersecting cylindrical


15 20 25

All

IC( I>I

0.1

0.01

0.001

0.0001a 5 10

11,,,

I

," ,I ~ f I

" '\ II" ,"\iI' .,' ~,., II 11'1,' r., ,I II ~ ,, I II.' I_ • ~ II ,I I I tc, • ~ II I' t I " "

I 111 .'. ,II',,.I "ll Iltl

t. 'I • ~ II II II'

t I II "", I,.

I' I'" " "" ~t ". ,,' I'tlI ," ; ', I. "~

I ~:: : ~ :: :: I,, I I . , t'"I I I ~

30 35 40

Figure 5: Velocity correlat ion functions for the diamond as a funct ion of t ime. The timeaxis is given in uni t s of the Lyapunov t ime I /AI. Dashed line: K = 0.25, Al = 0.809,and t he amplit ude decays rv exp{-0.236Alt} . Smooth line: K = 1/24

, Al = 0.404, andthe amplit ude decays rv exp{- 0.067Alt}.

surfaces, with rad ius R, which are arranged such that there is a one-dimensionalneut ral, non-d ispersive component parallel to the axes of the cylinders. Thisfeature makes the model semi-dispersive. However , thi s additional property doesnot seem to affect the scaling prop erties of the Lyapunov exponents with K, =1/R. Preliminary dat a indicate that (16) is obeyed with (3 ~ 1/2 [241 .

§4. The Lorentz Gas

If a bounded Sinai billiard with a circular or spherical scatterer is unfolded intoan infinite array of scatterers, one obtains the associated Lorent z gas. If a pathexists on which the point particle can move to infinity without colliding with thescatterers, the billiard is said to have an infinite horizon, and no upper boundexists for the time between successive collisions. If no such path exists and thetim e between collisions is bounded, the horizon is said to be finite.

4.1 The Galton Board or the Periodic Lorentz Gas. T his popul armodel , originally introduced by Galton who used it as a teaching tool as early

Simulatio n of Billiards and of Hard Body Fluids 289

Figure 6: Geomet ry of t he perio dic Lorentz gas with scatterers arranged on a t riangularlat t ice. The trajectory for t he field-free case consists of st raight segme nts betweensuccessive collisions. T hey ar e curved, as depicted , for t he driven Lorentz gas , whereE is t he externally applied field.


as 1873 [25], consists of a point particle moving freely and with constant speedv th rough a fixed ar ray of circular scatterers, on which it is elast ically reflected.In our work [1,2 ,5 ,26] the scatters are arranged on a tri angular lattice as inFig. 6, although other geometries have been considered as well by various authors[21 ,27,28] . For the field-free case the particle trajectory is a st raight line betweencollisions. In the following, the radius R of the scatterer, the wanderer-particlemass, and its velocity v are all taken unity, thus est ablishing our reduced units.The geometry is fully characterized by the numb er density P = 1/ A, whereA = v'3a2 /2 is the area of the hexagonal unit cell, and a is the lattice constant .The Lyapunov spectrum takes the form {>'1' 0, 0, - Ad.

In Fig. 7 the maximum Lyapu nov exponent Al is shown as a function ofthe density P [2] . The vertical lines mark, from left to right , (a) the criticaldensity Pc = (v'3 / 8)R- 2 ~ 0.2165R- 2 of the tr ansition from infinite to finitehorizon; (b) the close-packing density Po = (v'3 /6)R- 2 ~ 0.2887R - 2 , abovewhich the point particle is confined to a single cage formed by three scatterersand the diffusion coefficient vanishes; (c) the densit y Poo = (2v'3 /9)R- 2 ~

0.3849R-2 , for which the free volume accessible to the point particle vanishesand the collision frequency and, hence, the Lyapunov exponent diverges.

The confining transit ion at Po is analogous to the fluid - solid phase transitionof a gas and has been studied in Ref. [29] within the framework of a closelyrelat ed model , the correlated cell model invented by Alder et al. [30] . In thismodel, Al increases monotonously with p with the except ion of a small region

10

0.10.1

Pc Po

0.3

IIIIIIIIIIIIII1II1IIIII1IIII

Poo:II

0.4

Figure 7: Positive Lyapunov exponent for the two-dimensional field-free Lorentz gasas a function of the density p of the scatterer particles. R is the scatte rer radius, andv is the speed of the moving particle.

Simulation of Billiard s and of Hard Body Fluids 291

around the phase-transition density, where it exhibits a local maximum . Thisdensity is equal to the density at which the channels connect ing neighboringcells are closed, and corresponds to Po for the Lorent z gas.

For P > Po the syst em is a diamond-like billiard and the velocity correlat ionfunction oscillates with an amplit ude bound from above by an exponential.

Although the transition from an infinite to a finite horizon at the crit icaldensity Pc does not leave a mark in the density dependence of Al in Fig. 7, itsinfluence on the decay prop erties of the velocity correlat ion function IC(t)1 ofthe flow is most dramatic [27J . In the finite-horizon case, Pc < P < Po , C(t)oscillates, periodically for larger p and in a much more complicated fashion forsmaller [27], with an amplitude decaying, in essence, exponentially with time.An upper bound

IC(t)1< Aexp(-a to), A > O,a > 0,0 < <5 :::; 1 (17)

has been shown to exist [31 ,321, which is numerically confirmed in Fig. 8, wherewe plot IC(t)1for a density p = 0.25R- 2 . For systems with an infinite horizon,

1.00000

0.10000

lC(t)1

0.01000

0.00100

0.00010

0.00001o 5 10

p =0.25 R-2

I"J \' ,, \ .

1\' 1,.'-_I 1/,I ' , \: ~ \I II II II II Ir II I

20 25 30

Figure 8: Semi-log plot of the velocity correlat ion funct ion for th e field-free periodicLorent z gas for p = O.25R - 2

. The horizon is finite. T he t ime axis is given in units of th eLyapunov time I /Al . T he maximum Lyapunov exponent is Al = (2.352 ± O.OOl)v/ R.Smooth curves denot e posit ive, dashed curves negat ive parts of C{t). Th e correlat ionfunction was computed from a single tr ajector y.


however, theoret ical arguments have been given for an algebraic decay, namely[28, 331

IC(t )1 rv A/t (18)

with a computable A. Thus, the diffusion coefficient does not exist in thi s case.The prediction (18) seems to be confirmed by th e computer experiments

121,27,281 - to some extent ! In Fig. 9 we show C(t) for a density P = 0.1847R - 2 ,

slightl y below p.: The t ime is given in units of the Lyapunov t ime 1/)'1 , andAl = (1.3433 ±0.0001)v/ R . For the collision frequency we find v = 0.880755v/ R .This geomet ry corresponds precisely to the condit ions of Fig . 2 of the paper byFriedm an and Mart in [271 . T heir uni t of time is bigger than ours by a factor of2.5. In Fig . 9 an asympt ot ic decay consistent with (18) is found for A1t > 40.However , thi s time is too long for a correlation function to be trusted due tothe Lyapunov-induced growth of compute r round-off errors [21,34]. It has beenargued by Ruelle [341 that correlati on functions in the presence of noise are

1.0000

0.1000

\C(t)1

0.0100

0.0010

0.0001

10

p = 0.1847 R-2

100 1000

Figure 9: Velocity correlation function for the field-free periodic Lorentz gas for p =O.1847R - 2

. The horizon is infinite. The time axis is given in units of the Lyapunovtime 1/Al' where the maximum Lyapunov exponent is Al = (1.3433±O.OOOI) v / R. Thecorrelation function is an average over 104 trajectories with initial conditions sampleduniformly with respect to the uniform Lebesgue measure in Birkhoff coordinates.

Simulati on of Billiard s and of Hard Body Fluids 293

reliable only for times t « Iln~I /Al , where ~ is the noise due to finite computerprecision. For double-precision ar ithmet ic ~ ::::; 10- 16 . An exponential decay ofcorrelat ions, with a rat e correspo nding to the metric entropy of the syst em,should be expected for larger times. The corre lat ion function in Fig. 9 is anaverage over 104 tr ajectories with initi al conditions randomly selected from th eLebesgue measure in Birkhoff coordin at es. Exper imentally we have found th atthe fluctuations in the wing are domin ated by the square root of thi s numb er. Wemay conclude that the main contribut ion to t he fluctuations afflicting the tailof C(t) in Fig. 9 are of statist ical origin and not due to Lyapunov-induced errorenhancement of the finite computer precision. We have encountered th e samebehavior already in Fig. 5 for the velocity correlation function of the diamondbilliard.

A similar result is found in Fig. 10 for a density P = 0.10R- 2 well below p.:Cum grana salis, the Figs. 10 and 9 may be taken as a confirmat ion of thetheoretical expectat ion in (18) . It is surprising that the Lyapunov inst abili tydoes not spoil the tails of the corre lation functions completely [35J . At any rat e,a careful assessment of these errors is required as has been argued for in Ref. [21J .

1.0000

0.1000

cro

0.0100

0.0010

0.000110 100 1000

Figure 10: Velocity correlation function for th e field-free periodic Lorent z gas for p =0.lOR- 2

. The horizon is infinit e. The t ime axis is given in units of t he Lyapunov timeI /Al , where th e maximum Lyapunov exponent is Al = (0.6618 ± 0.0002)v /R. Thecorrelat ion function was computed from a single trajectory.

294 B .A. Posch and R. Hirschi

4.2 The Externally-Driven, Thermostated Lorentz Gas. The Lorent zgas has become a paradigm for the st udy of t ransport processes, and the fielddriven, thermost ated version of thi s model [36] is, arguably, the simplest modeldisplaying the characteristic features believed to be relevant [26] for a flow innonequilibrium steady states . The point particle carri es a charge c, taken to beunity, and is acted on by a homogeneous external field E of magnitude E , whichdr ives the system away from equilibrium. Between collisions the point-particle isaccelerated by the field. To achieve a stationary nonequilibrium stat e, th e kinet icenergy is const rained to be constant with a Gaussian thermostat [13, 37,3 8]. Theequat ions of motion for the streamin g are

q = p /m, j> = cE - (p, (19)

(20)

where the thermostat variable ( == c(p · E) /p2 is a momentum-like var iablewhich changes sign upon tim e reversal. The motion equat ions for the tangentvector 8r == {8q , 8p} are obtained by linearization.

The collision map (13) is st ill valid , but a term has to be added to thelineari zed map (14), which then becomes [1, 4, 5]

8qf = 8qi - 2 (8q i . n) n

8pf = 8Pi - 2 (8p i . n) n - 2 ((P i ' 8n) n + (Pi ' n) 8n)

+ 2c (n . E) 8qi . n [n + Pi ~n Pf] .Pi· n P

T he addit ional term is a consequence of the driving field and of the thermostat.Th e equations (19) are invariant with respect to the t ime-reversal transfor

mat ion t -t -t ,q -t q , P -t - P [391. The motion is best observed in a Poincaremap defined by the collision in slight ly-modified Birkhoff coordina tes in therectangle {O' 1\ sin ,8;0 < 0' ::; 21r , - 1 < sin ,8 < I}, where 0' is the angle between the normal vector n at the collision point on the scatte rer and the fielddirection, and ,8 is the angle between the outgoing momentum P f and n . In thefield-free equilibrium case the distribution of Poincare point s corresponds to theuniform Lebesgue measure. For small and moder atel y-strong fields, however ,the phase-space distr ibution becomes a multifr actal strange attractor, which issingular almost everywhere [36,40,411. The Hausdorff dimension Do is foundto be equal to the phase space dimension, and the system is ergodic [26] . Theinformation dimension [42], however , is smaller, D1 < Do, and th e reduct ion indimensionality is proportional to the square of the applied field. The existenceof the mult ifract al at tractor has been confirmed theoretically [43,441 .

The following chain of relat ions holds in the nonequilibrium steady state :

(oq oj>) 1 dB L

(d ln 8V/ dt) = oq + op = -(d - 1)(() = - kB

di = L AI ::; 0,1=1

where d is the spatial dimension of the system, and kB is the Bolt zmann constant . T he equal sign applies only at equilibrium. Here, 8V is an arbit rary phase-

Simulat ion of Billiards and of Hard Body Fluids 295

space volume element , co-moving with the How. We note that it is not constantbut shrinks on the average and, in the long-time limit , gives rise to the multifract al attractor. (- . . ) denotes a t ime average . The rate of ent ropy production,dB/ dt, is positive in agreement with the Second Law of thermodynamics. If wedefine a kinet ic temp erature by kBT = p2 / m and identify the microscopic current with j = cp/m, it follows from the definition of the thermostat variableth at

(() = (j) . E = IJ"cE2 .

kBT kBT(21)

In th e last equality, IJ"c is the conductivity defined by the constitut ive relation(j) = IJ"cE. It follows from (20) that th e relevant transport coefficient is proportional to the negative sum of all Lyapunov exponents . Thi s establishes oneexample for a general relationship between transport coefficients and the associat ed Lyapunov spectru m of stationary nonequil ibr ium steady-state systems[45-471 ·

4.3 The Random Lorentz Gas. In th e random Lorentz gas th e scatteringdisks (d = 2) or spheres (d = 3) are not arranged on a grid but are rando mlydistributed in space . The horizon is finite even for the lowest scat terer densi ties.We consider only the case that th e scatterers are not allowed to overlap.

The Lyapunov spect ru m was computed for the two- and three-dimensiona lmodels over the full ra nge of densities and for a large range of external fields.The results are tabulated in Ref. [4[ . As an example we show in the Figs. 11

a-0.1

-0.2

106 0.8

0.2 0.4 .p/Po

Figure 11: Deviat ion of t he two posit ive Lyapunov exponents At (solid lines) and At(dashed lines) from t heir equilib rium values for t he th ree-dimensional driven rand omLorent z gas . The dependence over a wide range of densities and fields is shown . po =v2/8R - 3 is t he close-packing density. The exponents are given in units of p/mR , andth e field in units of p2/ m R; (From Ref. [4]).


o-0.2-0.4-0.6

o0.2

0 4E . 0.60.8

106 0.8

0.2 0.4 .p/Po

Figur e 12: Deviat ion of th e two negati ve Lyapunov exponents Al (solid lines) and A;(dashed lines) from t heir equilibrium values for the t hree-dimensiona l driven randomLorentz gas. The depend ence over a wide range of densities and fields is shown. po =v'2/8R- 3 is the close-packing densit y. T he exponents are given in units of p/mR, andth e field in uni ts of p2/ m R . (From Ref. [41) .

and 12 the deviation of the exponents for the driven th ree-dimensional Lorentzgas from their equilibrium values. The density is given in units of the closepacked density Po = V2/8R3, and the field in units of p2/ mR. The deviationsfor the two positive exponents ),.i (smooth lines) and ),.t (dashed lines) arenearly identical, at least in the range of fields covered by Figure 11, and thesame is true for th e negative exponents ),.1,2 in Figure 12. However , for largerfields significant differences appear [481 .

At present , there is no theory capable of explaining the complicated behavior apparent in these figures, even for moderat e densities and fields. Forvery small densities accessible to Bolt zmann-equat ion methods 19], a theory hasbeen developed by van Beijeren, Dorfman and co-workers for the two-[6, 46] andthree-dimensional 1491 model. It is accurate to leading order in R/ fl. , where R isthe scat te rer radius and fl. is the mean free path of the moving par ticle , and tosecond order in the applied field. If in two dimensions the spect rum is written{A+,O,O,),.-} , one finds for the field-free equilibrium case

),.±(O) = ± 2pRv(1 - ln 2 - C -In(pR2 ) ) , (22)

with p the scatterer density and C Euler's constant . This is modified in thepresence of an applied field according to

)"+(E) = ),,+(0) - 41L£2 + 0(£4), )"-(E) = ),,-(0) - 4~V £2 + 0(£4) , (23)


0.0000

-0.0001

-0.0002

1...+(0.001) +A-(O.OOl) x

1...+(0.002) *1...-(0.002) 0

+

* +

*

*-0.0003

o 0.0004 0.0008 0.0012

£*2/ 2p*

0.0016

Figure 13: Field dependence of the Lyapunov exponents A+(E) and )"- (E) for the twodimensional driven random Lorentz gas. We show the deviations from the respectivefield-free equilibrium values )..±(O) as a function of the square of the field for thereduced densities p" == pR 2 = 0.001 and 0.002 . c* == cR /v = cE R/mv2 is a reducedfield, where R is the scatt erer radius and E is the field strength . v, m, and c are thespeed, the mass and the charge of the moving particle, respectively. The smooth linesare the predictions of Eq. (23), the points are the simulat ion data. (From Ref. [6]).

where v is the collision frequ ency, and f = cE / p is a field parameter. A comparison with compute r simulat ions in Fig. 13 shows excellent agre ement for smalldensities and small fields [61.

In three dimensions there are two positive and two negative exponents andthe spectrum is written as {At, At, 0, 0, A2,Ai} . With simil ar kineti c-theorymethods as in the previous case, Latz et al. [491 predict that

(24)Ai(E)

v p* 1 1 Vf*2=f-p*(ln- +C -ln2 + 1/2) - (- =f -)-

R 2 3 36 Rp* '

v p* 1 1 Vl:*2=fRP* (ln 2 +C + ln 2 -1/2) - ("3 =f 36) Rp* '

where p* = 7rpR3 and f* = cER/mv2 are dimensionless density and field parameters, respectively. One observes that At + Ai = (() is independent ofi E {1,2} , and where the equality is an immediate consequence of Eq . (19) .This is a formulation of t he conjugate pairing rul e [47,50-52] which is obviouslyexac t ly fulfilled in the Lorentz gas, at least up to second order in the appliedfield . These results are compared in Figs. 14 and 15 to computer simulat ions byDellago et al, [31 . In equilibrium and for small den sities, th e agreement for the

298

0.100

0.001

H.A. Posch and R. Hirschi

0.00001 0.0001 0.001

PR3

0.01

Figure 14: Density depende nce of th e two positive exponents At (0) (squares) and At (0)(crosses) for the three-dimensional random Lorentz gas in equilibrium. T he solid anddashed lines are the theoret ical predictions of Eq . (24) for E = O. (From Ref. [31) .

positive exponents in Fig. 14 is excellent . T he negative exponents may be obtained from A;(O) = - At (O), i = 1,2 , a consequence of the symplectic natureof the equations of motion in equilibrium. But also th e agreement for the fieldaffected deviations of the Lyapunov exponents from their equilibrium values isvery sat isfactory as is inferred from Fig. 15.

§5. Lyapunov Spectra for Hard Diskand Hard Sphere Fluids in Equilibrium

Even more challenging and rewarding is the computation of the full Lyapunovexponents for many-bod y hard ball systems. The algorithm outlined above maybe readily genera lized to this case [1 , 8]. For simplicity we give here only therelevant formul ae for the field-free equilibrium case, applicable to two and threedimensions. For exte nsions of thi s method to nonequilibrium steady states werefer to the References [1 , 53].

The systems consist s of N par ticles with diameter (J and mass m , located at% ,j = 1 . . . N , and moving freely between collisions with momentum Pj ' Th estate vector I' = {q. , q2, . .. , q N, PI , P2, . . . , PN } is a 2dN -dirnensional vector ,and so are the tangent vectors of = {Oql, Oq2, . . . , OqN, 0PI , 0P2, . .. , op N }.

Simu lation of Billiards and of Hard Body Fluids 299

0.0001 0.0002*2 *

E / P

0.00000

,--.., -0.00002:>

C2 -0.00004"-'.........,,--..,0"-' -0.00006-eI

,--..,~ -0.00008"-'-e'---'

-0.00010

-0.000120

'"~ ..... "

, -,, "

-,,-, -, -,

'A+ II

'A+ I )(

2~- I 0

'A- DI

0.0003

Figur e 15: Field dependence of the Lyapunov exponents for the three-d imen sionaldriven random Lorentz gas . 'liVe show the deviations from the respective field-freeeq uilibrium values as a function of the square of the field for the density p = O.OOlR- 3

.

p" == 7rpR3 = O.OOb and E* == ER/v = cER/mv 2 are reduced dimension less densityand field parameters . Here, R is t he sca t terer radi us , and E is th e field strength. v, m ,and c are t he speed, t he mass and the charge of the moving particle, respectively. T hesolid lines ar e the theoretical pred ict ions (24) of Latz et al. [491. T he points and th edashed lines are computer simulation results by Dellago et al. [3]. Th e dashed lineswere obtained with the DSMC method mentioned in Par. 8.

Clearly, the motion of the phase point during a streaming period is given by

qj (t) = qj (0) + Pj (0) tim, Pj (t) = Pj (0) ; j = 1, . . . ,N, (25)

and the respective tangent-vector evolution by

Jqj (t) = Jqj (0) + JPj (0) tim, JPj (t) = JPj (0) ; j = 1, . . . ,N. (26)

When two particles, say k and l , collide, their position docs not change , buttheir momenta change according to the collision map

where qi == ql- qt and p i == p i - p h, are the relative position and momenta immediately before the collision, respectively. As before, i and f refer to the initialand final states of the collision. The linearized collision map for the tangent-


vector components becomes [1 ,81

8q' = 8q~ + (8qi . qi) q i l(J2 , (28)

8q{ = 8qi - (8qi. s') q i l (J2, (29)

8p' = 8Pk + (8pi . qi) q' 1(J2+~ [(pi . 8qc) q ' + (pi . qi) 8qc] , (30)(J

8p{ = 8pi - (8pi . qi) q i l (J2 - ~ [(p i . 8qc) qi + (pi. s') 8qc] , (31)(J

where 8qi == 8qi - 8q~ and 8pi == 8pi - 8p~ are the relative position andmomentum displacements before the collision. The vector

(32)

denotes the infinitesimal displacement of the collision points of the perturbedtrajectory from the reference t rajectory. Of course, the components of 8f forparticles not partaking in the collision remain unchanged.

With these equations the dynamics in the phase and tangent spaces may bereconstructed . In d dimensions there are 2dN Lyapunov exponents, and as manytangent vectors need to be followed for the computat ion of the complete spectrum. Since the computat ional effort increases with N 2 , the numb er of par ticlesis restricted to a few thousand with present-day workst ati ons.

In all our numerical work reported here we use reduced units for which theparticle mass m, the ball diameter (J , the kinetic energy per particle K IN , andthe Boltzmann constant k» are uni ty. K = I: p2/2m is the total kinetic energy.With this choice the unit of time is (m(J2NIK)1 /2. For hard disk syst ems inequilibrium, to which we restrict our considerations in thi s section, the temperature is an irrelevant parameter since there is no potential energy and thedynamics scales strictly with /T. It is therefore sufficient to consider a singleisotherm , which corresponds to a kinetic temperature kBT = 2Kld(N - 1).The density is the only relevant paramete r. It is defined by p = NIV, whereV is the area, or volume, of the simulat ion box, dependin g on the dimension.In the two-dimensional case we use simulat ion boxes with various aspect rati osdefined by A = Lyl Lx, where Lx and Ly denote the size of the box in the xand y-direct ions, respectively. The close-packing density is Poo = (21V3)(J-2 intwo dimensions, and Poo = J2(J- 3 in three. Periodic boundary conclitions areused throughout .

The prop erties of a fluid of hard disks in two dimensions were studied in Ref.[1], those of a fluid of hard spheres in three dimensions in Refs. [8,91. Full Lyapunov spectra are reported there for densities ra nging from very dilute syste msup to th e close-packing density. The maximum Lyapunov expon ent Al and theKolmogorov-Sinai entropy litce , which according to Pesin's theorem for closedHamiltonian systems is equal to the sum of all positive Lyapunov expon ents [54,


55], increase monot onously with density, with the except ion of a nar row rangenear the fluid-solid phase-transit ion density. Both quantities display a local maximum at the phase-transition density. The maximum disapp ears, however , if )'1

and hK S are plotted as a function of the collision frequency instead of thedensity. Thi s demonst rates that the density-dependent collision frequency is thedetermining parameter and th at the collective motion characteristic of th e fluidto-solid phase transit ion hardly matters [29]. We expect that a Chapm an-Enskogtype of theory will be able to predict )'1 and hK S over a large range of densities.

The usefulness of the kineti c theory on the Boltzmann-equation level forthe computat ion of the maximum exponent and of the Kolmogorov-Sinai entropy of hard ball systems has already been demonstrated for dilute systems:van Beijeren et al. 171 formulated the first successful theory for the analyt icalcomputation of hK S for low-density gases in two and three dimensions, and extensions of th at work in terms of the B.B.G.K.Y. hierarchy have been developed[56] . van Zon et al. [57] were able to compute analytically the maximum exponent of such systems and compared them successfully to simulation result s. Fordetails we refer to the cont ribut ions of these authors in this book.

A much-discussed problem is the existence of the Lyapunov spect rum, and ofthe maximum exponent in particular , in the thermodynamic limit N -+ 00, P =canst . Theoretical arguments have been given by Sinai [581 for the existence ofthis limit for systems interacting with a pairwise-addi tive short -ranged potential. They are based on the assumpt ion that the thermodynamic limit N -+ 00

may be taken before the t ime-average limit t -+ 00. In previous work on shortranged but smooth interatomic interactions [591 and on systems of hard dumbbells [60,61] we have found that the phase-space expansion along the direct ionassociated with the maximum Lyapunov exponent is dominated by the tangentspace dynamics of only a few part icles at any instant of time [59]. In this senseAl must be seen as a localized property of a fluid, and it does not come asa surprise that the simulations reported in Refs. [1 , 81 do not show any sign ofa divergence of Al for large N , which may be ext rapolated to the thermodynamic limit with some confidence. However , Searles et al. [62] found a weak, butpersistent increase of Al with N for a two-dimensional Lennard-Jones system,which was interpreted as a possible logari thmi c singularity. More evidence isneeded to settle this quest ion.

§6. "Lyapunov Modes" for Hard Ball Systems

In Fig. 16 the Lyapunov spect rum for a gas of 1024 hard disks in two dimensionswith a density of p = 0.la- 2 and an aspect ra tio A = 1 is shown. The spect rum isdefined only for integer values of the so-called "Lyapunov index" I on the abscissa,which labels the exponents. Th e smooth line in the Figure is only for clarity.The full spect rum consists of 4N = 4096 exponents, the positive half of which isdisplayed. Th e negative exponents may be obt ained from the symmetry relati on

302 H.A. Pos ch and R. HirschI

1.4

5 12

A=l --+--

1024I

0.14

........-Hf 0.120.1

0.08 «0.06

0.04

0.02

o20082028

I

1536

1.2

0.8

«-

0.6 2048

0.4

0.2

02048

Figure 16: Positive branch of the Lyapunov spec trum for a square system of 1024-disksin equilibrium. l is the Lyap unov index lab eling th e exponents. The density p = 0.la2

,

and th e asp ect ratio A = 1. The inset is a magnification of th e int eresting range ofsma ll exponent s. The Lyapunov exponents are given in units of (Nma2/ K) -1 /2 .

A2 N +l = -A2N-l+l , a well-known consequence of the symplectic nature of theflow. Six exponent s vanish altogether as was explained before, three are includedin the "positive branch" shown in the Figure. In the inset the positive exponentsclosest to zero are magnified. They display an interesting step-like structure ofgroups of equa l-valued exponents for which the multiplicity alternates betweeneight and four. A similar structure was already observed by us in the first studyof hard disk syst ems in Ref. [11 , and in recent simulations of a planar fluidconsisting of hard dumbbells, a simple model for a linear molecule, in Refs.[60,61]. There it was already noted that there is a close connect ion of thisst ructure with the appearance of collect ive, long-wave-length patterns for thecomponents of the associated tangent vectors, to which we refer as "modes".The content of this section is meant to establish thi s connection for systems ofhard disks in two dimensions. We present here results for a modera tely densegas, p = 0.10"-2 . As a rule, th e step structure of the Lyapunov spectra is muchmore pronounced for dense syst ems. The density dependence will be the topicof a forthcoming publication [63].

Simulat ion of Billiards and of Hard Body Fluids 303

In the following we consider not only square but also rectangu lar systemswith aspect. ratios very different from unity. For practical reasons we specifythe size of our systems by the multiples Nx and Ny of a small rectangularunit cell with slight ly different side lengths L~ = (2 . 3- 1/ 2 p -l )1 /2 and Lg =

(2- 131/ 2p - l ) I / 2 in x- and y-directions, respective ly. For the density p = 0.100 - 2

this amounts to L~ = 3.39800 and Lg = 2.94300. The length and width of thesimulation box is therefore given by Lx == NxL~ , and Ly == NyLg . Since L~Lg =1/ p, the total numb er of particles is simply N = NxNy.

NI

A=-Y3/64 -t-A=-Y3/32 ---~ -_.

A=-Y3/16 ---*---A=-Y3/8 .. g. ..A=f3/4 _.-• .- .

0.25

0.2

0.15«-

0.1

0.05

02N-25 2N-50

I

0.8

«-

0.6 2N

0.4

0.2

02N

1.2

1.4

Figure 17: Lyapunov spectra for rectagonal syste ms with density p = 0.1 a2 and aspectratio A. l is the Lyap unov index. Th e inset is a magnifi cation of the interesting regimeof small exponent s. The side lengths of the simulat ion box for the various asp ect ratiosare:

A = V3/4 : L; = 8L~, Ly = 4L~ , N = 32;A = V3/8 : Lx = 16L~ , Ly = 4L~ , N = 64;A = V3/16 : L; = 32L~, Ly = 4L~ , N = 128;A = V3/32 : Lx = 64L~ , Ly = 4L~ , N = 256;A = V3/64 : Lx = 128L~, Ly = 4L~ , N = 512.

N is t he number of disks, and L~ = 3.398a, L~ = 2.943a are explained in the maintext . Th e Lyapunov exponents are given in units of (Nma2/ K) -1 /2 .


We show in Fig. 17 the Lyapunov spect ra for systems with a density p =0.1(7-2 , which differ in the aspect ratio and the volume. Since the numb er ofparticles varies, and consequently the numb er of exponents, the Lyapunov indexis renormalized to cover the same interval on the abscissa. The box size in ydirection is the same for all spectra , Ly = 4Lg , but Lx, the aspect ratio A , and

the particle number N vary by factors of two, from L x = 8L~ ,A = v'3 / 4, N = 32to L x = 128L~, A = v'3/64, N = 512. One observes that the vast majority ofthe larger exponents are very insensitive to the syst em size and the aspectratio, which is also confirmed by a comparison with Fig. 16. The inset , however ,which is a magnification of the interesting region but with the abscissa notrenormalized, shows a st rong dependence on A for the smallest exponents. Themultiplicity for the rectangu lar systems in Fig. 17 is smaller than in Fig. 16. Onlysequences of four and two are found , as compared to eight and four for A = l.

The step st ruct ure for the smallest exponents is det ermined by the longestside of the box, L x = NxL~ in our case. We show this in Fig. 18, where wecompare two systems with L x x L y = 32L~ x 32L~ and 32L~ x 4L~ , corresponding

to aspect ratios A = v'3 /2 close to unity and v'3 / 16, respectively. The smallestsix expon ents coincide for both spectra. We find that Lx has to exceed a certainthreshold for the steps in the Lyapunov spectrum to appear. It is apparent fromFig. 17 that they do not show up in systems which are too small in all directions.

Exponents for systems differing by factors of two in L x are also intim atelyrelat ed. The horizontal dashed lines in the inset of Fig. 17 provide examples,and others are also evident in the inset . For lat er reference, we list in Table 1a selected numb er of exponents for the spect ra displayed in Fig. 17.

A: v'3 /4 v'3/8 v'3 /16 v'3 /32 v'3 /64A Mode s. x N y: 8x4 16 x 4 32 x 4 64 x 4 128 x 4

A2 N - 3 L 1 0.124 0.103 0.051 0.027 0.012A2 N - 9 L 2 0.143 0.119 0.105 0.056 0.025A 2N - 15 L3 0.165 0.131 0. 115 0.089 0.038A2 N-2l L4 0.192 0.143 0.122 0.107 0.052

A2 N -7 T1 0.137 0.114 0.058 0.030 0.014

A2 N -13 T2 0.157 0.1 27 0.112 0.062 0.029

A2 N -19 T3 0.182 0.139 0.1 20 0.096 0.042

A 2N-25 T4 0.214 0.151 0.126 0.110 0.057

Table 1: Selected Lyapunov exponents for spectra of rectangular systems shown inFig. 17. The density p = 0.1(1- 2. The box size is NxL~ x NyL~ , where Nx x Ny isgiven in the table, and L~ = 3.398(1, L~ = 2.943(1. A is the aspect ratio. The totalnumber of particles is N = Nx Ny. In the second column the mode assignment interms of longitudinal (L) and transverse (T) modes is given, where the index gives theorder. Small-script numbers indicate exponents for which no mode structure has beenobserved. The Lyapunov exponents are given in units of N m(12 / K) - 1/2.


0.14

0.12

0.1

«?--0.08

0.06

0.04

0.02A=f312 --+

A=f3/l6 --*- -

2N-402N-302N-20I

2N-1O

OJIHH~-_....I.....-_-_--l-_-_-L_------'

2N

Figure 18: Lyapunov spectra for rectagonal systems with density p = 0.10-2 and aspectrat io A. I is the Lyapunov index. The side lengths of the simulation box for thevarious aspect ratios are: A = v'3/2 : L; = 32L~ , L y = 32L~ , N = 1024; A =

v'3/16 : L x = 32L~ , L y = 4L~ , N = 64. N is the number of disks, and L~ = 3.3980- ,L~ = 2.9430- are explained in the main text. The Lyapunov exponents are given inunits of (Nm0-2/ K) -1 /2 .

Another interesting observation is made if the smallest positive expon ent sfor all the spect ra displayed in Fig. 17 are plotted as a function of kx = 2rr/ Lx.It is seen from Fig. 19 that the lines connect ing th e exponents A2N -1 for variousspect ra with different box sizes in x-direct ion converge to zero with kx -+ O. Aswill be demonstrated below, kx is a wave vecto r. The straight-line portion inFig. 19 looks like an ordinary dispersion relation, although important differencesexist .

From the inset of Fig . 17 we expect th e numb er of converging exponents toincrease with Lx. Our data indicate that th e limit Lx -+ 00 for th e Lyapunovspectrum exists, if the index l is renormalized according to l = l/2N. They seemto suggest that the slope of this limiting distribution is finite at l = 1. This iscontrary to theoret ical intuition [65], according to which the positive branch ofthe spectrum approaches a positive lower bound in th e thermodynamic limitwith a discontinuous jump at l = 1.

306 H.A . Posch and R. Hirs chi

128 32 160.2

0.15

0.1

0.05

8 4

-- -)Eo--

...•.. .

······e ······

_.-...-_ . -0 - , -

...... ..

... ... ..

... .• . ..

--~--

... ~ .. .

......+ .

oo 0.1 0.2 0.3 0.4

2rrILx

0.5 0.6 0.7

Fig ure 19: Dep endence of all exponents of th e spec t ra shown in Fig. 18 as a fun ctionof the inverse box length Lx = NxL~ in x-di rect ion. The box length in y-direction isLy = 4L~ for all spec tra. The lines connect equivalent exponents, as ind icated by theLyapunov indi ces, for differen t spect ra.

It is obvious from these observations th at the Lyapunov expon ents of hardball systems closest to the abscissa measure the expansion (and convergence)prop erti es of collective perturbations. A natural explanation is provided by ananalysis if the individual particle contributions to the associat ed tangent vectors[60]. The components of Jf are given by the perturbation s JXi, JYi, JPx,i, Jpy,i ofall the particles i = 1, . .. , N . We consider in the following {JXi}, i = 1, . . . , Nand {JYi}, i = 1, ... , N , which amounts to select ing proj ections of Jf onto thecoordin ate axes belonging to the individual particles.

Let us consider square systems first . At the bottom of Fig. 20 we plot JXi'along the vertical axis, at the inst antaneous positions (Xi,Yi) of the par ticles

Simul ati on of Billiard s and of Hard Body Fluids 307

0.12~0.00.04

o2040 2025

X

"

Figure 20: Longitudinal mode of ty pe L, associated with th e exponent -\ 204 5 fora square system of density p = 0.la - 2 and 1024 hard disks. The location of theexponent in th e spectrum is marked in t he inset.Bot tom: P lot of the tangent-vecto r components OXi (along t he vert ical ax is) for allparticles i at t heir positions (Xi , Yi) in t he simulation box (horizontal plane)Top: Ana logous plot of 0Yi for t he same mode.

in physical space. The perturbation belongs to the smallest positive exponentA 2045 of a l024-di sk syste m with aspect rati o A = 1, as is indicated in theinset (2N - 3 = 2045). A wave-like pattern is observed, where the wave vectorpoints along the x-axis, with a wavelength equal to Lx, in accordance withthe periodic boundaries of the simulat ion. At the top of the same figure, ananalogous pattern is obt ained for a plot of 8Yi at (Xi, y;) for all i . Since inphysical space the perturbations 8Xi are parallel to the wave vector with a wavelength equal to Lx, and analogously for 8Yi, we refer to this perturbation asa longitudinal mode of order one, L 1 . The order is given by the multiples ofthe wave length in the periodic box. The exponent A 2045 is a member of thelowest group of exponents with multiplicity eight for A = 1. It turns out that allexponents of this group are of type L1. The inst antaneous patterns differ onlyby their phases.

At the bot tom of Fig. 21 we plot for the same square system as beforethe perturbations 8Xi, but this time for the exponent A 2035 (marked in theinset ), which is a member of the next group of exponents with multiplicityfour . It is seen that 8Xi generates a mode with a wavevector perpendicular to

308

0.1

-0. 1

0.1

H.A. Posch and R. Hirschi

D.12~0.080.04

~~IIIIII~~-< 0 2040 X 2025

y

X

Figure 21: Bottom: Transversal mode of type T] associated with the exponent A2035

for t he same square system as in Fig. 20. 6Xi (along the vertical axis) ar e plotted forall particles i a t their pos itions (Xi, Vi) in t he simulation box (horizontal plane) . Thelocation of the exponent in the spectrum is marked in th e inset .Top : Mode plot associated with the exponent A2033 for th e sam e square system as inFig . 20. 6Xi (along the vertical axis) are plotted for all particles i a t their positions(Xi, V;) in the simu lation box (hor izontal plane) . The location of the exponent in thespectrum is marked in the inset .

the perturbation and a wavelength equal to L y . This perturbation is thereforecalled a transversal mode of order one, T1 . All four exponents of this group areof the same type , differing only by their phases.

The following eight exponents are difficult to interpret, because the amplitude of the patterns is noisy and small. As an example we show at the top ofFig . 21 the mode plot for OX; associat ed with the expon ent '\2033 ' We refer tothem - tent atively - as mixed modes . They only show up in systems big enoughalong the x- and y axes to allow indep endent modes along both directions.

Next we turn to a rectangular systems with an aspect ratio A = J3/32.The density is again p = 0.10- - 2 . The relevant part of the Lyapunov spect rumis particularly simple and consists of alternating sequences of exponents withmultiplicities four and two. Here, L y is mall , and no modes in y-direction arepossible . Successive groups of expon ents with multiplicity four belong to longitu dinal modes of type L 1 , L 2 , L 3 etc. Similarly, successive groups of expon ent swith multiplicity two belong to transver sal modes of type T1, T2 , T1 etc. The

Simulation of Billiards and of Ha rd Body Fluids 309

1020 1005 990I

200150

10050

o-50 X

- 100- 150

-200

0.10.075

.-<,- 0.050.025

o

8yCA/)

0.15

0.1

0.05

o-0.05

-0.1

-0.15

Figure 22: Various t ra nsversal modes T1 for the expo nents lab eled 1017, 1011, 1005,and 999 as mark ed in t he inset for a rect an gular syst em wit h density p = 0 .10" - 2 andaspect rat io A = V3/ 32. T he tangent-vector components by; are plotted vert ica lly fora ll particles i at t heir posit ion (Xi ,Yi) in t he simula t ion box (in the hor izontal plan e).

latter is demonstrat ed in Fig. 22 for the exponents labeled 1017, 1011, 1005,and 999, which are specially marked in th e inset . the next group of exponentswith multiplicity four. A similar mode assignment may be carried out for thelow-density spectra in Fig. 17 and is given for selected exponents in Table 1. Itis interesting to note from these data, or from Fig. 19, that the exponents areproportional to their wavevector: If Lx is doubled, the exponent decreases bya factor of 1/2.

Unfortunately, our assignment of modes is not complete. We have just ment ioned that the patterns for a group of exponents with multiplicity four seemto differ only by a phase shift. All four tangent vectors are orthogonal, but thispropert y cannot be fulfilled by four sine funct ions differing only in their phase.

310 B.A . Posch and R. HirschI

For positive exponents the perturbations {Jpx,i , JpY ,d are found to be strictlyparallel to {JXi, JYi}, if they are interpreted as momentum offsets for i in physicalspace . Numerica lly, we find JPx,i ;::;; CJXi and Jpy,i ;::;; cJYi, where c > O. Fornegative exponents the constant c < 0 and the momentum offsets are antiparallelto the position offsets. This behavior is well understood and is a consequence ofthe linearized equations of motion for the offset vectors.

To complete our phenomenological description of the coherent perturbationsprobing the stability of phase trajectori es of hard disk syste ms and their interpret ation in terms of "modes", we note that the longitudinal modes prop agat ewith a speed of about one half of the sound speed in the gas. For each groupof positive exponents with multiplicity four, two pair s of modes move into opposite directions. Th e two modes moving in the same direction have a phaseshift of Jr / 2. The direction for two counterpropagating modes changes occasionally during a long simulat ion run. The transverse modes do not propagat e, andexponents with a multiplicity of two also have a phase difference of Jr / 2.

"Lyapunov modes" have been observed also in hard sphere systems in threedimensions [63] and were shown to have very similar properties as in two dimension. They were found in planar systems of hard dumbbells [60,61], where alsorotational degrees of freedom are present . They even persist if the systems aredriven away from equilibrium by an externally applied perturbati on. We shallcomment on these results elsewhere [63] . Very recently, Eckmann and Gat havedemonstrated the existence of hydrodynami c modes for a simplified spat iallyext ended random matrix model [641. Their arguments are based on th e localhyperbolic characte r of the interaction and on translation invariance.

§7. Hard Ball Systems in Nonequilibrium Steady States

Full Lyapunov spectra have been also computed for hard ball syst ems in nonequilibrium steady states. We mention in particul ar color conduct ivity, which wasstudied in det ail in Ref. [11 . The model consist s of N hard balls carrying colorcharges c = ±1 such that I: c = O. The charges interact only with a homogeneous ext ernal field, oth erwise the hard particles collide elast ically with eachother. To extract the heat continuously generat ed in the systems by the work ofthe external field, a Gaussian computer thermostat is introduced to constrainthe total kinetic energy. The algorithm outlined in Paragraph 5 for hard ballsystems in equilibrium has to be augment ed to incorporate the acti on of theapplied field and of the thermostat . We refer to Ref. [11 for further details. Numericall y also another complicat ion ar ises, since the par ticles do not move onst raight lines between collisions any more , but on curved trajectori es. Thus, theequat ions of motion must be integrated by molecular-d ynamics techniques andthe collision points determined iteratively.

The result s show that the conjugate pairing rule [47,50- 521 is obeyed, A/ +A2d n - /+l = - (C), l = 1, . . . 2dN . d is the dimension, and (() is the thermostat


variable, t ime averaged over a long trajectory of the system in the nonequilibrium steady state. We have encount ered this formula already in Paragraph 5.The pairing ru le allows, in principle although not advisable in pract ice, to determine transport coefficients from the knowledge of a single conjugate pair ofexponents , usually the most positive and the most negative [47]. It follows alsofrom th is results that the sum of all exponents is negative, indicating a collapseof the phase-space density onto a multifractal attractor. As has been pointedout before this is in agreement with the Second Law of thermodynamics andexplains, why these syste ms are macroscopically irreversible in spite of theirt ime-reversible equations of motion [1 , 26, 40]. The information dimension of theattractor may be determin ed with the help of the Kaplan-Yorke formula [42] andis found to be smaller than the phase pace dimension. The computation of thefull Lyapunov spectrum is the only practical method for the computation of thisimport ant prop erty for manybody syst ems in nonequilibrium steady states.

Another nonequilibrium flow, utili zing the so-called SLLOD equat ions ofmotion for planar Couette flow [13,66], has been st udied recently for hard ballsyste ms by Dellago et at. [53], where a Gaussian thermostat was used to compensate for the heating due to the driv ing shear rate. An algorithm for thecomputation of the Lyapunov spect rum was derived and applied to computethe viscosity and the associated Lyapun ov spectrum for a 256-disk system intwo dimensions.

§8. Conclusions and Outlook

It may be asked why so much effort has been spent to compute the Lyapunovspectra of hard ball systems by computer simulations. Is it wort h the effort ?We hope to have answered that question in the affirmat ive. We have mentionedabove the connect ion to the nonequilibrium case, to transport coefficients, andto the quest ions of irreversibility and the Second Law [26, 67, 68]. Qualitativeinsights have been gained by the simulat ions, and the interp lay between theoryand computer experiments has been ext remely useful over the last decade toshape and focus our views. There are some new ideas and developments whichpromise to be useful in the future. We mention only a few.

The computation of the full Lyapu nov spect ra for hard dumbbell moleculesoffers the possibility to study qualit atively the coupling between rot ational andtranslational degrees of freedom 160,61] . It was found th at for a given densitythere is a crit ical molecular anisotropy which must be exceeded for translationrotation coupling to become effective. For anisot ropies smaller than this threshold , the Lyapunov rotati onal and translat ional branches of the Lyapunov spectrum are clearly separated.

A very efficient numerical method for the computation of Lyapunov spect raof low-density systems was recently developed [3, 6], which is based on the directsimu lation Mont e Carlo (DSMC) method of Bird [69,701. The reference t rajectory is determined from a random sequence of collisions, the tangent-vector


dynamics, however, is purely deterministic, using the same sequence of collisions (and random numbers) as the reference traj ectory. T he performance ofa simulation of rarefied gases may be enhanced by almost two orders of magnitud e.

Theoretically, Boltzmann-equation methods have been applied to the computation of the maximum Lyapun ov exponent and the Kolmogorov-Sinai ent ropyof rare fied gases. It is to be expected that these methods may be refined andgeneralized to ext end the range of densities. In Ref. [91 the maximum exponentwas compared to the collision frequency v over the full range of densities fora hard sphere gas in th ree dimensions. It was found that limp-t o v / Al = 0 andlimp-tp"", v / Al = 00 , where Poo is the density for close packing. T he crossoveroccurs roughly at a density of 0.10" - 3. The lowest-order kinet ic theory (whichdisregard s corre lated collisions) is st rictly applicable only if v « AI.

The discovery of collective "modes" for the infinitesimal perturbations associated with the smallest (in absolute value) Lyapunov exponents in Paragraph 6opens the prospect of a new and useful link between dynamical systems theoryand the dynamics of fluids and solids. We may speculate on this connect ion byrecalling that the local thermal fluctu ations of the conserved quanti ties mass,energy, and momentu m are understood in te rms of the fundamental modes ofthe linearized hydr odynamic equations and are easily accessible to spectroscopicmeasurement , such as the famous Rayleigh-Brillouin triplet for polarized lightscattering from fluids [71]. T he autocorrelat ion funct ions of the fluct uating fields,such as the local density, are even functions of t ime, and their growth and decaytypically follows an exponential law, at least for low-enough densit ies. The timeconstants of the sponta neous fluctuat ions of the density, to give an example,are given by k2D T and k2 (r/V + 4T/s /3) , where DT is the thermal diffusivity,and T/s and ttv are the shear and volume viscosit ies, respectively. k is a wavevector of the fluctuations selected by the experiment . T hese finite but smallfluctu ations are very similar to the infinitesimal perturbations represented byJr. We conjecture that there is a close correspondence between the exponentialtime constants of these fluctuations and the smallest Lyapunov exponents (inabsolute value), which may be also connected to "modes" in the sense describedin Paragraph 6. We are working at present to make this correspondence moreexplicit.

During the recent work on the modes discussed in thi s paper one of us (HAP)had many stimulat ing discussions with colleagues, who helped to put the various topics into perspective and to whom we are most grateful. We would liketo thank, in particular , L. Bunimovich, E.G.D. Cohen, Ch. Dellago, J .R. Dorfman, J .-P. Eckmann, D.J. Evans, G. Gallavot ti , P. Gaspard , Wm . G. Hoover,D.J . Isbister , Lj. Milanovic, O. Penrose, Ya. G. Sinai, D. Szasz, T . Tel, H. vanBeijeren, and J . Vollmer. We also thank A. Vrtala from the Computer Center ofthe University of Vienna for his cont inuing effort s and his efficiency to improveour comput ing environment . The support from the Fonds zur Forderung derwissenschaftlichen Forschung , grant P11428-PHY, is gra tefully acknowledged.

Simulation of Billiards and of Hard Body Fluids

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The Lorentz Gas:A Paradigm for Nonequilibrium

Stationary States

C. P. Dettmann

Contents

§1. Why a Paradigm? . . . . . . . . . . . . . . . . . . . . . . . . . 317§2. Thermodynamics 318

2.1 The Second Law and Nonequilibrium Stat ionary States . 3182.2 The Clausius Entropy . 3202.3 Entropy Production . . . 321

§3. Stati stical Mechanics . . . . . 3223.1 The Boltzmann Entropy 3223.2 The Boltzm ann Transport Equation . 3233.3 The Random Lorentz Gas . 3243.4 The Gibbs Entropy . . . . . 327

§4. Equilibrium Molecular Dynamics . 3304.1 Numerical Methods . . . . 3304.2 The Periodic Lorentz Gas . 3344.3 Green-Kubo Relations . .. 335

§5. Nonequilibrium Molecular Dynamics . 3385.1 Introduction to Thermostats . . 3385.2 Gaussian and Nose-Hoover Thermostats 3395.3 The Nonequilibrium Lorentz Gas 3415.4 Symplectic Properties . . . 3465.5 Periodic Orbit Approaches 3505.6 Nonlinear Response . . . . 353

§6. Boundary Driven Systems . . . 3566.1 Open Boundaries: The Escape Rate Formalism 3566.2 Flux Boundari es . . . . . . . . . . . . . . . . . . 358

316 C. P. Dettmann

6.3 Bound ar ies With Thermostats§7. Outlook .References . . . . . . . . . . . . . . . . .

360361362

Abstract. Nonequilibrium stat ionary states form the building blocks of morecomplicated nonequilibrium systems as they define the transport coefficients appearing in the hydrodynamic equations. Recently, many connections have beenmade between the microscopic dynamical properties of such systems and themacroscopic t ransport . Although it is often difficult to visualise or understandthe dynamics of systems with many part icles, it turns out that the Lorent zgas, a system contai ning only one moving particle, provides a paradigm withwhich many of these connections can be exhibited and studied. This art icle surveys the prop erties of nonequilibrium sta t ionary states, from thermodynamicsto the computat ion of t ransport coefficients, demonstratin g how the Lorentzgas appea rs as one of the simplest models. A numb er of current approachesare considered, including linear response formulae applied to equilibrium systems, thermostatted systems and boundary dr iven systems. All of these and theconnect ions between them can be understood in some detail using the Lorentzgas.

The Lorentz Gas: A Paradigm for Nonequilibrium Stat ionary States 317

§1. Why a Paradigm?

Equilibrium stati stical mechanics has always been associated with dynamicalproperties such as ergodicity and mixing [1]' although proofs of such propertieshave been made only recentl y, and are mostly restri cted to billiard and hard ballsystems [21 . The advent of the computer has made visualisation and simulat ionof many kinds of systems possible, inspiring theoretical advances in nonlineardynamical syst ems, st atistical mechanics, and the relationship between them(and maybe also detriment al effects, Sec. 1.1 of [3]). Dynamical systems theoryhas benefited from Ruelle's thermodynamic formalism [41 and Feigenbaum 'srenormali sation approach to the bifurcation cascade [5], while dynamics in itsturn elucidates the foundations of stat istical mechanics [6, 71.

At the heart of the connect ion between dynamical syst ems and statisticalmechanics lies a paradox . Statistical mechanics is a theory of large systems, validin th e limit as the number of particles (or spins, etc) approaches infinity. Statist ical tr eatment s of small systems lack the ensemble equivalence and automaticaveraging characterist ic of large systems. On th e other hand , dynamical systemsare most understood in up to three phase space dimensions, due to easier visualisation and topolo gical prop erti es. Systems of infinite extent and number ofpar ticles are excluded from the usual definition of a dynamical system, and inany case are difficult to visualise and simulate.

Th e paradox can be resolved, at least par tly, in a number of ways: Realmacroscopic systems have a finite number of degrees of freedom, even if tha tnumber is large; many of the result s connecting dynamic and thermodynamicproperties (see below) apply to large as well as small syst ems; Gallavotti andCohen [81 conjecture that the physically relevant prop erties of syst ems withmany degrees of freedom are those of st rongly chaot ic dynamics in the sense ofAnosov; a turbulent fluid at the onset of chaos has effectively only a few degreesof freedom; numerical methods use a finite numb er of particles with periodicboundary condit ions to simulate an infinite homogeneous system.

Most macroscopic systems, however, have many effect ive degrees of freedom. The chaot ic prop erties of such systems can be difficult to visualise, andthe building blocks of a dynamical description such as Markov par titions andperiod ic orbits are all but impossible to const ruct . It is thus difficult to developa useful intuit ion and make predictions without some intermediat e example,sharing both the properties of chaotic dynamics and of the large syste ms of interest , without the complexity of many degrees of freedom. Another advantageof such an illustrat ion is th at it is possible to investigat e the distin ction betweenmacroscopic properties that are relat ed to chaot ic dynamics, and those that aredue to many degrees of freedom.

The discussion thu s far has included both equilibrium and nonequilibriumsystems. Thi s article focuses on nonequilibrium stationary stat es for which a natural parad igm is the Lorent z gas. The Lorent z gas can be represent ed as a twodimensional chaot ic map and also exhibits transport in the form of diffusion.

318 C. P. Dettmann

The main alternative, discussed in [9], is a class of models based on Baker mapswhich are exactly solvable, but have less relation to the physical processes theyare designed to mimic.

Section 2 outlines the physics of nonequilibrium steady st at es, introducingthe central concepts of entropy production and irreversibility. Sect ion 3 givesstatistical definitions of entropy, and how the random Lorent z gas app ears naturally as a model of dilute fluids. Section 4 explores computational techniques forsystems of many particles, from which the periodic Lorent z gas appears as thesimplest example. Section 5 discusses thermostatted models of nonequilibriumstationary st at es and how results for the Lorentz gas can be appli ed to suchmodels and hence to systems with many degrees of freedom. Section 6 discussesopen models of nonequilibrium stationary states, and their connection to thermost atted models . Finally, section 7 covers the limitations of the Lorent z gasparadigm and outlook for the future .

§2. Thermodynamics

2.1 The Second Law and Nonequilibrium Stationary States. An empirical observation is that it is impossibl e to convert thermal energy of a systeminto work without affecting the environment , the second law of thermodynamics. Conversely, there are many processes that convert work into thermal energywithout affecting the environment, so these are irreversible. It is possible to extract work from a warm er and a cooler subsyste m (this is frequently achieved inelect ricity generation) . Thus it is not possible to separate a uniform system intowarm er and cooler parts without the addition of work, as this would permit theext ract ion of work from the thermal energy of the original system. Conversely,the spontaneous flow of thermal energy from a warmer to a cooler subsystemwithout the extraction of work is also an irreversible process.

Many irreversible processes, including mutual diffusion of different particlespecies, electric current flowing through a resistor , shear flow of a viscous fluidand heat conduct ion can occur in such a way that macroscopic variables including the various forces and fluxes are independent of time in a region of interest.Such a syst em is said to be in a nonequilibrium stationary state. Two propertiesshould be noted immediat ely:

1. Due to the irrever sible processes, the region is necessarily in contact withan environment which is not truly stationary. For example, a resistor continually depletes its voltage source as well as heating its environment .Conceptual difficulties can arise when the environment is either ignoredor assumed to be infinite and hence unaffected by contact with the regionof interest .

The Lorentz Gas: A Paradigm for Nonequilibrium Stationary Stat es 319

2. The stationarity is of a sta t ist ical kind , as is usual when dealing withsystems with many degrees of freedom. The individual particles are notstationary, leading to stat istical fluctuations in macroscopic quanti ties,although these are often small when very many particles are involved.Statisti cal stationarity is quantified in terms of ensembles, or probability distributions on phase space, that may be stationary as determin edby the dynamics and boundary condit ions. This distin ction between theprop erties of individual realisations and ensembles is particul arly st rikingin systems with few degrees of freedom, such as the Lorentz gas, wherethe fluctuations are very large.

When the driving forces (concentrat ion gradient , electric field, shear st ress ortemperature gradient) in the above examples are set to zero, there are no longerany irreversible processes, and the steady state of the system is an equilibriumstate similar to that of an isolated syste m. The only difference would be dueto the interaction with the environment , which appears in th e ensembles ofequilibrium statistical mechanics . The microcanoni cal ensemble of an isolat edsystem is equivalent (in the limit of many particles) to the canonical ensembleof a system in thermal contact with its environment. An equilibrium stat e maynot be unique, for example a substance tha t is a crystalline solid at a certaintemperature may have its axes in many possible orient at ions.

When the driving forces are very small compared to relevant physical scalesso th at , for example, the relative variati on of all quantities is much less thanunity over a distance equal to the mean free path, the steady state is said to beclose to equilibrium. Linear response theory may be applied , leading to fluxes(particle current, electric curr ent , strain rat e or heat flux) proportion al to theforces. The constants of proportionality (diffusion coefficient , electrical condu ctiv ity, shear viscosity or thermal conduct ivity ) are known as linear transpor tcoefficients . They appear in macroscopic descript ions such as the Navier-Stokesequati ons. Mathematical proofs of their existence have been given for some smallsystems [10] .

A steady state need not be close to equilibrium, and such states show a richrange of phenomena, as we will see in the Lorent z gas. There may be nonlinear relationships between the fluxes and forces, but the concepts themselveschange as prop erti es no longer resemble those of an equilibrium system. A viscous system shearing sufficiently so that nonlin ear terms become important willalso be generating enough heat for thermal conduct ion effects to contribute.Higher shear ra tes correspond to an increasing Reynolds number, leading toturbulence. There is in general no guarantee of uniqueness or even existence ofa nonequilibrium steady state .

Another problemati c aspect of far from equilibrium steady states is the difficulty in defining a thermodynamic limit , where the number of par ticles, volume,and other extensive vari ables go to infinity such tha t their ra tios are finite. Thepresence of strong gradients rapidly causes huge varia tions in temperature, den-

320 c. P. Dettmann

sity, and so on, leading to physically unrealistic scenarios. This difficulty is solvedin the linear regime by demanding that the variations in such quantities remainfixed, so that their gradients approach zero in the thermodynamic limit .

There are a number of equivalent statements of the second law, and as manyapproaches to the related issues of irreversibility and entropy as there are textson thermodynamics. Some of th e more important ideas are sketched below withtheir relation to the Lorentz gas and nonequilibrium stat ionary states.

2.2 The Clausius Entropy. Just as the notion of temperature can beund erstood in a qualit ative manner from the direction of the flow of thermalenergy, it is clear that the existence of irreversible processes implies that there isa prop erty of the system, namely the ent ropy, that remains constant in reversibleprocesses and increases in irreversible processes. A unique state of maximumentropy then corresponds to equilibrium, because there are no more st ates towhich the system can go. We will usually assume (quite reasonably for fluids, atleast) tha t there is only one equilibrium state for given constants of the motion(energy, numb er of particles, volume) corresponding to maxim al ent ropy.

Historically the first quantitative st atement in thi s direction, due to Clausius,defines the change in entropy as a syst em able to exchange energy but notmatter with its environment moves quasistatically from one equilibrium stat eto another. Specifically,

D.Se = Ji (1)

where Se is the entropy (defined up to an additive const ant) , T is th e temperature, and q is the thermal energy injected into the system from a thermalreservoir at the same temperature as the system. "Quasistat ically" means a limitin which all time derivatives approach zero. It disallows processes such as thefree expansion of a gas when a partition is removed; such processes are inher ently irreversible. The temperature can be defined from the equation of state ofan ideal (in practice, dilute) gas, that is, proportional to the pressure times thevolume. Alternatively, if we equate Se with one of the statistical mechanicalentropies discussed below, the temperature is then defined from Eq. (1) or itsequivalent.

Note that the thermodynamic definition of entropy only makes sense fora system at or very close to equilibrium. Once we know the entropy of a particular subst ance as a function of temperature (or energy density) and pressure (ormass density) at equilibrium, the definition can be ext ended to syst ems in "localequilibrium", including stationary st ates close to equilibrium, by assuming theext ensive property, that is, the total ent ropy of a system is equal to the sumof the entropies of its subsystems, and that the subsystems can be consideredclose to an equilibrium state.

Extensivity is expected classically when interactions between the particlesare short ranged, which is usually the case. In the large syst em limit , the interactions reduce to boundary terms which are much smaller than the bulk effects .

The Lorentz Gas : A Paradigm for Nonequilibrium Stationary States 321

Notable exceptions to extensivity include some quantum systems (for example Bose-Einstein condensation) and gravi tational systems (for example blackholes) . When there are strong intera ctions between subsystems it does not makesense to consider the Clausius entropy of the subsyst ems.

2.3 Entropy Production. It is possible to apply the above prescriptionto nonequilibrium stationary states th at are close to equilibrium. The entropyof the region under consideration does not vary with time, due to st ationarity.However , the irreversible processes cause an overall increase, or production ofentropy, so that thermal energy released into the environment increases its totalent ropy. Thus we have

(2)

where <: corresponds to th e nonequilibrium steady state, Birr is the irreversible entropy production, and Bin is the (negat ive) flow of ent ropy in from theenvironment. Such ent ropy balance equations figure prominently in the Bakermap approaches [91 in various notations. The irreversible entropy productionand ent ropy flux are also independent of tim e from stationarity,

(3)

The origins of the ent ropy flux Bin depends on th e nature of the system. Anelectric current density J is driven by an electric field E th at does work butdoes not affect the entropy. This work is convert ed into an equivalent amountof thermal energy that then leaves the system, taking with it an entropy fluxgiven by (1). Thus we have

. . J ·EVSirr = -Sin = ---

T(4)

where V is the volume. Similar considerations hold for shear flow. In the case ofheat conduct ion, Bin contains contribut ions from heat (and hence ent ropy) flowin from a higher temperature and out to a lower temperature. The amounts ofheat are equal since energy is balanced, but more entropy flows out owing to thedifferent temperatures in the denominator. Entropy is also produced when twosubstances mutually diffuse; see the discussion on the Gibbs mixing paradox inSec. 3.4 below. The connection between mutual diffusion and heat flow is moredifficult to understand, but it is clear that work or a temperature differential isrequired to separate a mixture. Whatever the situation, the ent ropy productionis always the product of a force and a flux. The second law, which requirespositive entropy production, thus determines that transport coefficients (thequotient of a flux and force) are positive.

For steady states far from equilibrium, it is not clear how to calculate theentropy from (1) since there is no equilibrium state with which to compare thesystem. However , if it is possible to couple the system reversibly to a thermal

322 C. P. Dettmann

reservoir close to equilibrium, all of the above arguments remain valid so theent ropy production can be calculated from the forces and fluxes as above. Thereare some possible pitfalls to thi s approach, for example some steady states farfrom equilibrium have different effect ive te mperatures for par ticles moving indifferent directions. Thi s makes it difficult to imagine how to const ruct therequired thermal reservoir in principle, let alone in practice. Typic ally suchdetails are ignored, the above equations are applied , and an addi tion al postulat eis added to the theory.

§3. Statistical Mechanics

3.1 The Boltzmann Entropy. Now we turn to the statistical viewpointsof Boltzmann and Gibb s. The macroscopic thermodynamic variables fluctuatedue to microscopic movement of the molecules, with the except ion of exact lyconserved quantities such as the energy of an isolat ed system. This means that ,for example, the second law of thermodynamics is not always valid . The localtemperature (as measured by the average kinetic energy over a small region)of an equilibrium system fluctuat es, leading to a transition from a state withuniform temperature to a state with slight variations in temperature. However ,large fluctuations as measured by a large decrease in ent ropy are very rarelyobserved.

In order to quantify the frequency of certain fluctuations, and because we donot have precise information about the positions and momenta of macroscopicnumb ers of particles (and also for reasons related to quantum mechanics, whichwe shall ignore here) , it makes sense to describ e a system in terms of probabilities. Probabilisti c assumptions about the init ial conditions of the microscopicparticles can also explain the paradox of irreversibility, how th e second law ofthe rmodynamics is compatible with perfectly reversible Newtonian microscopicequation s of motion. The time reverse of a dissipative process shows large violations of the second law, but is not observed because the init ial condi tion s arenot very probable for some reason , depending on the physical or philosophicaljustification of the probabilistic assumptions of the theory.

For Hamiltonian systems with, say, N particles moving in d dimensions ,the most natural probability measure on the 2Nd - 1 dimensional surface rof const ant energy E is the (restricted) Lesbesgue measure dr = 8(H(x ,p) E)dxNddpNd /hNd, the postulat e of equal a priori prob ability. In the nineteenthcentury the only real justification for this was the theorem of Liouville that thismeasure is preserved by Hamiltonian dynamics (see Sec. 3.4). Ergodicity impliestha t this measure does indeed give t he correct time averages, but the t imerequired for a system to closely approach all points in the phase space with evenvery coarse precision is ast ronomical for systems with many degrees of freedom .See [1] for a more detailed discussion of ergodicity. Many of the current models

The Lorent z Gas : A Paradigm for Nonequilibr ium Stationary States 323

of nonequilibrium stationary st ates do not preserve Lebesgue measure, so otherinvariant measures are more appropriate , and will be discussed later. The abovemeasure is normalised by powers of Planck' s constant h for dimensional reasons.This particular norm alisation can be justified in quantum mechanics, but herewe note that it sets the (classically) arbitrary addit ive constant associat ed withthe entropy.

The Boltzmann definition of entropy considers that for each configurat ion ofmacroscopic system var iables X , there is a region in microscopic phase space ofvolume Ix di' . Then th e entropy corresponding to the configurat ion is

(5)

where kB is Boltzmann's constant and has dimensions of an energy divided bya temperature, see (1). The idea is that a system will be most likely to move toone of the very large regions of phase space corresponding to greater entropy.The probability of finding the system in a given state is thus proportional toexp[- (So- SB)j kBJwhich is virtually zero for a large system not in its maximumentropy state So since the entropy is prop ortional to the number of particles.The Boltzmann ent ropy of a unique equilibrium state is thus the logarithmof the volume of the whole surface of constant energy, and agrees with theClausius definition in the cases where they can be compared, th at is, equilibriumsystems with many particles. Boltzmann 's entropy and its relation to irreversibleprocesses is discussed in Ref. [111 .

In order to apply thi s to nonequilib rium steady st at es, decisions must bemade abou t the most natural phase space for a system in contact with its environment (discussed extensively below) as well as the correct measure to use. Inthis context it should be noted that recent pap ers of Rugh using the Boltzmannent ropy to define a dynamical temperature for Hamiltonian systems [12, 131have been applied to nonequilibrium systems [14J by means of the Hamiltonianformulation of the isokinetic thermostat (Ref. [15], see Sec. 5.4 below), and alsoto identify the heat flow in systems with inhomogeneous shear [16] . Apart fromthis , most applicat ion of entropy to nonequilibrium steady states seems to becloser in spirit to the Gibbs approach, Sec. 3.4.

3.2 The Boltzmann Transport Equation. Another type of st ati sticalassumpt ion appears in the Boltzmann equation which describes dilute gases at oraway from equilibrium. The quan ti ty of interest is the single particle distributionfunction f(x ,v , t) which gives the probability density of finding a particle withthe given position and velocity at a certain tim e. A st raightforward derivationbased on th e equations of motion gives

of 1-;:} + v . \lx f + - Fe . \lvf = r colidt m

(6)

324 C. P. Dettmann

where rn is the mass, Fe is the exte rnal force on each particle, and the term onthe right hand side denotes the effect of the collisions between particles.

For a dilute gas without long range interactions between the particles, onlytwo-bod y collisions contribute, however an exact t reat ment requires the twoparticle distribution function f2(Xi , X2 , Vi , V2 , t) which gives the joint probability of two part icles entering a collision. The assumption made by Boltzmann,called the stosszahlansatz, consists of replacing the two-particle distribution bythe product of two one-particle distribution functions , thus assuming that thepar ticles entering the collision are uncorrelat ed.

Boltzmann showed in his celebrated H-theorem that a certain quantity,

H(t) = Jdxdvf(x ,v,t)lnf(x,v,t) (7)

never increases as f evolves under (6) with the stosszahlansatz. In fact , - kBHcan be identified with the entropy (up to an additi ve const ant) , and Boltzmannargued that this was a derivation of the second law.

The Boltzmann equat ion describes a dilute gas approaching equilibrium well,but the addit ion of this statistical assumption has the effect of ignoring thefluctuations that are known to occur . The solution of the Boltzmann equat ioncan never return to a state of smaller entropy, despit e the fact that this isknown to happen occasionally. For this reason , th e statistical assumptions goinginto the Boltzmann equation, although useful for calculating the prop erties ofnonequilibrium gases, are not viewed as a fundamental explanat ion of the secondlaw. See Ref. [17] for further discussion.

3.3 The Random Lorentz Gas. The Boltzmann equation is a nonlinear integro-differential equation, and as such it cannot be solved in most caseswithout making rest rict ive and sometimes physically obscure approximations.One case that illustrat es the prop erties of the Boltzmann equation well, whileremaining simple enough to solve is the random Lorent z gas [18, 191 .

The random Lorentz gas can be motivated on physical grounds as follows:Suppose we have a dilute gas in equilibrium, consist ing of a mixture of twospecies. Both are spheres (or in two dimensions, disks) that are rigid (so anyinternal degrees of freedom are ignored) , and hard (so the range of interaction ismuch smalle r than other length scales) with one much larger and heavier thanthe other , and with numb er densities (numb er per unit volume) such that th ereare many more smaller particles, but almost all collisions are between smallerand larger particles, rather than among the smaller particles. Thus for massesrns and mi. , radii rs and rt. , number densities ti s and nL and dimension d werequire rns « me , r s « rt. . ive » «i. and nsr~-i « nLrt- i . The equipartit ionof energy implies that at equilibrium, the average kinetic energy of each particleis equal, hence the larger particles have much smaller velocities. In thi s limit wehave a large number of noninteracting pointlike particles colliding with fixed,ra ndomly placed spherical (or circular ) obst acles. The magnitude of the velocity


o

00oo

Figure 1: The random Lorentz gas. In the Lorentz Boltzmann equation (10) below,the part icle arrives in direction () from direction 7r + () + 2X.

is a constant of the mot ion, and changing the velocity is equivalent to scaling thetime, so we can restri ct ourselves to the case of unit velocity, averaging over thevelocity distribu tion later if necessary. Similarly, scaling the distance permi tsus to set the radius of the scatterer equal to unity (there are other convent ionspossible, such as setting the mean free path to unity). We thus have a modelwith one free parameter , the number density of scatterers n , called the randomLorentz gas. See Fig. 1.

The designation "random" comes from the placement of the scatterers; a periodic placement appears naturally from the methods of molecular dynamics, andis discussed in Sec. 4.2. From a mathematical point of view, it can be assumedthat a random placement ensures that there is no exact relation between thepositions of the scatte rers. It is possible to consider an average over all such random configura t ions, however this is generally unnecessary since any given finitearra ngement of scatterers appears to arbit rary precision somewhere in an infinite arrangement . A random configura tion drawn from the correct distributioncan be obtained on a computer by a variant of the Metropolis algorithm [20],in which any initi al arrangement (perhaps periodic) is modified by a large fixednumber of attempted random shifts of randomly chosen scatterers; any illegalshift resulting in overlapping scatterers is rejected, and the configuration is unchanged. Note that the number of attempted random shifts is fixed, not thenumber of successful shifts. As a model in its own right (but not for Boltzmannequation approaches), the dilute condition (not included above, nLrt « 1)may be relaxed. It is also possible to consider a model where the scatterers arepermitted to overlap.

326 C. P. Dettmann

Before returning to the Boltzmann equation, and hence the low density limit ,we note an exact result that holds for all densities, tha t is, the mean free timebetween collisions (or dist ance, since the velocity is one). The mean free time canbe computed exact ly for all finite billiard systems [21], and we assume it holdsfor infinite systems as well. Briefly, the argument is that the total volume ofphase space can be computed in two ways, one by subtract ing the volume of thescat terers from the total volume, and the oth er by considering the mean pathlength over each point on the boundary. Equating the two expressions gives, intwo dimensions,

and in three dimensions,

_ 1rIQI 1 1rr=-- =---

18QI 2n 2

_ 41QI 1 4r =-- =- --

18QI ttti 3

(8)

(9)

where IQI is the volume of the billiard and 18QI its boundary. The last equality ineach case corresponds to the non-overlapping Lorentz gas with n scatterers perunit volume. It is always positive; n can never be larger than the close-packedvalues. Th ese formulas are valid regardl ess of the locations of the scatterers,so they apply to both the random and the periodic Lorentz gas as long as thescat te rers do not overlap.

The Boltzmann equat ion for the Lorentz gas in the low density limit is linear ,because the prob ability distribution of one of the objects involved in a collision(the fixed scatterer) is constant . For example, in two dimensions we have forthe single par ticle distribution function f (x ,y,e,t) where e E R/21rZ is thedirection of the velocity:

(:t + cas e:x + sin e :y) f (x , y,e,t) (10)

= J 1r /2 f( x , y,1r + e + 2x )ncos X dX - 2nf(x , y ,e, t )- 1r/2

Here, the external force in (6) is zero, and the right hand side contains two termsgiving the rate of particles entering and leaving a given velocity direction, seeFig. 1. Without explicitly solving the equation, it can be seen th at the effect ofthe collision operator is to redistribute the 2nf to the other velocity directions,making the dist ribu tion flat ter and smoot her. We can immediately writ e downa solut ion in the form

00

f (x , y, e, t ) = L fmexp(ikxX + ikyy + ime - )'t ) (11)rn= - oo

The Lorentz Gas : A Paradigm for Nonequilibrium Stationary States 327

which is substituted to obtain

ik x +ky ikx - kybm- , )fm + 2 f m-I + 2 f m+1 = 0

with

8nm 2

,m = 4m 2 - 1

(12)

(13)

the decay rates of the modes in the homogeneous case (kx = ky = 0). Perturbingthe homogeneous zero mode with small kx and ky we obtain a "dispersion"relation

with the diffusion coefficient in two dimensions given by

3D2 =

16n

In three dimensions th e diffusion coeffi cient is

(14)

(15)

(16)

(17)

For general dilu te gases, in which the Bolt zmann equat ion cannot be solvedexactly, the diffusion coefficient is obt ained by the Chapman-Enskog methods ofstandard kinet ic theory [221 , for which the rand om Lorentz gas provides a useful pedagogical example. More calculat ions and relat ions involving the diffusioncoefficient of the Lorentz gas are given in later sect ions. A Boltzmann-like equation for the Lorentz gas has also been applied to a dynamical problem, that ofcomputing the Lyapunov exponents and the Kolmogorov-Sinai ent ropy, see [231for a detailed discussion.

At higher densities the Boltzmann equation is no longer a good approximation, and the physics changes due to the appearance of power law decay inthe correlat ion funct ions, the "long time tai ls", both for the random Lorentz gasand more general gases [19, 241. Th e Lorentz gas has a velocity autocorrelationfunction decaying as C d/HI , sufficient to lead to nonanalyt ic higher terms inEq. (14), see Secs. 4.3, 5.6 and Refs. [19,251.

We leave the random Lorentz gas at this point to cont inue our discussions ofent ropy in nonequilibrium stationary states, but it is worth noting th at manyof th e result s obt ained in connect ion with molecular dynamics and th e periodicLorentz gas in lat er sect ions also apply to the random case.

3.4 The Gibbs Entropy. T he other stat ist ical formul at ion of the ent ropywe will consider is due to Gibbs. Given an arbitrary smoot h probabili ty densityp(r) on phase space the Gibbs ent ropy is defined as

Sa = - kB Jp(f) In p(r)df .

328 C. P. Dettm ann

Thi s is similar to the Boltzmann H function of Eq. (7) except that p is defined on the whole of accessible phase space compared with the single particledist ribution function f. The accessible phase space I' could be the const ant energy surface of an isolated system, but will be generalised below when otherensembles are discussed.

The Gibbs entropy is also extensive (Sec. 2.2) if it is noted that when thereare N identical particles, th e phase space is a subset of R 2N d j SN where S N isthe permutation group of order N . In terms of the standard "unreduced" phasespace (which is easier to compute with) this means multiplying p by a factor N!when the par ticles are indistinguishable. The difference between the ent ropy ofidenti cal and distinguishable particles is called the ent ropy of mixing. It solvesthe Gibbs paradox which notes that mixing of ident ical substances has no effect , while mixing of different substances (without extract ion of work) is anirreversible process. In other words , self diffusion is not associa ted with an increase in ent ropy, and is not observable without art ificial means such as a "taggedparticle", whereas mutual diffusion is a true irreversible process, associated withan increase in entropy and directly observable.

This is relevant to the Lorentz gas in that when the Lorent z gas is consideredto be a mixture of two different species (as in Sec. 3.3) there is an ent ropy production associated with the diffusion coefficient, and when it is considered to bea model of one species (as in Sec. 4.2) there is no ent ropy production involved.The physics of entropy product ion is thus connected to the interpret ation of themodel rather than anything in the model itself, such as the equations of mot ion. Thi s illust rates the need for caut ion whenever establishing an equivalencebetween features of the model and physical reality.

The Gibbs ent ropy can be used to derive the ensembles of equilibrium statis tical mechanics as follows: The maximum ent ropy (subj ect to norm alisationof the probabil ity) corresponding to the equilibrium state of an isolat ed syste mis attained when p is a constant, consistent with the postulat e of equal a prioriprob ability, Sec. 3.1. If the system can exchange energy with the environment,the constant energy constraint on phase space is replaced by an average energyconstra int on p,

(E) = JE(f)p(f)df (18)

in addit ion to conservat ion of prob ability. The ext ra const raint when maximisingthe Gibbs entropy gives a Lagrange multipli er which turns out to be relat ed tothe temperature. In thi s manner the canonical ensemble

p(f) = exp (k~~) (19)

is derived . Similarly, when the system can exchange particles with the environment , the const raint of fixed N is replaced by a Lagrange multiplier which is

The Lorent z Gas: A Paradigm for Nonequilibrium Station ary St ates 329

relat ed to the chemical potenti al u, and the phase space is expanded to includeall numb ers of particles. For more details see for example Ref. [261 . In general ,for each constant of the motion (in a general sense) E, N and volume V thereis a thermodynamic conjugate var iable T , J-t and the pressure p respectively.

Given its success in equilibrium stat ist ical mechanics, the possibility of extendin g the phase space to allow for interact ions with the environment, and theappearance of conjugate variables analogous to the conjugate forces and fluxesof irreversible thermodynamics (Sec. 2.3), it would seem that the Gibbs ent ropyis the natural candidate for extension to nonequilibrium systems. Unfortunatelythere is one major obst acle, which we now discuss.

Suppose an isolat ed system with phase point I' has equat ions of motion

df = F(f)dt

then the Liouville equat ion for a prob abil ity density p(r) is

ap- +V' ·(Fp)=Oat

and hence (after two partial integrat ions)

d8c = lID J(V' . F)pdfdt

For a Hamiltonian system

(20)

(21)

(22)

so both the phase space volud and the Gibb s ent ropy are constants of themot ion:

dp ap- = - +F ·V'p=Odt at

d8c =0dt

(24)

(25)

The Gibbs entropy as it sta nds cannot explain t he second law of thermodynamics.

The reason behind this becomes clear as we realise that a Hamiltonian system(or any system with phase space volume conservat ion) moves probability densityaround, but does not alte r its initi al values. If the system has chaotic dynamics(say, mixing), an initially smooth distribution will be stretched and folded tobecome rap idly varying in the stable directions, but remains cont inuous for alltimes. The Gibbs ent ropy gives minus the amount of information (in the sense ofinformation theory) we have about the state of the system, and thi s informationdoes not change under incompressible t ime reversible deterministic dynamics.

330 C. P. Dettmann

It is clear that any observation of a real system is uncertain to some degree,so that from the point of view of measuring the system, a rapidly varying probability distribution may be replaced by its average over the scale of resolution.This procedure is called coarse graining, and the smoother distributions generated by such a procedure have a higher entropy than the initial distributions.Like the Boltzmann entropy, for which the definition of the st ate X is somewhat arbitrary, the coarse grained Gibbs entropy depends on the observer. Theparadox is that the second law of thermodynamics is valid however (and indeedwhether) the system is being observed. Quantum mechanics is not obviouslyhelpful in explaining thi s dilemma, since the second law is observed in classicalcomputer simulations. A critic al review of this issue as applied to recent workalong the lines of the thermostat ted and open models discussed below (see alsoSec. 5.3) concludes:

The above discussion on the coarse grained approach to a completedynamical theory of irreversible thermodynamics pointed out difficulties which we found in the current formulations . Th erefore it seemsthat a coarse grained entropy approach based on SG does not provide a satisfactory conn ection with irreversible thermodynamics. . . .further study of the connection of the dynamics of particle systemsin nonequilibrium states and irreversible thermodynamics is still required . . . [211

§4. Equilibrium Molecular Dynamics

4.1 Numerical Methods. At this point we move from statist ical to dynamical descriptions of many particle systems, in particular nonequilibrium stationary st at es. To construct mathematical models it is helpful to take inspirationfrom compu ter algorithms used to study such systems. Aggregates of millionsof particles can now be simulat ed on a computer. In this way, equilibrium andnonequilibrium properties of materials may be computed using any desired intermolecular forces and initial condi tions [28, 29, 30]. Compared to analyt iccalculat ions, many restrictions such as simplicity of the forces, approximat ionsand assumptions can be eliminated. Compared to experiments , the result s areonly as good as the model , but it is possible to simulate experimentally inaccessible regimes. Compared to mathematical proofs, the result s are usually notrigorous , however while a system may not be proved ergodic (for example), empirical limit s may be placed on non-ergodic behaviour, sufficient to determinewhether any such non-ergodic behaviour is physically relevant. A recent discussion of irreversibility using ideas from chaos and computer simulations can befound in Ref. [31] .

T he Lorentz Gas : A Par adi gm for Nonequilibrium Stat ionary States 331

It is difficult to put rigorous bounds on the accuracy of numerical simulat ion result s, par ticularly when the dynamics is exponent ially unstable. Oftenthere is a shadowing theorem stat ing that the numerical t rajectory is close tosome exact tra jectory of the dynamics, however this does not guarantee thatthe exact trajectory is typical with respect to the desired distri but ion of init ialcondit ions. Sometimes a simulated low dimensional attractor can result in a periodic orbi t due to the finite number of states accessible to the dynamics. Theaverages and other properties of this periodic orbi t are quite different to thedynamics as a whole. In this case, the addition of small amounts of noise to thedynamics often leads to more realistic tra jector ies, and can actually be used asa mathematical definiti on of an at t ractor [321. When the corre lation dimensionof the attrac tor is sufficient ly large (for example, 2) precision related periodicorbits are rarely observed, and standard test s such as varying the precision ofthe calculat ions usually indicate that the results have prob ably converged.

The results of numerical simulat ions are as good as the algorithms used.While at taining optimum speed and accuracy is somewhat of an art form, thereare a number of general methods and principles. Th e equat ions of motion forsimulat ions are Newton 's equat ions of motion, reducing in the simplest case ofN spherical identi cal par ticles to

. PiXi=

m

Pi = _~ a¢(r ij )Z: ax·)i-i t

r ij = IXi - x j l

(26)

(27)

(28)

interacting via a specified potential ¢, which can be calculated from pair correlation data obtained in diffraction experiments .

The Lennard-Jones potenti al,

(29)

is quit e realisti c for monat omic fluids such as argon. Th ere are of course moreelaborate models involving interactions between three or more particles for specific substances, for example carbon [33], and in principle there are also quantumeffects. Here (J and t are parameters setting the length and energy scales, respectively. In simulat ions mass, length and time are scaled so that m = (J = t = 1.When there is more than one type of par t icle it is possible to scale the positions,momenta and forces by appropriate factors of the square root of the mass inorder to remove m and t from the prob lem, however the differing radii remainintrinsic to the dynamics.

It is somet imes advantage ous to eliminate the possibility of bound statesgenerated by the negative par t of the potential , and also to make it finite range

332 C. P. Dettmann

(30)'" () { <PLJ(r) + 1,/,WC A r = 0

to shorten the computat ion. For this reason it is common to use a shifted andtruncated version, called the Weeks-Chandl er-Andersen [34] potenti al,

r < 21/ 6ar > 21/ 6a

which has a continuous first derivative across the boundary.Still simpler, and surprisingly realisti c at low to moderate densities is the

hard ball potenti al,

(31){

<Xl r< a<PHB(r) = 0 r >a

See Fig. 2. The great advantage of hard potenti als for simulat ions is th at thesolution of the equat ions of motion is known, so it is not necessary to use integration routin es which are much slower than substitution into an explicit formulaand often requir e relatively small steps for accuracy. T he disadvantage froma physical point of view is the absence of a characterist ic energy scale, lead ingto a trivi al dependence of thermodynamic qun ati ties on the temperature. Nevertheless, hard ball gases exhibit fairly realist ic phase transitions in terms ofpressure and density.

T he boundary condit ions are extremely important for both equilibrium andnonequilibrium simulat ions. For example, suppose we have 103 particl es in 3dimensions, so 10 particles in each direction. Suppose also that the boundarycondit ions are not treat ed correctly, affect ing a boundary layer of one particle.Thi s is a conservative estim ate , since a dilute system would have a longer meanfree path and hence a thicker boundary layer . The numb er of particles not onthe bound ary is 83 = 512. Thus almost half of the particles are affected bya poor choice of boundary condit ions in this example.

Often we are interested in th e bulk prop erties of a medium, far from anyphysical boundary. For th ese properties the natural boundary condit ions areperiodic, viewed either as a unit cell infinitely repeat ed (corresponding to aninfinite syst em with a special symmet ry) or as a finite syst em where particlesthat exit via one boundary reappear at the oppo site boundary. Both viewpointsare useful, dependin g on what is being discussed. The most common periodiccells for molecular dynamics simulations are either chosen for simplicity (square,cube), or based on a close packed array, particularly for high density (hexagonal ,rectangle with side rat io a rational multiple of J3, similar choices in threedimensions) . The hexagonal case is of specia l interest for the Lorent z gas, as itcan lead to a finite horizon, see Sec. 4.2.

It is clear that equilibrium prop erties can be calculated in this way, but thecorrect approach to nonequilib rium prop erties is far from obvious. T he manypossible schemes of theoretical and/ or practical interest may be broadly categorised as follows:

1. Linear response (Green-Kubo) formulae from which linear transport coefficients may be calculated from purely equilibrium simulat ions.


2.------- -.------T"1r---- - -,-- -------,

1.5

\

0.5

¢J(r) 0

-0.5

-1

-1.5

-20

- - Hard ball-------. Weeks-Chandler-Andersen

Lennard-Jones

0.5

r

1.5

.. ' -

2

Figure 2: Interparticl e potent ials, scaled so t hat a = t = 1.

2. Homogeneous molecular dynamics, where the contact with the environment is simulated by driving forces on each par ticle, thermostat "frict ional"forces, and (for shear flow) "sliding brick" boundary conditions.

3. Inhomogeneous systems driven ent irely by bound ary effects.

4. Inhomogeneous systems with a combinat ion of boundary effects and bulkeffects such as thermostats.

Th e most efficient met hods for calculating the linear transport coefficientsare the homogeneous thermostatted approaches, which is what they were designed for. The ot her approaches nevert heless have a great deal of theoret icalinterest , including a number of analyt ic relations between dynamical (microscopic) and thermodynamic (macroscopic) properties.

The calculat ion of nonequilibrium prop ert ies from equilibrium simulat ionsis clearly limited to situations close to equilibrium; beyond linear transportcoefficients the response is usually nonanalyti c as in Sec. 5.6. For the otherapproaches, the degree to which far from equilibrium predictions can be madedepends on the physics. A system far from equilibrium that remains homoge-

334 C. P. Dettmann

neous must usually radi ate heat (by phonons , photons, neutrinos etc. with longscattering length s) rather than conduct it . Similarly, boundary driven nonlinear effects are more strongly affected by the choice of boundary conditions thannear equilibrium. For reasons such as these, far from equilibrium situations needto be put on a more individual basis, not to say that they don 't share manyproperties in common.

The remainder of thi s article discusses a numb er of these schemes in detail ,specifically Green-Kubo formulae and some t hermostatted and bound ary methods. These are illustrated using the Lorent z gas, from which general propertiesof nonequilibrium steady states can be understood and discovered, and to whichwe turn now.

4.2 The Periodic Lorentz Gas. What is the simplest possible moleculardynam ics model? If we use periodic boundary condit ions, momentum is conserved, so a single particle moves trivi ally with const ant velocity. The simplestinteraction potential is the hard ball, Eq. (31). Two identical hard rods in onedimension exchange their velocities on collision, again leading to trivi al dynamics. Thus we need two hard disks moving in two dimensions under periodicboundary condit ions.

Assum ing there is no drift (that is, no cent re of mass motion) and moving torelative coordinates, we see that the problem of two hard disks is equivalent toa point particle colliding with a disk with twice the original radius in periodicboundary condi tions or (equivalently) on a periodic lat tice of such scatterers.This is the periodic Lorentz gas. As a model it differs only from the randomLorentz gas, Sec. 3.3 in the location of the scatterers and possibly wheth er theyoverlap , but the interpretat ion here is quit e different.

There are three possible regimes in the periodic Lorentz gas, dependingon the shape of the periodic cell and th e size of the hard disks , see Fig. 3.Because the reduction to relative coordinates has the effect of doubling theradius, it is possible for the disks in the reduced case to overlap , often leading toa trapp ed scenario where th ere is no diffusion. It is possible to define a viscosityhowever [101 . When the disks do not overlap , it is possible for a hexagonal cellto have an upp er bound on the t ime between collisions, and the Lorent z gas issaid to have finite horizon, and there is norm al diffusion defined by (x2 ) "-' t ,see (35) below and Ref. [351 . This is similar to the random Lorentz gas of Sec. 3.3which has zero probability of an infinite tr ajectory, and also normal diffusion.For square, rectangular , and three dimensional cells, non-overlapping disks havean infinite horizon, leading to anomalous diffusion of the form (x2 ) "-' tIn t (seeRefs. [36, 371for two dimensions) .

The periodic cell in two dimensions is usually square , rectangular or hexagonal. In each of these cases, the Lorentz gas is dynamically equivalent to a finitebilliard of the same shape and size and with hard wall boundaries . Thi s is because a billiard with reflections at the boundary can be extended by reflecting(rather than translating) the domain across each st raight boundary. In addit ion,

The Lorentz Gas : A Par adigm for Nonequilibrium Stationary St ates 335

00 0008008000 00

Figure 3: The periodic Lorent z gas with hexagonal lattice. Th e scatterers have un itradius. There are three regimes depending on th e spaci ng w : (a) Infinite hori zon ,w > 4/ J3- 2; (b) Finite hori zon, 0 < w < 4/ J3- 2; (c) Overlapping, J3 - 2 < w < O.

the square, rectangle and hexagon are the same whether reflected or translated,so reflecting boundary conditions are equivalent to periodic boundary conditions. Thus the Lorentz gas with a square periodic cell is equivalent to the Sinaibilliard, which contains a circular scatterer at the centre of a square billiard.

Common to many models with hard collisions, it is often convenient (alsofor nonequil ibrium extensions discussed below) to consider the natural Poincaresect ion det ermined by the collisions, replacing the flow in continuous time bya map from one scatte rer to the next , together with useful phase function s derived from the flow such as the time between collisions and the displacementbetween the cent res of the initial and final scatterers. For the two dimensionalLorent z gas this corresponds to a two dimensional map , with the variables givenby position on the scatterer and outgoing direction of the particle. For periodicmodels, the dynamics does not distin guish between scattercrs due to translational invariance, but it is necessary to keep track of the displacement of theparticle from its init ial position in order to calculate, for example, the diffusioncoefficient .

4.3 Green-Kubo Relations. The method of Green [38] and Kubo [391computes the linear transport coefficient s in terms of time correlat ion functionsof quantities computed in an equilibrium state . Th e relations can be derivedeither from linear response theory or an approach based on the Chapman-Enskogmethod of solving the Boltzmann equation (for example see Chap. 6 of Ref. [71) .Here we give a short derivation of the Green-Kubo relation for the diffusioncoefficient , then discuss various extensions.

336 C. P. Dettmann

We begin by solving the diffusion equation for the probability densi ty offinding a particle at a given position and time P(x, t) , which is the Boltzmanndistribution function f(x , v, t) integrated over velocity in the macroscopic limit(large t imes and dist ances) ,

(32)

We used a Fourier transformed version of this equation to define the diffusioncoefficient of the random Lorentz gas in Sec. 3.3 . The equat ion is linear , so thegenera l solution is an integral over the Green 's functions given by the solutionfor an initi al Dirac delta distribution P(x,O) = 8(x - xo) , that is,

(33)

where d is the spatial dimension. The mean square displacement of a particle isthus

(34)

We expect the diffusion equat ion to approximat e particle dynamics only atsufficiently large tim es, larger than typical correlat ion times since the diffusionequat ion contains no memory, and we have also neglected the velocity degreesof freedom. Thus we have

(35)

which is the Einstein relation for the diffusion coefficient . The diffusion coefficient is thus given (assuming the limit exists) by

D = ~ lim ~ ((Xt - XO)2)2d t--+oo dt1 .- lim ((Xt - xo)· Vt )d t --+ oo

-d1

lim r(Vt' . vt )dt't--+ ooi o

1100

- (Vt' . vo)dt'd 0

(36)

which is the Green-Kubo relation for diffusion. This relation has been used tocalculate the diffusion coefficient of the periodic Lorentz gas [35] .

A sup erdiffusive case where the mean square displac ement grows faster thanlinearly with the time , such as when there is an infinite hor izon, then corresponds

The Lorentz Gas: A Paradigm for Nonequilibrium St ationary St ates 337

to an infinite integral above, as when the velocity autocorrelat ion function decreases as l it or slower with a finite numb er of sign changes. Systems th at aresubdiffusive with a slower growth of the mean square displacement correspondto zero integral which is harder to observe, and not expected in the Lorentz gasunless a significant proportion of the disks are touching or overlapping.

Truncating the correlation integral at finite t ime gives (omit ting the limitsabove) a t ime dependent diffusion coefficient , propo rtional to th e time derivative of the mean square displacement at short times. Such a time dependentdiffusion coefficient is useful to describe the "t ransient" response , that is, beforecorrelations have died away.

Anisotropic systems can diffuse at different ra tes in different directions. Thediffusion coefficient is replaced by a real symmetric positive definite matrix D i j ,

which can then be diagonalised leading to d different coefficients along the coordinat e axes. The symmetrie s of the Lorent z gas in a square or hexagonal latticepreclude such an anisotropic diffusion coefficient, however it occurs naturallywith a rectangular lattice. As noted above, a rectangular lattice of one scattererhas an infinite horizon and hence anomalous diffusion, so at least two scatterers per unit cell are required to obt ain anisot ropic normal diffusion. Here wetypically refer to the hexagonal Lorent z gas and write simply D , however it iseasy to generalise most equat ions, for exampl e (32,49) to the anisot ropic case.Nonlinear response is more general, and leads to anisot ropic behaviour even forth e more symmet ric hexagonal case (see Sec. 5.6).

A more general macroscopic equat ion for P(x, t) would involve more spatialderivatives, corresponding to behaviour at shorter distances, and nonleadingterms in the dispersion relat ion (14). The coefficients of such terms are calledlinear Burnett and super-Burnett coefficients (not to be confused with nonlinearBurnett coefficients involving higher powers of the forces) . The time correlationfunction expressions for these coefficients [6] involve cumulants of the form

integrated over all times. They are in general less convergent, so are expected todiverge for the random Lorent z gas due to its power law decay of correlat ions [19,241 . This divergence corresponds to a nonanalyti c dispersion relation (14), seeRef. [25J . The map corresponding to the finite horizon periodic Lorent z gas hasexponent ial decay of (two time) correlations [40, 41J ; the current bounds on theabove (four t ime) cumulant are slightly slower than exponent ial, but sufficientto show the convergence of the Burnett (and all higher) coefficients [42, 43J . SeeRef. [6, 441 for the connection between this map and the diffusion and Burnet tcoefficients calculated in continuous time.

In genera l, all linear transport coefficients can be written in te rms of integralsof tim e correlation functions similar to (36), with the velocity replaced by therelevant thermodynamic flux. For example, the viscosity is computed in terms ofcorrelations of the shear stress, and thermal condu ctivity is computed in terms

338 C. P. Dettmann

of correlations of the heat flux. All correlations are computed at equilibrium.Details can be found in Ref. 1291.

There are a couple of limitations to the use of Green-Kubo relations for computing properties of nonequilibrium systems. The most obvious is th at these relations apply only to linear response; they cannot be applied to systems far fromequilibrium. The other limitation is that correlation functions being statisticalin nature are difficult to calculate to a high degree of precision, compounded bythe necessity of a numerical integration, often with a poor rate of convergence.Both of these difficulties can be alleviated using the rmostats, the subject of thenext section.

§5. Nonequilibrium Molecular Dynamics

5.1 Introduction to Thermostats. A thermostat , as its name implies, isa device constructed to control the temperature. In the context of moleculardynamics simulations , it is a term added to the equations of motion of a systemto simulate the effects of the environment. As such, thermostats serve two mainpurposes:

1. They allow simulat ion of nonequilibrium steady st at es. As noted in Sec. 2.1,nonequilibrium stationary states necessarily have contact with the environment. There are external forces and heat flows . In such situationsa thermostat is needed to keep the energy of the system const ant (eitherexact ly or in an average sense) , so that the system remains in a stationarystate despite external forces that tend to increase the energy.

2. They allow simulation of different ensembles. Sec. 3.4 describ es equilibrium ensembles in terms of contact with th e environment , and also pairsof conjugate variables . The Nose-Hoover thermostat (below) allows simulat ions in the canonical ensemble , by fixing the temperature and allowingthe energy to vary. Similarl y, thermostats may be designed to fix almostany system variable (for example kinetic energy, total energy, current ,pressure , enthalpy) while leaving conjugate variabl es to vary. The various thermostats can thus be understood as the ensembles of nonequilibrium stat istical mechanics. It is expected that they should lead to equivalent results in the thermodynamic limit (except for fluctu ations in thefixed quantities) as do the equilibrium ensembles, at least in the linearregime 129, 45, 46].

An alte rnat ive to a thermostat where environmental effects are put in theequations of motion is to simulate such effects at the boundary, which we discussin Sec. 6; an advantage of thermostats is that they permit the simulation toremain homogeneous, see Sec. 4.1.

The Lorentz Gas: A Paradigm for Nonequilibrium Stationary States 339

A common objection to the use of thermostats is that they add "unphysical" forces to Newtons "exact" equations of motion . The fact is that any schemeused to replace an unbounded environment by a finite number of degrees of freedom (including alternat ive boundary methods) must unavoidably make drasticapproximations. Some facts that inspire confidence in thermostatted methodsare their ensemble equivalence (above) and their agreement with Green-Kuborelations for linear transport coefficients. Far from equilibrium, thermostattedapproaches should apply whenever the bulk of the system is in contact withthe environment , either because it is two dimensional , or thermal transfer byradiation is sufficiently strong.

5.2 Gaussian and Nose-Hoover Thermostats. A simple method of ensuring that a nonequilibrium molecular dynamics simulation remains st ationaryin time despit e exte rnal forcing is to periodically renormalise the velocities ofall the particles to keep the (kineti c or internal) energy constant . If the timeinterval between successive renormalisations is reduced to zero, we obt ain theGaussian thermostat , discovered indep endently for the kinetic energy by Hooverand collaborators /471 and for the internal energy by Evans 148]:

P

F i + F e - ap (37)

Here the mass m = 1 and the particle indices have been suppressed, leading toa description in terms of N d-dimensional vectors. F , contains all the interparticleforces as describ ed in Sec. 4.1, Fe contai ns ext ernal driving forces, and a isa scalar thermostat "frict ion coefficient" which is the same for all particles anddirections .

The value of a is determin ed by the desired constraint : For constant kineticenergy we have the Gaussian isokinetic thermostat ,

(38)

where the dot product includes a sum over th e particles. It is easily verified thatthe kinetic energy K = p . p /2 is identi cally preserved by th ese equations. Theterm "Gaussian" applies to Gauss' principle of least constraint /49]' wherebythese equat ions may be derived by demanding the smallest const raint force(according to th e above dot product) at any time , see [29, 501 .

For constant internal (kinetic plus interparticle) energy we have the Gaussianisoenergetic thermostat ,

Fe·pa CIE =-

p .p(39)

which preserves K + cPi assuming that the internal forces are conservative, F , =-V'cPi '

340 C. P. Dettmann

In this notation the Nose-Hoover approach tre ats the thermostatting multiplier Ct as an additional dynamical variable with a feedback mechanism , so wehave for the Nose-Hoover (isokinetic) thermostat

. 2CtNHIK = Q(K - K o) (40)

where Q is a constant that determines the time scale of the feedback and Kois the desired kinetic energy, usually N dkBT/ 2. This is (apart from slight differences in not ation) Hoover 's reformulati on of the Nose thermostat discussedin Sec. 5.4, see Refs. [51 , 52, 53J. The feedback operates as follows: Supposethe initial kinetic energy becomes too high, then 0: is positive, leading to moredamping in (37) which then decreases the kineti c energy, and similarly if th ekinetic energy becomes too small. It is also possible to replace the K 's above bythe internal energy to construct a Nose-Hoover isoenergetic thermostat.

All of these thermostats simulate the exchange of thermal energy betweenthe system and its environment. On the average thi s flow is outward (zero atequilibrium), corresponding to positive (o), however there is no reason that Ct

should not become negative occasionally, unlike macroscopic frictional forces.To be more precise, we can compute the amount of heat being removed by thethermostat and use irreversible thermodynamics (Sec. 2) to writ e

(41)

from which we deduce th at the non-negativity of Ct on average is guara nteed bythe second law.

In the limit of large systems, we expect th at all the various processes thatthe par ticles undergo tend to average out , leading to a more or less constantvalue of o , as well as various macroscopic vari ables[54]. Thi s is consistent withthe very low probability of a decrease in entropy in a large dissipative system. Because the fluctuations of all thermodynamic quantities are smaller inlarge systems (except near phase transitions) , different thermostat s approachthe same thermodynamic state, another stat ement of ensemble equivalence.

At equilibrium, th at is, with no external force F e it is clear that the isoenergetic thermostat multiplier CtGIE is identi cally zero, while th e other thermostat svary around zero. The Nose-Hoover thermostat is special in that the equationsgenerate the canoni cal ensemble [51, 52, 53], that is, assuming the dynam ics isergodic (a reasonable assumpt ion in practice for all but the smallest syst emsbased on numeric al work [55]), the probability distribution of x and p (averaging over Ct) is given by (19). This means that the Nose-Hoover th ermostat canbe (and is) used to simulate an equilibrium system at fixed temperature, ratherthan fixed energy.

There are many other typ es and uses of thermostats. There are specific algorithms for computing all possible transport coefficients . For example, shear vis-


cosity can be computed using "sliding brick" boundary condit ions, thermostatted such tha t the temp erature is measured relative to a linear velocity profilecharacterist ic of Couette flow. Thermal conduct ivity can be computed by including forces that accelera te hot and cold particles in different direct ions. Bothof these examples are homogeneous, with no dependence of distribut ions onposition. There are also inhomogeneou s algorithms, where different parts ofa system (for example particles sufficiently close or belonging to the walls) arethermostatted at different temperatures, or at th e same temperature relativeto different velocities. Finally, it is possible to apply thermostats to enforceother ensembles, for example constant pressure (hence fluctuating volume). Allthese examples and more are described in texts on nonequilibrium moleculardynamics 128, 29, 301. A more recent review of the Gaussian and Nose-Hooverthermostats with a discussion of Gauss' principle and applicat ion to the Lorent zgas is given in Ref. [50] .

5.3 The Nonequilibrium Lorentz Gas. Just as the random Lorentz gasappears as one of the simplest applicat ions of the Boltzmann equat ion (Sec. 3.3)and the periodic Lorentz gas appears as one of the simplest exampl es of equilibrium molecular dynamics (Sec. 4.2), so the (more precisely "a") nonequilibriumLorentz gas appears as one of the simplest examples of nonequilibrium molecular dynamic s. We begin with a description of the "colour diffusion" algorithm forthe self diffusion coefficient , see Ref. [29] . Thi s is arguably the simplest nonequilibr ium molecular dynamics algorithm, as it is homogeneous, involves only theusual periodic boundary conditions, and the external force on each par ticle isa constant .

Th e self diffusion coefficient is the limit of the mutual diffusion coefficient ofa mixture of two species th at become identi cal. In the colour diffusion algorithm,each particle is assigned a positive or negati ve "colour charge" which (unlikeelectric charge ) has no effect on the interparticle forces, but determines theinteraction with an exte rna l "colour field" Ee . Thus the external force on particlei with charge Ci is Fe = Ci Ee . The response to such an external field is the"colour current", J c = L CiPi!m. The diffusion coefficient (equivalent to "colourconduct ivity") is then proportional to the ra t io I(JC) I/IEc l in the limit IEel -7 o.In order for th e time average of the curre nt to make sense, a thermostat mustbe applied. From the point of view of calculat ing the linear response of a manyparticle system , it doesn 't matter which thermostat is applied, or whether it isappli ed to the whole system or to the two types of particles separately.

The simplest such case of the colour diffusion algorithm is thus two particles(one of each colour charge) interact ing with a hard ball potenti al in two dimensions with a Gaussian th ermostat (the Nose-Hoover thermostat has an extraphase space variable a ; it has recentl y been studied in Ref. [56]). The isokineticand isoenergetic thermostat s are equivalent here, since the internal force is zerooutsid e collisions, and the collisions are not affected by either thermostat . Asin Sec. 4.2 we consider relative coordina tes, which reduces the problem to that

342 C. P. Dettm ann

of a single point particle moving in a periodic cell under the influence of a constant field F and a thermostat, and colliding with a single circular scatterer:The nonequilibriurn Lorentz gas [57J .

The thermostat ensures th at twice the energy of the particle, p 2 /m is constant, so as before it is possible to set th e magnitude of the momentum, themass and the rad ius of the scat tercr equal to unity by appropriate scaling. Theequat ions of motion are thus

PF - F · pp

(42)

(43)

Note that the denominator of (38) for (XCI K may be set equal to unity due to theconstancy of the kinetic energy. The equat ions for the nonequilibrium Lorent zgas, generalised to arbitrary dimension and position dependent exte rnal forcesapply to many par ticle systems const rained by the Gaussian isokinetic thermost at , and hence approximately to other thermostatted systems when ensembleequivalence holds . Thi s close connection between the nonequilibrium Lorentzgas and many par ticle systems in a nonequilibrium steady st at e is ext remelyuseful in discovering and demonstrating genera l properties of the latter .

The solution of the isokinet ic equat ions for the Lorentz gas is most easilyexpressed in terms of the angle {} between the direction of mot ion and thefield, which is assumed to be in the positive x direction and have magnitude F .Specifically Px = cos {} and Py = sin {} in two dimensions, leading to iJ = - F sin {}.Given initi al condit ions with a subscript 0, the solutions are

tan({}12) ( t - to) (44)tan({}o /2) exp - -p

1 ( sin{) ) (45)x Xo - - In -.--F sm{}o

({) - (}o ) (46)Y Yo- --F

with direct genera lisations to higher dimensions. Note that the displacementtransverse to the field y - Yo cannot exceed 7rIF ; the particle rapidly approachesthe direction of the field. The transcendent al functions make it difficult to determine analyt ically when a collision with the circular scatte rers takes place ;one possible numerical approach is to put a lower bound on the tim e to thenext collision using a circular approximation to the trajectory, moving forwardth is time step, and iterating to convergence [58J . In spite of thi s difficulty, it ispossible to obtain analytic expressions for the linear integrat ed equat ions usedto compute the Lyapunov exponents in terms of the initi al and final angles ofeach free path between collisions, see Refs. [50, 59].

In response to the externa l field F and collisions with the (usually hexagonal)lattice the particle drift s with a current given by

J=x

assuming finite horizon. Using Eq. (38) with Ipi = 1 for a we find

J·F=a

(47)

(48)

in agreement with (41). In the limit of small field the average current is thesame for almost all (Lesbesgue) initi al conditions [611 and is given by

(J) = DF + o(F) (49)

where D is the diffusion coefficient , or tensor in the anisotropic case. For the caseof infinite horizon , there are two possibiliti es: When the field is along one of theinfinite horizon directions the particl e almost always ends up moving withoutcollisions along this direction , otherwise the current appears normal. The zerofield limit is thus not defined, and in any case would correspond to anom alousdiffusion.

The equations of motion of the nonequilibrium Lorentz gas have the followingimmed iate properties, which also apply to more general thermostatted systems:

1. Tim e reversib ility: Reversing the direction of t ime is equivalen t to replacing p by - p , as in Newtonian (unthennostatted) mechanics . This has theeffect of changing the sign of a . On the Poincare section determined by thesurface of the scatterers this corresponds to a reflection in the outgoingangle across the norm al to the scatteror, th at is, replacing eby e+ 2X inFig 1.

2. Phase space contraction: Liouville's equation (21) implies that the rateof growth of a volume element <5V , which is inversely proportional to theprobability density p evolves according to

<5V = _ ~ dp = _ ~ (ap + F . "Vp) = "V . F<5V pdt p at

(50)

Evaluating "V . F for the equations of motion, (37) with (38-40) we find

{

-(Nd - l)acIK"V. F = -(Nd - l)acIE

- N daNHIK(51)

which reduces to -a in the two dimensional Lorent z gas. The collisionsof the Lorent z gas (or other hard ball systems) are instantaneous andpreserve phase space volume, so they do not affect the above formulae.

344 c. P. Dettm ann

The phase space cont raction has a numb er of effects, namely that the sumof the Lyapunov exponents is negative, and related to the average value of 0:,

(52)

(53)

For the case of the two dimensional Lorentz gas , the Kaplan-Yorke relat iongives the information dimension D 1 [60] of the attractor for the Poincare mapfor sufficient ly small field [61],

D_ )'1 _ 2>'1 + (0:)

1 - 1 + - - - - -,.:--'...1>'21 >'1 + (0:)

Thi s is less than the dimension of the map (two) since the phase space contraction requires the density to concent rate on a small set most of the t ime. Nevert heless, the attractor is dense in phase space for sufficient ly small field [61],leading to a box dimension Do of

Do = 2 (54)

Numerical evidence for what "sufficient ly small" implies in pract ice is givenbelow, Sec. 5.6 . The concentrat ion of the density onto multi fractal distribut ionsmeans that for the steady state, the density becomes a distribution, and isst udied by means of more general techniques, such as Sinai-Ruelle-Bowen (SRB)measures or periodic orb it measures, Sec. 5.5.

It is clear from Eq. (41) that 0: is related to the rate of entropy production,so now phase space contraction can also be related to entropy production. TheGibbs ent ropy, Sec. 3.4, which was constant for equilibrium (specifically phasespace volume conserving) systems, now decreases to negative infinity!

lim ddSc = lim / (V' .F )pdr = - (0:) < °

t -too t t-eoo(55)

Needless to say, this has been the source of a large amount of confusion inthe literature. The correct resolution is probably along the following lines: TheGibbs ent ropy is telling us (appro pria te ly) that ent ropy is being removed fromthe system via the thermostat ; it does not take into account irreversible entropyproduct ion in the system, as it did not do so for isolat ed systems; it cannot tellus about the entropy increase in the environment since the phase space doesnot include these degrees of freedom. There have been a numb er of at tempts(most ly in connection with Baker maps) to coarse grain the Gibbs entropy ofa nonequilibrium system, in orde r to take into account the irreversible ent ropyprodu ction. Thi s is a very act ive area of discussion at present , see Refs. [9 , 271 .

T he accumulated phase space cont raction along a trajectory is easily foundto be

JV( t) _ - f~ o (t' )dt 'JV(O) - e (56)

The Lorentz Gas: A Paradigm for Nonequilibriurn Stationary States 345

for the nonequil ibrium Lorentz gas (with obvious extensions to all the thermostats considered above) , corresponding to a probability density of

p(Xt, t) = eJ~ a (t' )dt'

p(xo,O)(57)

assuming continuous distributions. This is an example of a Kawasaki distribution funct ion [29, 621 . The argument of the exponent ial gives the total amountof entropy removed by the thermostat ; for the Lorent z gas th is is

l t

a(t')dt' = l t

J . Fdt' = JF· dx = -b.cI>

assuming the external force is conservative,

F= -V'cI>

(58)

(59)

For the Lorentz gas, cI> is a linear funct ion of the coordina tes. Thi s expressionfor the accumulated phase space cont raction provides a motivation for the symplectic st ructure of th e next section , as well as a basis for a discussion of timereversibility.

Newton's equations are tim e reversible, so one of the difficulties in understanding the second law of thermodynamics is that for any system observed toincrease in entropy, it is possible to set up a time reversed system with a decrease in entropy with time. Boltzmann 's solut ion (Sec. 3.1, Ref. [11]) is thatthe most likely st ate s (corresponding to large regions of phase space) are thosewith high entropy; the initi al st at e of the Universe has very low entropy for somereason, but the final state is not const rained in this way. In the same way, forevery tr ajectory in a therrnost at ted system with positive a and hence positiveentropy production , there is a time reversed trajectory with the opposite. However a uniform initi al distribution, or in fact any smooth initi al distribution, has(at long tim es) a great er probabili ty of posit ive a leading to a positive (a) . Thi sis because the volume in phase space is bounded, and so only an exponenti allysmall proportion of t ra jectories can grow with a positive exponent ial, while theremainder are forced to contract to make room for th e growing t ra jectories.A more quantitative description can be given in terms of periodic orbits , seeSec. 5.5 and Ref. [63] .

If, in addition to phase space contraction, sufficient ly strong chaot ic propert ies ("Anosov-like") can be assumed, th e ratio of the probabilities th at a trajectory of length T will have entropy production b.S (as measured by the phasespace contraction above) or -b.S in the limit T -+ 00 approaches et:. s . The limitis taken keeping the entropy production rate b.S/T constant. This result , calledthe fluctuation theorem was first observed for shearing flow in Ref. [64] andproved in Refs. [65, 661 . It appli es to the Lorent z gas if the field is not too large;although it is not st rict ly Anosov due to the collisions, it nevertheless ret ains

346 C. P. Dettm ann

(61)

very strong chaotic properties . The fluctuation th eorem and its generalisation sare an active area of investigation at present .

5.4 Symplectic Properties. Another of the unexpected properties of thermostatted systems (in particular those with isokinetic thermostats) is that , despite phase space contraction , it is possible to express th e dyn amics in termsof Hamiltonian equations which are by definition (23) phase space conserving.The first such formulation was the origin al Nose thermostat [51, 52, 531,

N 17I"il2 p2

HN(x , s; 71", Ps;A) = 82mis2 + ¢(x) + 2Q + 2Ko In s (60)

Here s and its conjugate momentum Ps are supposed to represent the "heatbath". If we interpret the time vari able A as relat ed to physical t ime t by dt =dAjs then we can derive Hoover 's form of the equations, Eqs. (37,40) usingPi = m idxd dt = 7I"ds and a = Ps/Q (and in our case, setting th e massesequal to one) . Note that rewriting the equations in thi s manner has reduced thenumber of dimensions of phase space by one , since the equat ions of motion for x ,p and a do not cont ain s. Thi s also means that th ere is no manife st constant ofmotion (given by H N ) for th e new form of th e equa t ions. The new equat ions arephase space contracting because they are written in different var iables - thephysical momentum p differs from canonical momentum 71" by a factor s, whichkeeps t rack of th e entropy production since its equat ion of motion is ds / dt = as .

Another Hamiltonian for a th ermost atted system is th at of the Gaussianisokinet ic thermostat , which in contrast to the Nose-Hoover thermost at hasa manifest constant of motion, namely the kinetic energy. Thus it is natural forthe Hamiltonian to be some function of the kinetic energy, written so th at thephysical and canonical moment a vary by the accumulated phase space cont raction, eLl.<l> (see th e end of the previous section) . In fact , the Hamiltonian [151

71"2Hc(x;71" ; A) = e2<l> 2

with the interpretation dt = e<l> dAleads to the Gaussian isokinetic equations (37,38) with p = e<l> 7I" and F = - V'cfl when the const ra int p2 = 1 is imposed. This isalso a Hamiltonian th at generates geodesics [67] on the space with conformallyflat met ric

(62)

leading to variational approaches based on finding the stationary (usually minimum) geodesic length, and an interpretation as light passing through a mediumwith refractive index n = e- <l> .

There is a thi rd th ermos tat with a Hamiltonian description, namely theisoenergeti c thermostat (39) restricted to the case where th e internal and ext ernal forces are proportional , that is Fe = -I'F , F , = (l- I')F with I' consta nt

(63)

The Lorent z Gas: A Paradigm for Nonequilibrium Stationar y Stat es 347

and F = - \7ll>. The momentum equat ion is th en

dp F ·p- =F- ,--pdt p2

with the conserved energy E; = p2/2 + (1 - , )ll> . This restri cted isoenergeticthe rmostat is not real istic from the point of view of internal and external forcesbeing proportional; rather it allows a continuous interpolation between the caseof no thermostat , = 0 to that of the isokinetic thermostat , = 1. Becausethe kinetic energy is no longer constant, the denomin ator cannot be ignored ,in fact an additive constant is added to ll> to ensure th at E: is zero, then p 2can be replaced by -2(1- ,)ll>. Noting th at FIll> can be written - \7 ln lll> l, theaccumulated phase space contraction (56) is thus 1ll>1 -y / (2(1 - -y)). Paralleling theisokinetic thermostat , we then arrive at the "restricted Gaussian isoenergetic"Hamiltonian [68J

(64)

which, coupled with the constraint HR G1E = 0 and the tim e scaling dt =1ll>1 - -y/ (2(1 - -Y))d,\ leads to the above equat ions of motion.

It was noted in [15] for the Gaussian isokinet ic th ermostat , in [69J for theNose-Hoover thermostat , and in [68J for the restr icted Gaussian isoenergeticthermostat that the somewhat arbitrary time scaling may be obviated by addinga constant to the Hamiltonian to make its numerical value zero, and then multiplying by an appropriate factor, namely c1> for the Gaussian the rmostat and sfor the Nose-Hoover th ermostat . In general, the Gaussian isokinetic Hamiltonianwith a t ime scaling of dt = e{31> d,\ becomes

71"2 e ({3 - 1)1>H a(x' 71" '''\) = e({3+ 1)1> - - - - -

" ' , 2 2(65)

with the isokinetic constra int simply H{3 = O. These Hamiltonians apply tothermostatted systems with arbitrary conservative forces and arbitrary numbersof particles. The Lorent z gas version of the case f3 = - 1 corresponding to thefamiliar kinetic plus potential energy Hamiltonian was noted by Hoover andcollaborators eight years previously [70J .

The Hamiltonian gives an alt ernative derivation of the solutions of the equations of motion of the nonequilibrium Lorent z gas, Eqs. (44-46). The pot ent ialll> = - Fx does not depend on y , so 1ry = eFxpy is a constant of the motion . Px

is determined by the constraint p; + P~ = 1, allowing an immediate solution incartesian coordinates by integra tion.

While the equations of motion of these thermostats can be derived froma Hamiltonian, the global structure including the periodic boundary conditionsis not st rictly Hamiltonian . This is because the potential ll> (for example) isnot periodic; for th e Lorent z gas it is a linear function of position. The lack of

348 C. P. Dettmann

a global Hamiltonian allows the st eady state distributions not to be uniform onsome energy surface; they are typically multifr act al. In spite of this , the localsymplectic structure is sufficient to ensure the pairing of Lyapunov exponents,discussed next. The isokinetic Hamiltonian has also been applied to a definitionof temperature using the Boltzmann ent ropy in [14] . Choquard [71] has a furtherexpos ition of the variational prop ert ies of the isokinet ic thermostat, includ inga Lagrangian approach and a link with the conforrnally symplect ic formalismused in Ref. [721 for a proof of the pairing rule, below.

We have already seen the Lyapu nov sum rule (52), which relates the entropyproduction, a macroscopic quantity, to the sum of the Lyapunov exponents,a microscopic quantity. The pairing of Lyapunov exponents , also called the conjugate pair ing rule or symmetry of the Lyapunov spectrum, is a much strongerproperty, relatin g individual pairs of Lyapunov exponents. It is proved using thesymplect ic property of the dynamics, and appears to be limited in validity tosystems admitt ing a Hamiltonian description.

It has been known for some t ime that the Lyapunov exponents of a Hamiltonian system come in ± pairs, that is, they may be split into groups of two,each of which sums to zero [73] . In 1988 Dressler [74] showed that for a COnstant frictional coefficient a , the sum of each pair of Lyapunov exponents is -a.Incident ally, the constant a "thermostat" can also be derived from a Hamiltonian [75], obtained as for the isokinetic thermost at above, with the accumulatedphase space cont ract ion e'I> replaced by e- at = l /(a>.) . In contrast to the usualthermostats, thi s Hamiltonian is explicit ly tim e dependent .

Meanwhile, numerical simulat ions of many particle systems where Lyapunovexponents were computed began to show evidence for a similar law [29, 76,77, 781 . Ironically the first observations of Lyapunov exponent pairing were inshearing systems, where more detailed recent computations have ruled out exactpairing [79] . Initially the result s were explained in terms of the large numb erof particl es 180]. In systems of many particles it is often easier to computethe largest and smallest Lyapunov exponents than the whole spect rum, so thepairin g rule if it holds can be used to relate these measurable exponents to theentropy production and (also measurable) transport coefficients .

In order to clarify the role of the system size, and also because it is possible tocompu te Lyapunov exponents more precisely in small systems, the author andtwo collaborators studied the Lyapunov exponents of the simplest th ermostat tedsystem with more than one nont rivial pair of Lyapunov exponents, t he threedimensional Lorent z gas [581 . The results, that two pairs of Lyapunov exponentseach sum to - (a) whether positive or negative and th at a trivi al pair is zero dueto the conservat ion of kinetic energy, were extre mely helpful in understandingthe condit ions under which pairing occurs. In this case at least , pairing does notdepend on a large system limit , or on chaotic prop erties associated with posit iveLyapunov exponents , so it must be derived from the equations of motion. T hedegrees of freedom corresponding to the direction of the flow and the conservedkinetic energy give zero exponents not summing to -(a) , so they must somehow

The Lorentz Gas: A Paradigm for Nonequilibrium Stationary Stat es 349

be excluded from consideration. With these point s in mind , we move on toa st atement of the result and a sketch of the proof.

The conjugate pairin g theorem states that for th e isokinetic thermostat andthe restricted isoenergetic thermostat discussed above there are two zero Lyapunov exponents, and the remainin g N d -1 pair s of exponents sum to - (a) - A.The Nose-Hoover thermostat is the same except that there is one zero exponentand N d pairs . The Lyapunov exponents and average values of a are computedusing the same invariant measure, which may be any trajectory or invariantmeasure of the system. In particular, the theorem holds irresp ective of chaot icproperties such as ergodicity or positive Lyapunov exponents, and irrespectiveof the size of the system.

The main ideas of the proof are sketched below; details can be found for theisokineti c thermostat in Refs. [72, 81, 82], the restricted isoenergetic thermostatin Ref. [83] and the Nose-Hoover thermostat in Refs. [69, 72] . Refs. [72, 82Jexplicit ly include the collisions, and the isokinetic thermostat on a curved manifold. Numer ical evidence excludes pairin g in shearing systems [791 and a moregeneral isoenergetic thermostat [83J .

Hamiltoni an dynamics can be written most simply using a matrix

J=(O I)-I 0

(66)

where I is th e unit submat rix, and the block form corresponds to x , tt . Wehave the transpose JT = - J and j2 = - 1. Then Hamilton's equations arer = J'VH and the equat ion of motion for perturbations is Jr = T(t)Jr whereT = J'V'VH . The matrix T satisfies the equation TTJ +JT = 0 (where a superscrip t T denotes t ranspose) due to derivatives of H commuting, compare withLiouville's theorem (23). The first step to prove the pairing rule is to show thatthe appropriate mat rix T sat isfies a generalised equation,

TTJ + JT = -aJ (67)

For the case of constant a this is straightforward, but for the other thermostatsit is first necessary to reduce the space to exclude the zero exponents by rulingout perturbations th at are parallel to the flow, and for the isokinetic thermostat ,those that violate th e constant energy condit ion. The T matri x then containscoefficients of the constrained perturbation equations. Refs. [72, 821 also provean equivalent condit ion for the hard collisions.

Th e equat ion (67) for the perturbation evolut ion equations can be extendedto finite evolutions Jf(t) = L(t)Jf(O) using the equation for the L matrix,L = TL with initial condit ion L(t = 0) = 1 to obt ain

(68)

where fL = exp(Joadt) . Consider th e eigenvalues of M = LT L , which obeysfL2M T J M = J following from (68). Straightforward matrix manipulations of the

350 C. P. Dettmann

eigenvalue equation leads to the result th at the eigenspace of an eigenvalue A2 istransformed by J into an eigenspace with eigenvalue 1/(A 2fJ2). The Lyapunovexponents are the infinite tim e limit of the logarithm of the eigenvalues, dividedby twice the t ime. Thus the spect rum is symmetric with the pairs summing to-(a), and the theorem is proved .

5.5 Periodic Orbit Approaches. It was noted in Sec. 5.3 above that invariant measures of thermostat ted nonequilibrium systems (including the Lorentzgas) are multifractal. Thi s means in particular that the concept of a smoothprobabili ty density p(r) must be replaced by a more general description.

Th e most primitive approach is to coarse grain the space into arbitrary part itions (say, of equal size) and count the numb er of times a long (hopefully typical) trajectory passes through each cell. This does not depend on st rong chaoticprop erties; ergodicity is sufficient to define a unique measure. The disadvantagesare that there are few mathematical results for such a general framework, thepar ti tion does not take into account the natural struc ture of the dynamics, andit is not immediately clear how to define measures on repellers of open systems,which almost all trajectories leave after a finite (typi cally rather short ) t ime,see Sec. 6.

It may be possible to prove (or make a plausible hypothesis) that the dynamics is sufficiently hyperbolic that there are invariant measures smooth along unstable (expanding) directions in phase space; these are called Sinai-Ruelle-Bowen(SRB) measures. While it is possible to prove a numb er of result s pertaining tosuch syste ms [32, 82], a proof of the existence of (for example) a Markov partition does not necessarily show how to construc t it efficient ly, and is of no use ifthe required dynamical prop erties have not been shown. For the nonequilibriumLorent z gas, rigorous results are mostly restricted to the case of small field andfinite horizon, see for example [61].

Period ic orbit theory 13, 84] provides both the mathemati cal justification(given sufficientl y st rong hyperboli city [85]) and also gives explicit expressionsfor multifractal measures that can be applied to many systems (with apparentsuccess, although sometimes slower convergence [86]) for which enough periodicorbits can be located, but rigorous proofs are not available . In addi tion , theperiodic orbits are coordin ate invariant , make use of the dynamics in a naturalmanner , and are applicable to open systems. We refer here to classical periodicorb it theory; there are similar theories applicable to quantum systems in th esemiclassical limit [3, 87] and more recentl y to stochast ically perturbed classicalsystems [88, 89].

It may seem strange th at the prop erties of a system can be determined froma set of zero measure orbi ts such as the periodic orbits; to make an analogy,numerical integration schemes often use only rational points at which to evaluatethe integrand. The main question is whether the set of zero measure (rationalpoints or periodic orbits) is dense in the measure (phase space or some lower

The Lorentz Gas: A Paradigm for Nonequilibrium Stat iona ry States 351

dimensional att ractor). For the case of per iodic orbits , this is usually eitherproven or a reasonab le assumption.

Periodic orbits arise naturally when syste m properti es are computed fromthe spectrum of evolut ion operators. The desired property is first expressedin terms of a generating function that is mult iplicative in tim e, for examplethe current (for the nonequilibrium case) and the diffusion coefficient (for theequilibrium case) are expressed as

J;e

(69)&(3; 8(13) 113=0

1 1 o oD = - trD;j 2d L &(3; &(3; 8(13) 113=0 (70)

d ,

8(13) lim ~ In (e13·Lh ) (71)t-+oo t

using the Einstein relation (35) where 13 is a dummy variable, ~x = x(t ) - x(O)and 8(13 ) gives the rate of exponent ial growth of the average, and is thus thelead ing eigenvalue of the Liouville oper ator weighted by the exponenti al.

Th e leading eigenvalue of an evolution operator (such as a weighted Liouvilleoperator) may be computed in a number of ways. Some of the most common,namely the long time asymptotic form of its trace, Ruelle's dynamical zetafunctio n, and the Fredholm determinant lead to expressions in terms of periodicorb its [3, 4, 7, 84, 90]. For example the most rapidly convergent expressionsusually come from the Fredholm determinant of a discrete time system (forexample using the collisions of the Lorentz gas to define the dynamics) , det (l z.c) where z = e- s and .c is the weighted evolut ion operator. The determinant isthen expanded using the general matrix relat ion det M = et r In M to a maximumorder in z . The result ing expression involves tr.c n which counts the ways thesyst em can ret urn to its starting point after n iterations, th e periodic orbits oflength n . Specifically,

e13 ·t:.x

tr .cn = ""L- Idet(l - J(n)(x ))1

x :f n (x )=x

(72)

where J is the Jacobian matrix of derivatives of f'" , the nth iterated Poincaremap.

The denominator is often approximated by IAI, the product of the expandingeigenvalues of J , that is, those with a magnitude strict ly greater than one.IAI is also given by eT L A+ , the exponential of the period tim es the sum ofthe posit ive Lyapunov exponents along the periodic orbit . Approximating thedenominator of (72) by IAI is exact in the limit of long orbits and affects therate of convergence but not the result of the periodic orbit expressions for theleading eigenvalue and derived quantities. They lead to the two most often used

352 C. P. Dettmann

closed expressions for the diffusion const ant , one obtained directly from thetrace ,

and one obtained using dynamical zeta functions ,

1 L:{p} (-I)k(6.Xl + + 6.xk? / IA1 .. . AklD = 2d L:{p} (-1)k(T1 + + Tk) /IA 1 .. . Ak l

(73)

(74)

Here, 6.x is the displacement of an orbit that is periodic in the elementary cell.It might be zero, corresponding to a periodic orbi t in the extended phase space,or it might be nonzero , finishing at an equivalent point on a different scatterer .T is the period , in terms of the continuous t ime. p indicates prime cycles, thatis, thos e periodic orbits that are not repeats of shor ter orbits. For the first expression, the sum is over all periodic points, whether belonging to a prime cycleor the repeat of a prime cycle; in the limit n --+ 00 almost all cycles are prime,so this does not matter . The second expression is a sum over all sets of distinctprime cycles containing k = 1,2 ,3 . . . cycles. The alternat ing sign (-I)k usuallyleads to partial cancellations between longer cycles AB and a combination ofshorte r cycles that approximate them, A and B , thus making the zeta functionmore rapidly convergent than the t race formula. The zet a function expressionis usually ordered by topological length, that is, all combinations of cycles witha total numb er of collisions less th an a maximum Nm a x are counted, with anassumed limit Nm a x --+ 00.

The current is computed by similar expressions (omit ting 2d and the powersof two) , and in fact any phase variable a(x) may be averaged in this manner ,replacing 6.x by fa dt computed along the periodic orbit. The trace formula (73)thus leads to a sequence of increasingly det ailed measures supported on theperiodic orbi ts given by Dirac delt a functions weighted by the inverse orbitst ability. The zet a function expression (74) gives a more complicated but oftenmore quickly convergent (in a weak sense) sequence of measures on the samesets.

There have been a numb er of applicat ions of the above formulae to th ehexagonal Lorentz gas [91 , 92, 93, 94] numerically searching for period ic orbitsup to typically ten collisions and computing the current or the diffusion coefficient . There are a number of technical difficulties , such as making sure all of thetens of thousands of orbits up to this length are found and making maxim al useof the symmetry. The conclusions are that the formulae work, although not yetto the level of precision of alt ernative methods; the symbolic dynamics (allowedsequences of collisions) is very complicated and depends strongly on the externalfield; the t race formula may converge more quickly than the zeta function forthis system. A zeta function approach with ordering by stability Am a x rather


than topological length Nrnax appears to work bet ter when there are many almost stable cycles at high field [691 (see Sec. 5.6) and in other syste ms withweak hyperboli city 186] .

Finally, there are general arguments made using periodic orbi t measuresconfirming a numb er of physical result s. It is clear from (73) th at the diffusioncoefficient must be nonnegative, in agreement with the second law of thermodynamics. Combining a periodic orbi t and its t ime reverse (with negat ive displacement and Lyapu nov exponents ) and using the Lyapunov sum rule (52), itis possible to show that J . F and hence the entropy production (41,48) mustalso be nonnegative out of equilibrium. This argument was given in Ref. [95] forthe Lorentz gas and extended to syste ms with many part icles in Ref. [96] . Thi sleads to the following explanation of the second law in thermostatted systems:periodic orbits corresponding to increasing ent ropy are more stable and havesmaller values of A th an their tim e reversed counterparts, hence tho se withincreasing ent ropy are weighted more strongly, leading to an average ent ropyproduction which is nonnegative. Rondoni and Cohen [971 have used periodicorbi t measures for thermostatted systems to derive the Onsager reciprocal relations which state that th e full linear response matrix connect ing all possiblefluxes and forces is symmetric.

5.6 Nonlinear Response. Diffusion in the Lorentz gas (or indeed othersystems [98]) is a linear process. In Sec. 3.3 the point particles are noninteracting, so the prop erties of a dist ribution of many point particles can be obt ainedby a linear superposit ion of many single particle trajectori es. Until the noninteractin g, pointlike approxima t ion fails, there is no density at which the systemceases to be linear. On the other hand , the nonequilibrium Lorent z gas hasa natural scale, determined by when the curvature induced in the trajectoriesby the field is comparable to the distance between the scatterers, at which thecurrent is no longer approximately proportional to the field.

One approach to nonlinear response is to define nonlinear Burnet t coefficients. Linear Burnett coefficients which form an expansion for the par ticle fluxin terms of higher derivatives of the densit y were briefly described in Sec. 4.3. Wecould also envisage nonlinear Burnett coefficients forming an expansion for thecurrent in terms of higher powers of the field, or vice versa. In realistic systemssuch an expansion usually involves nonanalytic terms. For example, in threedimensional shear flow, the viscosity TJ is well described in terms of the shearrat e y (not too large) by TJ = TJo -TJn 1/ 2 [29] . The nonequilibrium Lorentz gas isst ill more problemati c, with J most likely nondifferent iable almost everywhere,although this has not been proved and numerical evidence is not conclusive,see Fig. 4 and Ref. [99] . It is also unknown whether th e diffusion coefficient isa differenti able function of the spacing between th e scat terers. Discontinuousone dimensional maps are known to exhibit nondifferentiable diffusion coefficients [100], however the Lorentz gas dynamics viewed as a flow is continuousso the diffusion coefficient is probably somewhat smoother .

354 C. P. Dettmann

0.8

0.6

J

0.4

0.2

32.521.50.5

O IL:.... -'--- --'--- ---L ----'- ----J'---__--J

o

F

Figure 4: Current versu s field for th e non equilibrium Lorentz gas . The lat tice is hexagonal with w = 0.236 and finit e hori zon, see Fig. 3. Th e field is directed along th eline between neares t neighbours. At small field the current is proportion al to field according to (49) with a diffusion coefficient of approximate ly 0.18. The support of th eat tractor collapses to a fract al set at about F = 2.2, but t his has no appa rent effecton the cur ren t. For some fields above 2.4 and all fields above about 2.5 th e at t rac toris a stable periodic orbi t. The speed of the particle is fixed , so the current can neverexceed unity.

We observed in Sec. 4.3 that the symmetry of the hexagonal Lorentz gasrequires that an isotropic conductivity, and hence to linear order th e averagecurrent is parallel to the field. There is no such restriction for th e nonlinearresponse; except for the cases when the field points along th e line bet weennearest or next nearest neighbours (hence a reflection symmetry) , the averagecurrent is not in general parallel to the field [59].

It is known th at for sufficient ly small field the two dimensional nonequilibrium Lorentz gas with finite horizon is ergodic [61]. Together with timereversibility and the cont inuity of the dynamics (in cont inuous time, not inth e Poincare map) , th is implies that while almost all initial conditions leadto th e same average current , there are arbit rarily large deviations for short


tim es. This is because almost every trajectory must pass arbitrarily close to thet ime reverse of a normal tra jectory, that is, a trajectory with negative ent ropyproduction.

At larger field strengths, the ergodicity is observed to break in one of twoways, depending on the spacing of the scatterers and the orient ation of thelattice with respect to the field [1011 . One possibility is that a marginally stableperiodic orbit app ears , surrounded by an elliptic region separate from the restof the hyperboli c phase space , first observed by Moran and Hoover [571 . If theinitial condit ion is inside this region, the part icle always moves between thesame two scatterers, and the average current is zero. Out side the region, thedynamics is similar to that at lower fields.

In the other mechanism, the final state (and hence average current ) is th esame for almost all initial condit ions, however it is no longer dense in phasespace, and has a box count ing dimension less than that of phase space. It is nowcomplete ly disjoint from its tim e reverse (the "repeller"), and deviations fromthe second law are limited to a single collision. Thi s implies that the distributionof fluctu ation s (both parallel and perpendicular to the field) is quit e differentto that of small field . The transition to this state , described in Ref. [1011 istermed crisis induced intermi t tency, and corresponds to a discontinuous changein the box countin g dimension of the at trac tor, but the current , Lyapunov exponent , and informat ion dimension are continuous. Not all periodic orbits nowlie in the att ractor, so it is imperative that periodic orbit calculat ions (Sec. 5.5)only contain those cycles actually in the at tractor. This can be accomplishedby searching a long typical t rajectory (rather th an the whole phase space) forperiodi c orbits, often a useful approach in any case.

Typically, both mechanism s are observed at different field strengths for thesame spacing, and as the field is further increased, further crises occur , creat ing,destroying and removing periodic orbits from the at tractor. Eventually one (ormore 1102]) periodi c orb its becomes stable, at tracting all or at least a posit ivemeasure of initial condit ions. There is a range of fields over which stable windowsand chaotic at t ractors alterna te in a complicated fashion 159] . At sufficientlylarge fields there is always a stable orbit , and at infinite field, the limitingbehaviour is that of an orbit creeping along a disk until it can move in thedirection of the field to the next disk.

While it is clear that many similar features occur in the three dimensionalLorentz gas [58], the Nose-Hoover thermostatted Lorentz gas [56], and various molecular dynamics simulat ions driven to very high fields 129], th e detailsdepend to a large extent on the model at hand . While it might require unreasonably strong forcing to generat e stable configurations with no positive Lyapunovexponent s, it is sufficient to let only one of the positive exponents go negativeto expect that the attractor and repeller are disjoint , and therefore a dynamicaland tim e reversible st ructure qualit ati vely different to th at near equilibrium. Itis also possible tha t measurements of large systems ignore and hence averageover many degrees of freedom, which may tend to wash out the multifractal

356 C.P. Dettmann

structure of phase space. In any case, there is much more to be understoodabout the dynamics of a many particle system in a far from equilibrium steadyst at e.

§6. Boundary Driven Systems

6.1 Open Boundaries: The Escape Rate Formalism. Now we turn tononequilibrium systems with Newtonian equat ions and no phase space contraction , with nonequilibrium effects generated by the boundaries. Syst ems withboth boundary effects and thermostats are considered in Sec. 6.3.

Suppose we consider a Lorentz gas, either random or periodic (with finitehorizon) , in a bounded region of space. Trajectories in the system can then bedivided into four classes, dependin g on whether they remain in the system atlate or at early times. Almost all (Lebesgue measure) trajectories remain in thesystem for only a finite time. Thos e that remain in the system at both earlyand lat e tim es form the repeller , which in this case is the closure of the periodicorbits. Trajectories that are in the system at lat e but not early tim es form thestable manifold of the repeller, and those in the system at early but not lat etimes form the unstable manifold of the repeller.

A smooth distribution of initial conditions will converge (weakly) to a distribution over the repeller and its unst able manifold th at is steady except thatit decays in time as the measure escapes through the boundary. In the langu ageof Sec. 5.5, a generic initial distribution acted on by the Liouville evolution operator will be dominated at late times by its leading eigenfunct ion. The rat e ofdecay, the escape rate " is directly given by the leading eigenvalue; the numb erof particles in the system given an initi al uniform distribution decays as

N(t) rv N(O) e- ,t (75)

This exponent ial decay rat e and its calculation as an eigenvalue using standard periodic orbit theory [3, 103] depends on the uniform hyperbolicity of thesystem. Nonuniformly hyperbolic systems have recently been treated in thismanner , but with more care due to the appearance of a power law decay anda branch cut in the spect rum [1041. For hyperbolic systems, the escape rate isalso related to other dynamical quantities, the sum of the positive Lyapunovexponents, and the Kolmogorov-Sinai ent ropy by [32]

(76)

and in the two dimensional case, also to the partial information codimensionCl [105]

(77)

The Lorentz Gas : A Par adi gm for Nonequilibrium Stationary States 357

where Cl is the dimension of phase space minus the informati on dimension D1 ofeither the stable or the unst able manifold. So far we have related the exponent ialescape rate of a hyperbolic system to period ic orbits , the positive Lyapunovexponent(s) and a dimension of the repeller.

Suppose now th at the dimensions of the system are so large (specifically,much larger than the mean free path) that the evolution of phase space densityis well described by the diffusion equation (32). Open square boundaries correspond to the condit ion P = 0 on x = 0, y = 0, x = L and y = L (for simplicity ;other geomet ries are possible, alte ring the constant 11"2 below), leading to thegeneral solut ion

from which we find the decay rat e of the leading m = n = 1 mode,

211"2D

1= [;2

(78)

(79)

Equating the escape ra tes of the dynamical and hydrodynamic approaches inthe limit of large systems, we obtain escape rat e expressions for the diffusioncoefficient [106],

. I L2 . L2 . L2 . L2D = lim --2 = lim - 2('" A+ - hK s ) = lim - 2A+ C1 = lim - 2A+CH

L -4 OO 211" L-4OO 211" ~ L-4 OO 211" L-4OO 211"(80)

where the last equality involving the partial Hausdorff codimension in the largesystem limit is found in Ref. [107]. Thi s is useful since CH can be computed moreeasily than either tvtcs or Cl [1081. Unfortunately none of the above quantiti es canbe calculated efficient ly enough in the large system limit for these equations tocompete with the thermostatted approach as a means of computing the diffusioncoefficient . They can be used to check the consistency of the approach, however,and remain of great theoreti cal interest. Compare Eq. (53) where the informationcodimension in the thermostatted two dimensional Lorentz gas gives a verysimilar expression for the diffusion coefficient:

D = lim A+ClF-4 0 p 2

(81 )

Note that the thermost at ted Hausdorff codimension is exactly zero up to reasonably st rong fields (see Sec. 5.6). The escape rat e 1 plays the same role forthe open system as the multiplier a plays for the thermostatted system in determining the rat e of decay of phase space volume occupied by an initially smoothdistribut ion of particles; in one case part icles are lost through the boundaries,while in the other the volume cont racts due to the equations of motion.

358 C. P. Dettmann

T he escape rate formalism applies not only to diffusion , but also to oth erlinear tr ansport coefficients . The idea is that each Green-Kubo expression (36)can be transformed into an equivalent Einstein relat ion (35) containing themean square difference of a quanti ty other than displacement . Such a quan ti tyis called a Helfand moment , for exa mple, the Helfand moment correspondingto shea r viscosity is (up to a constant factor) L i Xi Piy where the sum is overparticles. T he escape condit ion then corresponds to a bound on the Helfandmoment. In this way, all linear transport coefficients may be related to escapein an appropriate system with a large size limit . The small size limit correspondsto a steady state far from equilibrium, however it is quite different to the thermostat ted system at st rong field , and it is not clear what physical system itcould represent. More details on the escap e rate formalism and its applicat ionscan be found in Refs. [6, 7, 106, 107, 1091 .

6.2 Flux B ounda ries . A Lorent z gas in a finite domain need not haveabsorbing boundaries; it is also profitable to consider th e possibility of injecting particles into the system from the boundaries. The most common (butby no means the only possible) geometry considered for this situa tion is thatof a Lorentz gas (random or periodic with finite horizon) in a slab given by- L/ 2 < x < L/2 and - 00 < y < 00 . At the left (right) boundary, particles areinjected in all directions with a density f - U+). T his is ana logous to numericalsimulations where boundary conditions at a certain temperature are maintainedby injecting particles at the boundary with a Maxwell-Boltzman n distrib ution,ignoring correlat ions.

In the ste ady state, the particles fill the whole phase space except the repeller and its unst ab le man ifold which const it ute a set of zero measure. Sincephase space volume is conserved, the density of particles at a given posit ionand velocity is either f _ or 1+ depending on the boundary through which thepart icles ente red. T his means that the phase space density is piecewise constant(hence piecewise smooth) wit h a fractal set where the density is und efined .

This prescription for the phase space density can be coded by the followingformu la [n0I:

f + f ( r T (x ,v ) )f(x ,v) = - 2 + + g . x + i o V t dt (82)

where g = exU+ - f -) /L is the density gra dient across the slab, and - T isthe time th e particle ente red the system. The term in the large parenthesesevaluates to the position the part icle entered th e system, with an x-componentof ±L/ 2; combined with g it provides the necessary increment to obtain thedensity f ±. The term g . x gives a linear density profile; after int egrat ing overth e velocity directions to obtain P(x ) from f (x ,v ), this is a trivial solut ion ofthe diffusion equation (32). Th e integral then determines how far the actualdensity f ± differs from th e average behaviour.


As in Sec. 6.1 above, we are really interested in the large system limit,L -t 00 . Th e gradient g is kept finite, while f + - f _ tends to infinity. The firstterm (f+ + f - )/ 2 can be ignored, as it gives only a constant shift , the averagedensity at x = O. Th e time the particle entered the system goes to -00. We findthat the result ,

III (x, v) = g . (x +1-00

Vtdt) (83)

diverges for all x and v . This is perh aps not surprising given that the phasespace density for the nonequilibrium steady state is multifractal in the thermostatted approach, Sec. 5.3. In any case, it does not cause a problem , sincethe average with respect to the nonequilibrium distribution One of an arbit raryphase variable a(x ,v) can be naturally defined by

(84)

If a is the current J , this leads directly to the expected relation J = - Dg withthe diffusion coefficient D given by its Green-Kubo formula (36).

Distributions of this form were originally introduced by Lebowitz 11111 andMacl.ennan [1121. It is possible to represent III by its cumulative distributionfunction , which is cont inuous 16] . It is one of the main tools used to apply Bakermaps to the understanding of nonequilibrium steady st ates and entropy production , where the cumulat ive distribution function becomes an exactly selfsimilarTakagi function 16, 7, 91 . There is a natural extension to other transport processes in a similar fashion to the open case, Sec. 6.1. See also Ref. [1131 wherethis approach is used to describe hydrodynamics outside local equilibrium.

We conclude our discussion of flux boundary conditions with a connectionto the th ermostatted approach. Suppose we coarse gra in III to some resolution E,

ignoring smaller variat ions. We can approximately compute III in some region ofsize E in phase space by tracing back in time until the chaot ic dynamics amplifiesthe initi al uncertainty to the point at which th e particle could have come fromany direction with roughly equal prob ability, time -T. We can then write forthe E smoothed distribution,

1lI,(x, v) ~ g. x .,; (85)

Compare thi s with an E smoothed distribution using a field and thermostat .For sufficiently small field, the trajectory remains close to a trajectory withoutfield over such a time T. The thermostatted case has no overall variation indensity, so the average densit y at time -T is roughly unity. However , phasespace contraction increases the average density to approximately eF .Ll x , whichreduces in the limit of small field to 1+F ·~x. Thus the nonequilibriurn steadysta te distribution obt ained using flux boundary conditions is the same up to

360 C. P. Dettmann

a multiplicative constant as the deviation of the distribution from equilibriumin the weak field thermostat ted case. The distribution III is directly proport ionalto the gradient g , so it cannot exhibit any nonlinear feature s, as expected fordiffusion in the Lorent z gas.

6.3 Boundaries With Thermostats. T here are also a few approachescombining elements from both thermostatted and boundary driven nonequilibrium models. Chernov and Lebowitz 1114, 115] use wall collision rules that areenergy conserving, time reversible and phase space contract ing (on the average)to drive a many par ticle system into a sheari ng steady state. This can be madeequivalent to a thin layer where the particle is subject to a st rong oblique forceand a thermostat , and thu s belongs with th e methods mentioned at the endof Sec. 5.2. Rat eitschak and collaborators use phase space contracting collisionrules to thermostat the Lorent z gas with an elect ric field [116] .

Tel and collaborators [9, 117, 118, 1191 consider open syste ms with an external field. They focus on Baker map approaches, but much of their discussion onthe relationships between escape rate, ent ropy product ion and dimension appliesequally to the Lorent z gas or many particle systems. There are now two limits ofinterest , F ---+ 0 and L ---+ 00 . If the latter is taken first it is necessary to imposea thermostat to keep the velocity und er control. Nevertheless, the phase spacecontract ion is bounded, since the repeller is in a finite domain, see (56). Thismeans that the Lyapunov exponents add to zero as in a Hamil toni an system.

Th e analysis proceeds similarly to that of the field free case, Sec. 6.1. Eqs. (75-77) pert ainin g to the escape rate, Lyapunov exponent and the partial information codimension of general open two dimens ional systems remain valid . Thehydrodynamic equat ion now contains both a diffusion and drift term,

aP H- = \7 . (D .\7P - JP)at (86)

H

where D and J depend on F according to the microscopic dynamics; for theusual case of a homogeneous system J does not depend on position. For smallfield we have J = DF from (49), where D is the (usually isotropic) zero fielddiffusion coefficient . The equation is easily solved in a st rip 0 < x < L byseparation of variables leading to the escape rate

D x x 7r2 J'; ( )

,=~ + 4Dx x

87

reducing when the zero field limit is taken first to (79) and when th e largesystem limit is taken first to another expression for the diffusion coefficient ,

. 4, . 4A+ClD = lim F 2 = 11m -F2 (88)

F--+O F --+O

The factor of four difference from Eq. (81) was noted in Ref. [118] and is due tothe different (here semi-infinite ) geometry. In all cases the information codimension of the relevant measure can be associated with the transport coefficient,

The Lorentz Gas: A Paradigm for Nonequilibrium St ationary States 361

and hence the entropy production. The thermostat ted methods and open systems, alone or in combination , describe the same nonequilibrium processes, atleast in the linear regime .

§7. Outlook

Many of the connections between dynamical and statistical descriptions andbetween microscopic and macroscopic properties of equilibrium and nonequilibrium stationary states have been addressed using a very simple model , theLorent z gas. It is remarkable that most of these connect ions and properties donot depend on the number of particles, but apply to both the smallest andlargest systems. There are undoubtably many more connections to be made onthis level. One of the chief aims of the present work is to bring a diversity ofideas together to catalyse progress in this direction. For this purpose, it is alsohelpful to keep in mind a few limit ations of the Lorent z gas paradigm.

In the Lorent z gas it is necessary to distinguish between real space densityP and single particle density 1. Similarly, in many particle systems there is anadditional distinction between single particle density f and phase space densityp. A significant conceptual difficulty is that macroscopic entropy, defined asan extensive quantity according to Sec. 2.2 is a function of real space, whilethe microscopic descriptions of Sec. 3 involve the phase space. The effect ofthis, which is not app arent from the Lorentz gas, is that the thermostattingmultiplier a and the escape rate I are not local quantities in general; theydepend on a simultaneous description of all the particles. These distinctions arealso important with regard to Baker map approaches [9], where concepts suchas real space and phase space do not obviously play the same roles and need tobe carefully delineated.

There are some inst ances where the same chaotic properties act differentlyin larger systems. While we expect systems of many par ticles to have hyperbolicprop erties [8], some of the fractal structure might be washed out by measurement s th at average over many of the degrees of freedom. It is also not clear towhat extent such averaging can be simulat ed by, for example, random placementof the scatte rers in the Lorentz gas.

Conversely, some chaot ic properties of large systems are different to thoseof lower dimensional systems. A numb er of result s, par ticularly those relatingdimensions and Lyapunov exponents have only been proven for two dimensional systems. Higher dimensional result s may be more difficult to prove, or thestructure may be more detailed th an in two dimensions. The three dimensionalLorent z gas, corresponding to a five dimensional flow or a four dimensional mapand thu s having two nontrivial pairs of Lyapunov exponents, has already provided a useful example of the conjugate pairing rule [58] and may well containmuch st ructure characteristic of higher dimensional dynamics. An alternat ive isthe six dimensional map corresponding to three hard disks in two dimensions .

362 C. P. Dettmann

Th ere remain a number of challenges in the theory of stationary states farfrom equilibrium. Not the least of these is the difficulty defining a useful anduniqu e entropy, despit e the observation that the irreversibility of th e second lawapplies universally, near or far from equilibrium. Another issue is th at many ofthe approaches such as various thermostats or boundary condit ions are equivalent only in the linear regime. The nonlinear prop erties of the Lorentz gasgiven in Sec. 5.6 are only the beginning of what can be understood about suchnonequilibrium syste ms.

The author is gra teful for helpful discussions with N.!. Chernov, E.G.D.Cohen , J .R. Dorfman, P. Gaspard and W.G. Hoover , and for collaborat ion onmany of these subjects with G.P. Morriss.

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The Lorent z Gas: A Paradigm for Nonequilibr ium Stationary States 365

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Entropy Balance, Multibaker Maps, andthe Dynamics of the Lorentz Gas

T. Tel and J. Vollmer

Contents

§1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369§2. Coarse Graining and Entropy Production in Dynamical Systems 371

2.1 Exact and Coarse-Grained Densities 3712.2 Gibbs and Coarse-Grained Entropies 3732.3 Irreversible Entropy Production . . 3762.4 Entropy Balance. . . . . . . . . . . 3772.5 Entropy Balance for Steady States 3772.6 Closed Volume-Preserving Systems 3782.7 Missing Ingredients to a Thermodynamic Description 379

§3. From the Lorentz Gas to Multibaker Maps . . . . . . . . . 3803.1 Mapping Relating Subsequent Scattering Events

for an Unbiased Dynamics with Periodic Boundary Conditions . 3813.2 Adding an External Field and a Reversible Thermostat . 3843.3 The Spatially-Extended Lorentz Gas . . . . . . . . . . . . 3863.4 Symbolic Dynamics and Pruning 3893.5 Piecewise-Linear Approximation of the Lorentz Dynamics 3903.6 The Next-To-Nearest-Neighbor (nnn) Multibaker Map . . 390

§4. Transport and Entropy Production in the (nnn) Multibaker Map . 3944.1 Time Evolution of the Entropies . . . . . . . . . . . . . . 3954.2 The Macroscopic Limit for Transport

and the Advection-Diffusion Equation. . . . . . 3964.3 The Macroscopic Limit for the Entropy Balance 3974.4 Entropy Production in the Macroscopic Limit 398

§5. Results Obtained with (nn) Multibaker Maps . . . . . 4005.1 Invariant Densities and Takagi Functions . . . . 4005.2 Particle Transport and Entropy Balance in Isothermal Systems 402

368 T . Tel and J . Vollmer

5.3 Green-Kubo Relation in the Isothermal Case 4035.4 Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 4045.5 The Entropy Balance in the Presence of Temperature Gradients 4055.6 Thermoelectric Cross Effects 4065.7 The Irreversible Entropy Production as the Average Growth

Rate of the Relative Phase-Space Density in Steady States . . . 4085.8 Fluctuation Theorem for Entropy Production in Steady States

with Density Gradients . . . . . . . . . . . . 409§6. Discussion . .. . . . . . . . . . . . . . . . . . . . 411

6.1 Deviations from Dynamical Systems Theory 4116.2 Interpretation of Coarse Graining . . . . 4126.3 Interpretation of the Macroscopic Limit. 4136.4 Outlook and Open Problems. . . . . . . 414

Appendix . . . . . . . . . . . . . . . . . . . . . . . 414A. Trajectories of the Thermostated Lorentz Gas 414

A.l Time Reversibility and Phase-Space Contraction . 415A.2 The Form of Trajectories 415

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

Abstract. We extend and review recent results on nonequilibrium transportprocesses described by multi baker maps . The relation of these maps to the dynamics of the Lorent z gas is discussed . Special emphasis is put on the concept ofcoarse graining and its use in defining the analog of thermodynamic entropy andin deriving an entropy balance. A full analogy with Irreversible Thermodynamicscan only be obt ained if at cert ain points we deviate from traditional dynamicalsystem theory, and allow for open boundary conditions which make the systemto converge to a 'forced' stationary measure. This measure differs from thenatural SRB measure which can be realized with periodic boundary conditionsonly.

Multibaker Maps and th e Lorentz Gas

§1. Introduction

369

Entropy production is a measure of deviation from thermal equilibrium [1-6]. Itis a surprising result of the last years that a consistent definition of this centralconcept of Irr eversible Thermodynamics can be given for general dynamicalsystems. For closed systems with periodic boundary conditions, the descriptionof the effect of an applied external field and of a reversible thermostating mechanism has been pioneered by Hoover and Evans and their respective coworkers[3]. A rigorous treatment of the same problem was first suggested by Chernovet al [7], and was further worked out by Ruelle [8]. These approaches modelthe transport process by dissipative dynamical systems and relat e the transportcoefficients and other physical observables to properties of the invariant SRBmeasure. A concept of steady-state ent ropy production in dynamic al systemshas been based on the average phase-space cont raction rate [3, 9, 10] . In an alternative approach, open volume preserving dynamics are considered subjectedto absorbing boundary condit ions. In such cases the escape rate from the nonattracting invariant set responsible for transient chaos is relat ed to the transportcoefficients [11~14], and to the entropy production [15].

The heart of Irreversible Thermodynamics [2] is to set up an entropy balancefor the thermodynamic ent ropy S . This balance is commonly written in the form

(1)

Its right hand side contains the sum of the external and internal changes ofent ropy, i.e., changes due to an entropy flux out of the considered volume anddue to irreversible entropy production. The lat ter is non-negative, while theformer one can have any sign. For noisy dynamics this ent ropy balance wasderived by Nicolis and Daems [16] for the Gibbs entropy of systems subj ectedto (arbit rary) small noise. In the present paper, we show ~ by extending theresults of [15] - that such a relation can also be found in determ inistic systemsif a suitably chosen coarse-grained entropy is considered. Stat ements about general dynamical systems unavoidably refer to global prop erties, i.e., they containthe full system's ent ropy, and cannot provide a comprehensive thermodynamicdescription of t ransport driven by spatial inhomogeneities of the thermodynamicvar iables. To this end local relations are needed.

Multib aker maps have been introduced to model transport in open systems,i.e ., systems where transport can be indu ced by suitably chosen boundarycondit ions. In the original model of Gaspard [17- 20] the dynamics is fullyHamiltonian and describ es a diffusion process compat ible with Fick 's law. Lat erit was extended to model chemical reactions, too [21]. The invariant measurebelonging to these problems is not the Liouvillian , but a fractal distributionforced on the system by different densities prescrib ed at the two ends. Theirreversible ent ropy production is due to a kind of mixing entro py, which canonly be understood by using the concept of coarse graining. Multibaker models

370 T. Tel and J . Vollmer

can also describe tra nsport dr iven by boundary condit ions and external fields[22- 33]. The lat ter dri ving typically requires a t hermostating mechanism, anddue to t his, a deviation from local phase-space volume preservation. The irrevers ible ent ropy production then contains cont ribut ions from both t he averagephase-space cont raction rate and t he mixing entropy. Early versio ns of t his classof models describ e par ticl e t ra nsport by t he t raditiona l piecewise-linear map[22- 25]. Later on the cha nges due to a nonlinear form were worked out [26],and the linear version was used to derive fluctuation relations [27]. The mostrecent versions have been extended to deal with simultaneous par ticle and heatt ra nsport [28- 32].

One of the appealing features of t his approach is that it permits a comparisonwith a local thermodynamic entropy balanc e

(2)

Here s is the entropy density, and and (J (irr ) represent the ent ropy flux densityand the rate of irreversible ent ropy production per uni t volume, respectively. Forsystems with gradients of t he t hermodynamic fields , multibakers are up to nowt he only models for which a consiste ncy with the local entropy balance could befound.

Some authors (e.g., [33], and Dettmann in this volume) consider mul tibakermaps as abstract models of little physical relevan ce. In cont rast, we show herethat t hey mim ic the evolut ion of independent particles in t he Lorentz gas dri venby an external field and subjected to nonequilibrium boundary cond it ions. Theyform t he only analyt ically access ible class of systems that allows us to findconstraints on t he dynamics needed to achieve a consistent description of theentropy balance, and of t he tra nsport equations and transport coefficients.

We first discuss how far one can go towards an ent ropy balance in a classicaldyn amical system (Sect. 2), and in later sections we describe which additionalfeatures have to be imp osed on t he dynamics to permit a consiste nt thermodynami c description. In Sect. 3 we show how, via a sequence of t ra nsformationsand approxima tions , one can 'derive' a five-strip multibaker map mimickingt he dyn amics of independent particles in the driven Lorentz gas. The requiredparticle trajectories of the Lorentz gas are given in the App endix. The entropyproduction for the resulting multibaker map is derived in Sect . 4. In Sect . 5results found with the earlier st udied three-strip multibaker model are summarized. The paper is concluded in Sect. 6 by a discussion and outlook.

Multibaker Maps and the Lorentz Gas

§2. Coarse Graining and Entropy Productionin Dynamical Syst ems

371

We treat invertible and hyperbolic [34, 35] dynamical systems whose phase space1

is either closed or open. In the former case we consider the full phase space(accessible to the particles in the presence of some general constraints ). In thelatter , we restrict our at tent ion to a fixed finite phase-space volume of interestcorresponding to the motion in a finite region of the configuration space. Particles can escape this volume with some escape rat e K, [36]. In both cases thedimension of the phase space is denoted by d.

On general grounds, one expects that closed Hamiltonian systems can onlybe used as models of thermal equilibrium, or fluctu ations around it (cf. Sec. 2.6).Models for nonequilibrium processes based on low-dimensional dynamical system must therefore either be dissipative [3] or, if phase-space-volume preserving,their phase space has to be open. The long-tim e dynamics is then associatedwith either an SRB measure on a chaotic attractor [34,35] or with a condit ionally invariant measure [36~38] located on the invariant manifolds of the openvolume-preserving system. (For simplicity we call both measures natural. ) Asa consequence, the nonequilibrium processes are associated with fractal phasespace structures, which have no volume with respect to the Liouville measure.The set of average Lyapunov exponents characte rizing the relat ed invariant setwill be denot ed by '\1 ~ '\2 ~ .. . '\d, and the locat ion dependent eigenvalues ofthe linearized dynamics are accordingly >'1 ~ A2 ~ .. . Ad.

The key observation in trying to model irreversible processes by dynamicalsystems is the ever refining phase-space st ructures associated with the convergence towards the fractal measure. It is impossible to describe the asymptoticst at es by smooth st ationary densities in phase space. Instead , we suggest toconsider a coarse-grained description which is inte rpreted as an approach wherethe ever refining structures in phase space are followed with a finite resolution.Comparing the t ime evolut ion in this coarse-grained description with the exact one, gives new insight in the dynamics and into the possible st ructure ofa macroscopic description of transport. For illustrational purposes, we confineourselves to discuss only the simplest possible coarse graining, which consist s individing the adimensionalized phase space into identical boxes of linear size E

much smaller than unity (E« 1). The phase-space volume of the boxes is thenEd .

2.1 Exact and Coarse-Grained Densities. We use two different phasespace densities belonging to the same smooth initi al condit ion

• e(x, t) , the exact phase-space density at phase-space location x and timet, and

1 By phase space we mean here the phase space of a dynamical system without yetthe structures required to set up statistical mechanics.

372 T. Tel and J. Vollmer

• OE(i ,t) , the coarse-grained density of box i at time t , which is obtained byaveraging O(x, t) over box i .

The averaging on the set of boxes defines the coarse graining. (For simplicity we use the same notations for the conditionally invariant density and itscoarse-grained version characterizing open syste ms. Both types of densities arenormalized to unity.)

After long times there is a qualitative difference between t he exact and thecoarse-grained densities: the exact density keeps developing finer and finer structures and has no time-independent limit . It becomes undefined as a density andits asymptotic distribution is to be described by t he natural invariant measure f.l[34, 35]. On the other hand, the coarse-grained density converges to a (piecewiseconstant ) station ary distrib ut ion iJE (i) such that the natural measure f.li (e) ofbox i equa ls iJE (i )c:d . The asymptotic temporal dep endence of o(x, t) can bewritten as

while

O(x, t) rv e<7(x) t , (3)

(4)

Here O"(x) is the local phase-space contraction rate [3,5] at point x . Equation(3) follows from the fact that t he phase-space volum e around x is behaving likeexp[- O" (x )t] and the measure of a given volume is not changing in time du e tothe conserva t ion of probability. In closed systems the phase-space contractionrate is the negative sum of all the local eigenvalues [3]

d

O"(x) = - L Aj(X) ,j = l

(5a)

in open volum e-pr eserving systems this role is played by the escape rate [15J

O"(x) = K,. (5b)

This last equa t ion expresses the fact t ha t the effective volume in a fixed portionof the phase space is decreasin g du e to esca pe, in spite of the preservationof volume in the full phase space. The escape rate is an effect ive phase-spacecont ract ion rate for open systems. T he state in which the coarse grained densityis stationary we call a steady state.

The qualitative difference between the behavior of the exac t and the coarsegrained densities ref. (3) , (4)J is a hallmark of irreversibility: every macroscopic description of transport is based on coarse-grained densiti es and hencedoes not resolve all information on the fine details of the system's phase-spacedynamics.

Multibaker Map s and the Lorentz Gas 373

How quickly the two densities differ depends on the type of initial condit ions.

A. For extended initial density profiles, the difference is in the beginning onthe order of the box size E and is negligible. Strong deviations start to developonly after a crossover time to set by the fact that the support of the densityin the stable directions becomes of the same order as the box size, due to thecontraction with the negative Lyapunov exponents. An upp er limit to this timescale is

(6)

where ~_ is the largest amongst the negative average Lyapunov exponent (thesmallest one in modulus). In typical dynamical systems this Lyapunov exponentis of order unity. Thus, the crossover time is also on the order of unity , anddepends logarithmically on the box size.

B. When the initi al condition is sharply localized, concentra ted in a single boxonly, so that i?(x ,O) = E- d, the st rong deviati on is immediate since the coarsegrained density does not feel any contraction in the stable directions. The changeof the phase-space volume is due to the expanding directions and the volumestarts to grow in t ime as exp (kt) , where k = LA .>o .xj is the sum of the positive

)

local eigenvalues .xj > O. Open syste ms soon start to feel the effect of escapeleading to a crossover of the growth rate towards k = L A.>o .xj - /"1" i.e., the

)

difference of the sum of positive local eigenvalues and the escape rate. Therefore,initially the coarse-gra ined density behaves in all occupied boxes approximatelyas

i?o(i, t) ~E-d exp( - kt). (7)

Due to Pesin 's theorem [35] and its generalization for escaping systems [38-40]the met ric (or Kolmogorov-Sinai) entro py hK S for the motion on the invariantset is hK S = L >. >o ~j -/"1,. For very localized initial conditions the average value

)

of k is thus the metric ent ropy Ie = hK S ' The coarse-grained density typicallystarts to grow expon enti ally with the metri c entropy as its rat e.

2.2 Gibbs and Coarse-Grained Entropies. A natural choice for theentro py characte rizing the state of the system at t ime t is the informationtheoret ic ent ropy taken with respect to a phas e-space density at that tim e.Since we simultaneously consider the tim e evolut ion of the exact and the coarsegrained densities, two ent ropies are defined:

• The entropy S(G) is evaluated with respect to the exact density:

(8)


where g* is a constant reference density. S (G ) is commonly referred to asthe Gibbs entropy.

• The coarse-grained entropy So is defined in an analogous way as a sumover boxes of size e

( ) _ '"' (.)d ( go(i , t ))So t = - Z:: go z, t c: In ~ ,,

(9)

where the notation expresses that the coarse-grained entropy depends onthe box size e.

The tim e evolution of the ent ropies immedi at ely follows from that of thedensities.

A. For extended initi al condit ions, S(G) and So nearly coincide before thecrossover time to is reached. Typically they both decrease since the distributionsstart approaching the one on the invariant set and hence the information contentis increasing. Thi s tendency does not change for S (G ) which keeps decreasingafter to. In view of (8) and (3) we find:

S (G)(t) = -a(t)t + const , (10)

where a(t) is the average of the phase-space cont raction rate (5) evaluated withrespect to g(x, t ) at time t . The asym ptotic behavior is a linear decay

(11)

with ii as the long-time average of phase-space cont ract ion rate taken withrespect to the natural measure. T he coarse-grained entropy, on the other hand,goes into saturation since with the given resolution the localization of the invariant set does not change any longer afte r to. Asymp totically, the coarse-grainedent ropy tends to a constant So which depends on the box size but is independentof the initial condition:

(12)

Thi s expresses the convergence of the coarse-grained density to a stat ionaryvalue. In contrast to S (G ), the coarse-grained entropy has a dissipative dynamicswith a fixed point aiiracior.

B. For sharply localized initi al condit ions localized to a single box, both Sand S (G ) are init ially In (g*c:d ) and start increasing because the dist ribution isspread ing out . Since the coarse-grained density behaves according to (7), itsinitial growth is described by

(13)

Multibaker Maps and the Lorentz Gas 375

Figure 1: Time evolution of the entropies in dissipative or open volume preservingdynamical systems relaxing towards a steady state. The black and gray lines/ symbolscorrespond to the respective initial conditions A and B described in the text. Thearrows represent the shift of the coarse-grained entropy curves, when decreasing theresolution from e to c' < c.

Later the coarse-gra ined entropy approaches exactly the same limit (12) asin the case of extended initial conditio ns. The growt h of the Gibbs entropydeviates from the linear law since the support of the exact density also feels thecontraction towards the unstable foliation. Due to this effect, it starts decreasingand asymptotically it shows the same linear decay (11) as with extended initialconditions, jus t shifted downwards (Fig. 1).

In the following subsect ion we discuss in detail t he strong temporal differencein the dynamics of these two entropies, and how the rate of irreversible ent ropyproduction can be related to this observat ion.

Before, however , it is worth briefly discussing the dependence of the ent ropyon the resolution c. To thi s end, we assume (cf. [41]) that th e reference dens ityis e-dependent according to the law

(14)

The s-dependence of th e asymptot ic coarse-grained entropy can then be expressed by a numb er: the information dimension DI of the coarse-grained steadystate distribution, i.e. , of the natural invariant measure u. This quantity hasbeen introduced in th e context of the multifractal characterization of the natural measures of chaotic at tracto rs and of other fractal distributions [42-44] .For every stationary measure characterized by boxes of very small (but finite)linear size E, which carry probab ilities Pi(E) , t he information dimension is defined

via the asymptotic relation - L i Pi(E) InPi(E) rv - DI In E. In view of this, thedep endence of SEon E for fine enough resolutions is

SE= -D1 InE (15)

where DI < d is the information dim ension of the natural measure. If werefine the box size from E to E' < E, the saturation value is shifted upwardby D1 In (E/ E') (Fig . 1). At the same time the initial value of the ent ropychar acterizing the initial condit ion is also shifted upward but by an amountof dIn (E/ E'). After all , the information dim ension of smoot h measures is t hedimens ion of t he phase space. This implies that for a high er resolution (smallerE) the steady-st ate distribution will be reached at lat er times, as already st atedby Eq . (6) . It is int eresting to note that besides t he act ual t ime evolut ion ofthe Gibbs and the coarse-grained entropies, two traditional, ent ropy-derivat ivetype quantities, which are well-known in dynamical syst em theory, also appearin Fig. 1. The met ric entropy hK S is the temporal derivative of the informationtheoretic ent ropy taken with respect to the symbol-sequence distribution ina symbolic encoding [35] (ef. Sect . 3.4 below for an example of a the symbolicencoding) , and D1 is t he In (l / E) derivative of the information theoretic ent ropytaken with respect to the box probabilities.

2.3 Irreversible Entropy Production. The difference between the coarsegrained and the Gibbs ent ropy characterizes the information on the exact st ateof the syste m which cannot be resolved in the coarse-grained description. Thete mpora l change of this quantity measures thus t he rate of the loss of informationon the exac t state of the syst em . In view of this , we suggest (see also [22- 24]) toidentify this temporal change at any instant of time with t he rate of irreversibleentropy production ~~irr) of the dynami cal system,

~~irr)(t) == ddt (SE(t) - S (G)(t)) . (16)

It is defined for every dynam ical system, and for sufficiently lar ge t it is typicallynon-negative (ef. Fig . 1)

(17)

in accordance with the analogous thermodynam ic quantity. Note that in spite oft he s-dependence of the coarse-grained entropy, the entropy production dependsonly slightly on the coar se graining. Latest upon reaching the st eady state wherethe coarse grained density and SEbecome t ime-independent, t he s-dependencefully disappears from the ent ropy production: ~~irr)(t) -+ t (irr).

The difference in slopes between t he coarse-grained and the Gibbs ent ropybecomes pronounced aft er the crossover time tE , leading to a substantial ent ropyproduction for t > t Eo This clearly shows that we are lacking information on t heexact asymptotic st ate when applying a description with a finit e resolution.

2.4 Entropy Balance. Interestingly, a decomposition of the time derivativeof the coarse-grained entropy, very similar to Eq. (1), can be given for generaldynamical systems by taking the time derivative of the identity S£ = S(G) +(S£ - S(G)):

and interpreting

d~£~t) == (t) + I: ~irr)(t),

dS(G)(t)(t) == dt

(18)

(19)

as an ent ropy flux through the system which does not cause irreversible changes .Note that this is defined as the change of the Gibbs entropy. From (10) we deducethat

(20)

where in the last step we used the approximation that the time derivative ofthe average phase-space contraction rate tends to zero much faster than l it .Eq. (20) directly relates the entropy flux to the average phase-space contract ionrate at any instant of tim e. Note that it is independent of c.

The definitions of I:~irr) and show that it is impossible to find an ent ropybalance by merely considering the Gibbs entropy. An analogy with thermodynamics can only be obtained by simultaneously following the exact timeevolution characterized by the Gibbs entropy and comparing it to a coarsegrained description.

2.5 Entropy Balance for Steady States. From (10), (12) and (16) wefind for the stationary entropy production f;( i rr ) that

(21)

irrespective of the box size. For every dynamical system [8,15] the rate ofirreversible entropy production in a steady state is the average phase-spacecontract ion rate.

Since in steady states the coarse-grained entropy is constant, the flux indeedcompensates entropy production and

(22)

This formula explains the result obtained via heuristic arguments by severalauthors [3,5 ,7,8] stating that the derivative of the Gibbs entropy is the negative of the average phase-space contraction rate in a steady state, and hence

378

ss .... --

&

o

T. Tel and J. Vollmer

s

Figur e 2: Time evolut ion of t he entropies in a closed volum e-pr eser ving dynam icalsyst em relaxing towards an equilibrium state. The black and gray lines/ symb ols corresp ond to the respective initial condit ions A and B describ ed in th e text . The arrowsrepr esent t he shift of the coarse-grained entropy curves when decreasin g t he resolutionfrom E to E' < E .

of the steady-state ent ropy production. Note that this is not a property ofthermodynamics but of dynamical-systems theory.

Using again that hK S = L >. >o~j - K for general open systems, we obtainJ

et> (i r r ) = L ~i + lixs .

>' i<O

(23)

Thus, in a steady state we find a relati on between the entropy flux and the metricentro py. The first term is negative and expresses the decrease of the ent ropydue to the convergence towards the invariant set , the second one represents theincrease due the chaoticity of the dynamics.

It is interesting to see that in two-dimensional chaotic maps the irreversibleentropy produ ction essentially determin es the information dimension of thenatural measure. Since in general D[ = 1 + (K - ~d/~2 [38], we find [15] that

(24)

The deviation of the natural measure's dimension from the phase-space dimension (d = 2) is thu s propor tional to the irreversible ent ropy production in thesteady st at e. T his is valid for both closed dissipative and open volume-preservingsystems.

2.6 Closed Volume-Preserving Systems. According to the above considera t ions, a vanishing irreversible entropy production in a steady state is onlyconsistent with a volume-preserving dynamics (for instance, the Hamiltonianequat ions of motion in classical mechanics). To see how the vanishing of the

ent ropy production comes about in the steady st at e of such a system, we brieflyconsider the evolut ion of the Gibbs and the coarse-grained ent ropies again. Thebasic difference to the cases considered above is that the Gibbs entropy cannotchange in t ime due to Liouville's theorem, i.e., due to u(x) == 0 (see also [45]) .The time evolution of the coarse-grained entropy depends again on the initi alconditions.

A. With extended (spatially non-const ant) initial condit ions the coarse-grainedentropy first slightly changes but afte r tE it increases and eventually saturatesat a constant according to (12) (Fig. 2).

B. For a sharply localized initi al condition the coarse-grained ent ropy starts toincrease linearly as lites! and approaches asymptot ically the same SEas withthe other initi al condit ions.

Entropy is produced only until reaching the st eady stat e, where t (irr) = o.By increasing the resolution, both the non-zero initi al value and the asymptoticvalue is shifted by din (c/ c f

) , in accordance with the fact that D1 = d. Afterall, the asymptotic distribution is homogeneously space filling.

We conclude that the same approach of following the exact and the coarsegrained densities that we used in the presence of asymptotic fractal structures inthe phase space of open and/or dissipative systems, also describes the relaxationtowards a uniform density in closed, volume-preserving syst ems. Similarly tothe difference in the behavior of syst ems approaching thermal equilibrium andnonequilibrium steady states, the temporal change of the entropies is markedlydifferent in these cases (d. Figs. 1 and 2).

2.7 Missing Ingredients to a Thermodynamic Description. Theentropy production and entropy balance derived above are valid for every dynamical system. The balance equat ion already has the structure well-knownin thermodynamics, but more than this cannot be achieved in such generality.A full agreement with thermodynamics would require in addition that and~~irr) can be expressed by coarse-grained currents. In particular , the entropyflux, which is based on the exact phase-space density and on the Gibbs entropyshould be relat ed to the divergence of a coarse-grained entropy current .

An important condition to pursue such a further comparison is the existenceof coordinates representing the (macroscopic) spat ial exte nsion of the syste mand the focus on (macroscopically) large observat ion t imes. The up to nowmost commonly used approach of this type is that of dissipat ive (thermostated)dynamical systems with periodic boundary conditions [3,9,46- 56]. Periodicitymimics a large spati al extension. In the st eady st at e an agreement with thethermodynamic ent ropy production is then obtained in the sense of (22), evenwithout explicitly referring to coarse graining. However , this approach is byconst ruction unable to deal with boundary conditions ensuring sustained density gradients in the system. Moreover , one cannot address local relations sincecharacteristics of dynamical systems are global in nature.


3N

3I......

3 3+......

3+N

Figure 3: Arrangement of the scatters on a triangular lattice; a denotes th e latticespacing and R the radius of scatterers. The distance between neighboring disks is2aj V3. Subsequent columns of scatterers ar e lab eled by the index m . We will allowfor an exte rnal field E parallel to th e x ax is. The numbers 0, . . . , 11 inside th e scatterersdenote the values of the symbols s of those disks which can be hit imm ediately afterleavin g the one in the cent er. For E of. 0, in addit ion to these also the light shadeddisks can be reached .

In the framework of multibaker maps [17,22] flux and absorbing boundaryconditions can be implemented, and the time evolution of coarse-grained andexact densities can be followed. Agreement with local thermodynamic relationscan be achieved in a suitably chosen macroscopic limit , which comprises the limitof large linear size and long observation times [13]. It turns out then that theweak resolution-dependence of the entropy production and of the time derivativeof the coarse-grained entropy, which are present in general dynamical syst ems,can disappear in this limit , leading to a full consistency with thermodynamics[24]. On the other hand, in this class of models the agreement is only achievedon expense of deviations from dynamical system theory in certain points, as willbe discussed lat er.

§3. From the Lorentz Gas to Multibaker Maps

The above considerations concerning the relation between thermodynamic properties and an underlying chaot ic dynamics can conveniently be explored bystudying diffusion of independent particles scattering elastically from a periodicarray of circular scatterers (Fig . 3). To avoid technical difficulties arising fromtraje ctori es which travel infinitely far between collisions (i .e., an array of scatterswith an infinite horizon) , we choose a tri angular arrangement of scatterers and

Multibaker Maps and th e Lorentz Gas 381

Figur e 4: Definition of th e impact parameter bn and th e direction of the t ra jectoryOn, as well as th eir respective images (bn+1 , On+r) for a collision with the scattererencoded by Sn relati ve to t he disk of th e nth collision. (a) T he depend ence of bn +1

on b« and On, and (b) th e one of On+l on bn+l and On. The cente rs of the scatterersare indicated by grey bullets; all ot her symbo ls are explained in t he text. Note tha t bnand bn+l are negative here for our convent ion for the sign of the impact param eter.

fix the lattice spacing a to be twice the radius of the scatterers R = a/ 2, i.e.,we fix a to the largest value where the horizon is finite. We set the mass and thecharge of the par ticles to unity. The momentum vector p of the par ticles doesnot change its modulus upon collision.

3.1 Mapping Relating Subsequent Scattering Events for an Unbiased Dynamics with Periodic Boundary Conditions. In the field freecase, the t rajectories proceed along st raight lines segments between subsequentscattering events. Each of these segments is uniqu ely characterized by two realnumbers. The cont inuous time evolut ion can therefore be reduced to a mappingM from any of these pair s to the next one. A convenient form of the mappingis found by choosing these numb ers as the angle eof the trajectory with the yaxis, and as the impact parameter b == (Rxpy - Rypx)/ R (d. Fig. 4). Here,(Rx , Ry ) denot es the vector from the center of the disk to the location ofimpact on its circumference. We specify the scatterer hit in collision n + 1(d. Fig. 3) by Sn = 0" " ,11 , and introduce l (sn ) for the distance of thisscatterer and the scat terer hit at collision n . The latter takes the value Za] >/3and 2a for even and odd values of Sn, respectively. From the marked trianglein Fig. 4a one verifies that sin (en - ~ Sn ) = - (bn+1 - bn )/l (sn). Moreover , byobserving in Fig. 4b that sin 0: = - bn+I/R and f3 = tt - 20: , and using that theangles of the dashed t riangle add up to n , one immediat ely determines en+1 .

382

This leads to

T. Tel and J . Vollmer

(25)

By definition b E [- R, R] and () can be taken in [0,21T] such that M is definedon the fundamental domain [-R, R] x [0,21T] . The symbol Sn = 0"" ,11 labelsdifferent branches of the mapping. They are separated by the lines (Fig. 5a)

b = ~ sin(() - ~s)±R SE {0, 2, 4, 6,8, 1O}, (26)

where the impact parameter takes one of its extreme values b = ±R in thenext collision, i.e., where the trajectori es become tangent to the scatterers. Upto a trivial shift in the angle () the action of the mapping is the same on allbranches carrying an odd (even) label s. A straightforward calculat ion showsthat M bijectively maps the fundamental domain onto itself (Fig. 5). Moreover ,the dynamics locally preserves the area of volume elements such that an initiallyuniform density stays uniform at all tim es.e Since the Lorentz gas has a mixingdynamics (ef. [34, 57] for details) every smooth initi al density will asymp toticallyapproach thi s uniform density.

The motion of a particle in the field free Lorentz gas can be traced backwardby inverting its velocity at any given tim e. At a collision this corresponds to theact ion bn+1 >-+ -bn+1 and ()n +l >-+ 1T + ()n which leads to the involution

(28)

as a tim e-reversal operator of M . Indeed, by straight forward calculations oneverifies that

TT =1,

M- 1 = TMT.

(30)

(31)

By this mapping time reversibility is thus connected to a pairwise interchangingof possible initial conditions, i.e., to a geometrical operation on points in thefundamental domain. We stress that T is area preserving . The determinant ofits Jacobi matrix identically takes the value -1.

2 This can be checked by verifying that the Jacobi matrix MJ of the mapping (25)takes t he form

(27)

which has unit determinant irrespecti ve of the values of bn+l and On . A graphicalillustration is given in Fig . 7a.


(a) (b) (c)

8n 8n+1 8n+1

21t 21t

1t ----. nII

0 0

-R 0 +R -R 0 +R

bn+1 bn+1

Figure 5: Graphical illustration of the action of the mapping M (25). (a) Location ofthe branches. The ones with odd numbers are not labeled. (b) Images of the respectivebranches. (c) Images in the fundamental domain (b,8) E [- R,R] x [0,211"], as obtainedby using the 211" periodicity of 8n+1.

The mapping M is a composition M = M 2 oM 1 of the operations M 1 andM 2 . The first one

M 1 : ( ; ) H ( b - l(s) Si; (() - ~ s) ) (32a)

(32b)

amounts to a strong shearing in horizontal dire ction, bringing the lower boundary of every branch to the right (i .e., to b = R), and the upper boundary to theleft (b = - R), resp ectively. The second operation is

M 2 : ( ; ) H ( () +1I" +2barcsin ( ~ ) ).

The act ion of the arcsin-funct ion adds an additional phase of 211" to the imageof (), when changing b from - R to +R. Thus M 2 represents a shearing in thevertical dir ection, which leaves the b= - R axis invariant . For instance, the strips = 0 completely traverses the strips s = 3,' . . , 9, has overlap with s = 1,2 ands = 10,11 , and only touch es s = 0 in its two fixed points (b,()) = (± R,O) . Thelacking overlap reflects t he "difficulty" to reach t he scatte rer s = 0 in the secondcollision, when starting from s = 6 (d . Fig. 3). Analogous statements hold forthe other branches.

The action of the mapping is schematically described in Fig . 6. For claritywe restrict there to the branches carrying even labels, which are shown asparallelograms , omitting the nonlinear correct ions leading to the curved form ofthe boundari es (27). The piecewise-linear transformations take the form

384 T . T el and J . Vollmer

- -

~

R:-~_~--

---+ ---+"" II , ,,11

2

bn+1 bn+1

8n+1

2 1t

~1t

" II = ,,112 " II ,

0 +R

bn+1

Figure 6: T he action of t he piecewise linear approximation to the map M . T he upperpart of the figure shows the action of t he linear ized M l and M 2 given by (33) . Theircombined action on t he fund am enta l domain is shown in t he lower part . The two fixedpoints of t he bran ch s = 0 are ind icated by bull ets.

M 1 : ( ~ ) H ( b - I(s) Je - ~s) ) , (33a)

and

(33b)

As a result of the combined act ion of M 1 and M 2 , the initi al parallelogramis squeezed in the vert ical and st retched in the diagonal directi on.

Finally we remark that by using the collision map we have lost information onthe collision times. They might change in a range but their order of magnitudeis set by the lattice dist ance a and the par ticle velocity p to T = alp. Oneapplicat ion of the map therefore corresponds to a t ime ste p of length T .


3.2 Adding an External F ield and a R eversible Thermost a t . Thecurrent of particles in response to an applied external field E has extensive lybeen discussed for the Lorentz gas [3] . In order to avoid an unbounded growth ofthe energy, the system is typically subjected to a deterministic thermostat fixingthe energy (d. App. A and the review by C. Dettmann in the present volume) .The net effect of the field and the thermostat is to change the trajectories inbetween collisions from straight line segments to curves of the form (d. App. Afor a derivation)

( ) _ _ p2 [ cos [80 - E (y(t) - YO) j p2] ]x t - Xo E In 8 'cos 0

(34)

but leaving the modulus of their momentum untouched in spite of the field.Here, (xo,Yo) is the position of the particle at time t = 0, 80 is the angle of itsinitial velocity vector to the y axis, and E is the strength of the app lied field,which is taken along the x-direction.

Also in this case the time evolution can be reduced to a mapping relating theinitial condition of trajectory segments immediately after subsequent collisions.In this case, however, the angular momentum b of a trajectory segment andthe angle 8 it has with the y axis are no longer preserved between collisions.As a consequence, the map describing the collisions M has to be augmented byanother one £ describing the evolution of (b ,8) between collisions. The resultingmapping

(a) (b)21t ., . "; . :

; ' 'ft~'t' ''.e ,.....f:'_ •

''f ~ r'n"" . .. ',. '}.. .r" .

It ':r ~

" ':• : ' ~• • • r.. .:: 4v-

. "0

.~

-R 0 R -R 0

b bn n

(c)

R -R ob

n

(35)

R

Figure 7: (a) Boundaries of the different branches and their respective images for the

mapping M(E) for (a) E = 0, (b) E = 0.4p2 ja , and (c) E = 0.6p2 ja. The dots inthe figures give an impression on the respective invariant densities by showing 20000iterations of the initial condition (b,O) = (0,0.14571r).


remains one-to-one on its domain (cf. Fig. 7) as long as the field is not very strong(E ;S 4.5p2 [a, d. [58]). However , the boundari es of t he different bran ches canno longer be calculated ana lyt ically. The num er ical solution for t he shape of t hebranches and t heir images for different field strengths is shown in Fig. 7. Wemention that collisions wit h the six unmarked disks of Fig. 3 also occur but inso tiny regions that t hey cannot be resolved in Fig. 7b and c. Since t he modulusof t he momentum has not changed, the t ime uni t remains T = al p.

The dyn amics of t he Lorentz gas in t he presence of an external field is st illinvariant under reflecti ons at t he horizontal axis par allel to the field , whichamounts to the symmetry (b, B) -+ (R - b,27r - B) observed by t he map M (E ) .

Moreover , since the collision operator M , as well as the evolution, L ar e timereversal invariant (d . App, A), the involution T generates the time reversedmoti on also for M (E ) , at least for sufficiently small E.

On the other hand, the field breaks the discrete rotational invariance of themapping M (E). As a consequence, the branches of the map become different inarea. From Fig. 7, one checks that t he branches to be mapped in t he dir ectionof the field (8 = 7 · · ·11 ) are now larger t han those mapped oppos ite to t he field(8 = 1· ·· 5) . However , since t he t ime-reversal operator is area preserving, thearea of the image of the branch 8 exactly mat ches t he area of branch [8+6]mod12.Consequently, M (E ) expands (contracts) area when moving in (oppos ite to)the direction of the field , i.e., t he mapping M (E ) does no longer preserve t hephase-space volume. This is indeed expected , since t he evolution operator Lexpands (cont racts) area, when moving in (oppos ite to) the direction of thefield (as shown in App . A). As a consequence of t he phase-space cont raction andexpansion t he natural invaria nt density is no longer uniform. This is exemplifiedby t he complicated asymptotic st ructure of the density sampled by a singletypical t rajectory in Fig. 7. A typical trajectory is more often map ped int o thedirection of the field than oppos ite to it (d. Fig. 7, where the strips 7- 9 carrymore particles than 1- 5). This leads to a particle cur rent induced by the appliedfield , and a closely related average phase-space cont rac t ion. The properties oft his response have been st udied in Refs. [7, 58, 59].

3.3 The Spatially-Extended Lorentz Gas. In addition to syste ms withperiodic boundary condit ions, our int erest is the modeling of spat ially extendedsystems, where transport can also be induced by imp osing flux boundary conditions. These are called open syste ms. In this situation not even t he stat ionarydensities ar e uniform in the dir ection of t ransport, i.e., along t he x axis. Thereis only translat ion invari an ce par allel to the y axis, and one explicit ly has tokeep track of the moti on of t he par ticles in t he x dir ecti on. An appropriate wayof doing t his in t he Lorentz gas is to view the array of scatterers as a sequence ofcolumns par allel to t he y axis (d. Fig. 3). Wi thin t hese columns the scatterersare not disti nguished. We consider a system of fixed length L == N a in thex direction. Also in t his sit uation the t ime evolution of a t rajectory can bedescrib ed by a mapping, bu t now its domain cons ists of N + 4 columns. These

Multibaker Maps and t he Lorent z Gas 387

-+11 [/

-+11 [/

-+" II [/

a,

-+1( [/

m-3 m-2 m-1 m m+1 m+2 m+3

Figure 8: Act ion of t he mapping M (O) for E = 0 where the spatial information of t heposition of the scatterer is kept for the x-d irection only. All init ial cond itions lie in cellm , and under the mapping they are redistributed among neighboring cells. The initialconditions which are mapped to cell m - 2 (m + 2) in the first t ime step are coloredblue (green), t hose mapped to cell m - 1 (m + 1) are colored yellow (red) and t he onesst aying in cell m are black.


columns will be called cells in the following. They are labelled by the indexm = - 1, 0, 1, ' . . ,N ,N + 1, N + 2, such that their sequence reflects the progression of the columns of scatterers . The columns m = - 1, 0 and m = N +1, N +2are needed to implement the boundary condit ions. In addit ion to the pairs(bn,en) one now has to specify the column m n, where the n th collision takesplace. A convenient way to describe this dynamics is to extend the mapping Mto a chain of mutually connected cells (Fig. 8) , where the map with index mn isapplied when the n th collision takes place in cell m n . The horizont al coordinateof the mapping is thus X n == a mn + bn , while the vertical coordinate remainsen' In the cells m = - 1, 0, N + 1, and N + 2 special rules are employed torealize the boundary conditions. The new mapping M (O ) describing the timeevolut ion between the nth and (n + l )st collision

(36)

comprises a displacement V between the cells in addit ion to the mapping M (E )

relating the coordinates (b,e) of successive collisions. The displacement V actsas follows in the different branches of M (O ) (d . the upper two rows of Fig. 8for an illustration of the act ion of M (O )) :

s = 3 This branch is mapp ed from cell m to m - 2. The space taken by theimage of the s = 3 branch in Fig. 5 is taken now by points with theirpreimages in cell m + 2.

S E {I , 2, 4, 5} The branches are mapp ed from cell m to m - 1. The corresponding space taken by their respective images in Fig. 5 is taken by points withtheir preimages in cell m + 1.

S E {0,6} The t rajectories proceeding to scatterers labeled by S = 0 or S = 6 arenot displaced in the direction of transport. The corresponding branchesare mapped into the initi al cell m itself.

S E {7, 8, 10, 11} The branches of the map are mapp ed from cell m to m + 1. Thecorresponding space taken by their respective images in Fig. 5 is taken bypoints with preimages in cell m - 1.

S = 9 The branch of the map carrying the label s = 10 is mapped from cell mto m + 2. The space taken by the image of the s = 10 branch in Fig. 5 istaken now by points with their preimages in cell m - 2.

In Fig. 8 the resulting dynamics is followed over four tim e ste ps for initi alcondit ions in cell m. Points start ing in different branches of M (O ) are indicatedby different colors. In the first ste p the images of the various branches of themap can st ill clearly be distinguished, but afte r only a few time steps they arest retched and mixed up so strongly that this is no longer possible in a graphicalillustration of this type. The motion of the shown initial condit ion appea rs to


be diffusion like from this representation. This prop erty is a direct consequenceof the displacement operation V act ing on top of the chaot ic dynamics M (E )

(as initially shown for spatially exte nded one-dimensional maps [60]).

3.4 Symbolic Dynamics and Pruning. The observat ion of diffusive-likemotion of particles in the exte nded Lorentz gas can be encoded by a symbolicdynamics. To this end the trajectories are represent ed by symbol sequencesspecifying the order a trajectory visits the scatterers. By this approach onerigorously links the time evolut ion M (O ) to the one of a random walk (withan exponentially decaying memory). The particle trajectori es are encoded byspecifying the scatterer hit at collision n + 1 [at approximat e time (n + 1)7] bythe symbol Sn. The code can take on any of the symbols defined in Fig. 3. Thus,one arrives at the symbol sequence· .. S - 3 S-2 S -l e S o S l S2 .. . . In this way,the full sequence describes how a trajectory proceeds forward and backward intime from a given disk. Application of the map M (O ) corresponds to shiftingthe e one position to the right. Regions in the fundamental domain obtained byintersecting the k th preimages of the branches with their l th images contain byconst ruction all tho se (allowed) sequences which share a given middle part of thesymbol sequence with k (l) symbols before (after) the e. For hyperbolic systemsthese regions contract to points in the limit k , l ---+ 00, i.e., every trajectory inthe syste m is uniqu ely characterized by a symbol sequence.

We st ress, that not all sequences represent allowed tra jectories. For inst ance,it was argued above that for E = 0 a trajectory cannot proceed to S = 0 intwo successive collisions. As a consequence, no pair s of two neighboring identical symbols can appear in an admissible sequence. Thi s necessity to rule outunphysical symbol sequences is called prun ing in the physical literature [44].A full characte rizat ion of the dynamics requir es to specify the set of allowedsymbols as well as the grammatical rules specifying the admissible sequences.There are changes in these rules whenever th e syste m undergoes a bifurcationsin the course of changing parameters such as the external field E . The prevalenceof bifurcations leads to a complex (fractal) dependence of transport prop ertiesof the Lorentz gas on the parameters, which is a typical feature of transportin low-dimensional dynamical systems. These complicated dependences are inour eyes non-thermodynamic features, which we will not further discuss in thefollowing. They have been studied in detail in Refs. [61 ,62] .

The importance of the symbolic dynamics lies in the relation it establishesbetween the invertible microscopic dynamics of the particle system and a stochast ic, random-walk like process defined by a Markov graph on the space of admissible symbol sequences (d. [44,63] for details ). By this approach the diffusivespread of the initial conditions observed in Fig. 8 can thus directly be relat edto the one of random walkers on the line. The cont inuum limit of this discretestochastic process, we relate to the transport equations. This allows us to gofrom the deterministic microscopic dynamics to the relat ed thermodynamicdescription of transport in a well-defined sequence of ste ps. We emphasize that


a full thermodynamic description requires not only the transport equat ions buta consistent entropy balance, too . A discussion of this relation will be t he aimof the next Section. First , however , we introduce a simplifying condition whichmakes analyti c computat ion possible.

3.5 Piecewise-Linear Approximation of the Lorentz Dynamics. Details of the long-time dynamics and of the t ransport properties of the Lorent zgas are difficult to deal with analytically due to the complexity of the mappingM (G ). In order to understand generic features of models based on dynamicalsystems it is therefore helpful to consider the spatially extension of the linearizedversion (33) of the map introduced in Fig. 6. In Fig. 9 the resulting dynamicsis followed over two tim e steps for initial conditions in cell m. The branches areindicated by the respective values of the (even) symbols s. The evolut ion of theshown init ial condition appea rs to be diffusion like from this represent ation, too .Again , this property is a direct consequence of the displacement operation Dacting on top of the chaot ic dynamics, but in this case the dynamics is describedby a piecewise-linear mapping. Note that there is still pruning, however.

3.6 The Next-To-Nearest-Neighbor (nnn) Multibaker Map. Theoccurrence of pruning and of correlat ions between successive jumps are technicaldifficulties, which need not be considered when trying to clarify conceptual problems. A slight geometrical change of the map of Fig. 9 leads to the disappearanceof pruning without modifying the overall features. This change corresponds tochoosing the boundaries of the map 's branches to coincide with the local stableand unst able manifolds . The new coordinate orthogonal to the x axis (the analogof B) is denot ed by p and can take on values in the interval [0, b] where b is anarbit rary positive numb er. The variable p is moment um-like. It is perpend icularto the transport direction and genera tes fractal structures. Note, however , thatp is not conjugated to x in the sense of classical mechanics. T he contracting(expanding) direction is now parallel to the p (x) axis. The map B obtained inthis way is called a next- to-nearest-neighbor (nnn) multibaker map since thereare next-to-nearest-neighbor transit ions of particles. It is defined graphic ally inFig. 10 and 11. It s tim e unit T is the same as for th e previous maps.

Similarly to the extended Lorentz gas the phase space of the multibakermap B consists of a chain of identic al cells of linear size a and area r = a x b(cf. Fig. 10). The dynamics is the same on each cell, except for possibly theout ermost ones (-1 , 0, N +1, N + 2), where boundary condit ions are imposed.

The map B is piecewise linear and defined on a set of branches partitioningeach cell. Every branch is compressed in the vert ical and stretched in the horizontal direction. The images are t ranslat ed to neighboring cells or are rearrangedin the original cell (d . Fig. 10). More precisely, every cell is divided into 5 verticalcolumns (see Fig . 11). The rightmost (leftmost ) column of width r2a (l2a) ofeach cell is mapped onto a st rip of width a and height r2b (l2b) in the second cellto the right (left) . The inner right (left) column of width rl a (it a) of each cellis mapped onto a st rip of width a and height rlb (ilb) in the neighboring cell to

en


-+..., 11o

m-2 m-1 m m+1 m+2

Figure 9: Map M(G) = MV based on the piecewise-linear approximation of the fieldfree case M introduced in Fig. 6, which neglects trajectories to next-to-nearest neighbors. All initial conditions lie in cell m, and under the mapping they are redistributedamong neighboring cells.


~ a-t

.11

Tb P

1 x

cell : m-3

displacement:

m-2

-2

m-1

-1mo

m+1

+1

m+2 m+3

+2

Figure 10: Sketch of the next-to-nearest-neighbor multibaker map . The mapping is thesame in each of the inner cells m = 1 · · · N , and acts aft er t ime units T . Its branchesare marked by different shades of gray and indicat ed by th e letters I, L, S, R , r , inorder to demonstrat e the act ion over two tim e steps on initi al cond itions in cell m . Themapping connects t he cell m with its neighbors m ± 1 and next- to-n earest neighborsm ± 2 as describ ed in the text (also d . Fig. 11). Displacements are measured in unitsof a.

I S

I----- a ----i

=:~f - lbs b

b~ 1~12 .

height

_~'+:2 __+1

o

CUi''0tiloCD3CD;a.

Figure 11: Sketch of the next-to-neares t-neighbor multib aker dyn amics. Five vert icalcolumns of height b are squeezed and stretched to obtain horizontal strips of width a,which are displ aced to neighboring cells. The width and height of the respective stripsis given at the margin of th e plot. The respective displacement, measured in units ofa, is given to t he right . The letters (l,L,S,R,r) are meant to illustrat e the action of themap on the different columns; they do not corr espond to t he widths of the columns.

Mul t ibaker Map s and the Lorentz Gas 393

the right (left) . These columns are responsible for transport in one t ime step T .

The middle column of widt h sa stays inside the cell, thus modeling the motionthat does not cont ribute to transport dur ing a single iteration. Thi s column ismapp ed onto a strip of width a and height sb. T he internal dynamics is chosento be area preserving, i.e., s = s. In analogy to the Lorentz gas we require globalphase-space conservation, which implies the sum rules

- -11 + rl + 12 + r 2 = 1 - s = 1 - s = h + 1'1 + 12 + 1'2· (37)

Altogether , the multi baker dynamics is governed by the mapping (for convenience x is measured here with respect to the left corner of cell m)

( xl~m + (m - 2)a,

for 0 < ~ - m < 12

( x-a(;;+l2) + (m - 1) a,

for 12 < ~ - m < 12 + II

B: (x,p) I--t

for h + 12 < ~ - m < 12 + h + s

( x- a (m~~2 +l 1 + 8) + (m + 1) a, b(l2+ [1 + s) + rIP)

for 12 + II + S < ~ - m < 1 - r 2

(38)

( x- a(rr;.; I- r2) + (m + 2) a,

for 1 - r2 < ~ - m < 1.

Using the Frobenius-Perron operator [4,44] this completely specifies the tim eevolut ion of the phase-space density r( x,p).

It is worth introducing a short-hand not ation Pi for the transit ion prob abilities from a given cell to its ith neighbor (d. Fig. 10). Since the dynamics isassumed to be translati on invariant along the chain, these rates do not dependon m. The t ransit ion probabilities Pi ta ken with respect to a uniform cell densityfrom cell m to cell m + i are prop ortional to the width of the columns . In viewof the definition of the map , we have


p, ~1l 2 for i =-2i. for i = - 1s for i = O (39)r1 for i = 1r2 for i = 2.

In a similar fashion , the quantity Pi is a shorthand notation for the parameterscharacterizing the height of the horizontal st rips in (38).

Similarly to the approach taken for the Lorentz system, we call the (x, p)dynamics of the multib aker map t ime reversible if there is an involution Ts suchtha t

(40)

which fulfill the additional constraints that (i) it acts only locally, i.e., it mapspoints only within cells, (ii) it is area preserving, and (iii) it is independent ofthe bias r1 - h or r2 - h modeling the presence of an external field. Theseconditions can only be fulfilled provided that the area fPi of the st rip mapp ed icells to the right exactly mat ches the area fp- i of the image of the st rip mapp edi cells to the left , i.e., if

Pi = P-i for every i . (41)

The tim e reversibility of the multi baker map can in this case be expressed likein Eq. (29) by applying the time-reversal operation

Te . (x ,p) H (a(1 - ~ ) ,b (1- ~)) (42)

in every cell (x is taken here mod a to avoid writing out the trivi al dependenceon the cell index). Equation (41) const itutes the only choice of parameters ofthe considered multibaker map , where a time-reversal operator complying withthe above requirements can be found . It covers the area-preserving dynamicswithout bias, but even the biased area-preserving dynamics is excluded fromthe class of t ime reversible dynamics.

§4. Transport and Entropy Productionin the (nnn) Multibaker Map

In multibaker chains it is natural to carry out coarse graining within the cells.Let

em == ~1 dx dp e(x ,p)cell m

(43)


be the coarse-gra ined density in cell m . It will be called cell density in the following. The conservat ion of particle number leads to a master equat ion relatingthe density after one time step {}'m to th e initial densities {}m [64]:

(44)

Here and in the following primed quantities denote quantities taken afte r onetime ste p T .

4.1 Time Evolution of the Entropies. The cell density {}m is a coarsegra ined density in the spirit of Sect . 2 taken with respect to boxes correspondto the cells of the multibaker map . The resulting coarse-gra ined entropy is

s; = - r {}m In ( ~7 ) , (45)

where o: is a constant reference density. By definition, t he coarse-gra ined ent ropy afte r time Tis S'm = -r (}'m ln ({}'m / {}* ).

We assume that in the initial configuration the phase-space density is const ant in each cell: (}o(x, p) = (}m . Consequently, the coarse-grained and the Gibbs

ent ropy S~G) agree initially. In order to compute the Gibbs entropy S~)' afte rtime T , we observe that the number of par ticles evolving with a given branchof the map is preserved . Thus, the density changes due to the changes of thephase-space volume only, which are described by th e contraction factors pdPi.This leads to the new densities

, Pi{}m i = -::- {}m- i, Pi

(46)

on the strips of volume Pir of cell m . Note that the density is preserved forvolume elements mapped wit hin cell m , i.e., {}'m ,o = (}m . The Gibbs ent ropy isthus

S (G )' = _ ""r-. ' . I ({}'m,i)m - L......i P' {}m , n

i ' {}*

L r I ({}m- i Pi)=- p ./) . n ---

H~m-t * "-i o Pi

r [ , I {}m "" I ({}m- i Pi)]= - {}m n ----;; - L......i Pi{}rn- i n ---::-{} i (}rn P,

(47)

(48)

where (44) has been used to arr ive at the last equation. The irreversible ent ropychange, the discrete t ime analog of (16), becomes

f1·S = (S' - S(G)') - (S - S(G)) = S' - S(G)'tm - m m m m m m

[ ({}m- i Pi ) , ({}'m)]= r LPi{}m- i In ---::- - {}rn In - .

i {}rn P, {}rn

396 T . Tel and J. Vollmer

We remark that these expressions are valid even for more general multi bakermaps, where non-vanishing transition probabilities Pi and corresponding phasespace contraction factors pdpi appear for arbitrarily long jumps. For concreteness, we restrict ourselves , however, to treating the next-to-nearest-neighbormap only, for which Pi and Pi vanish for Iii > 2.

4.2 The Macroscopic Limit for Transport and the Advection-Diffusion Equation. After having derived exact results for the coarse-graineddensity and the irreversible entropy production, we evaluate them in the macroscopic limit. In thermodynamics one is interested in the large-system limita « L. This corresponds to observing the transport process on a macroscopiclength scale, comparable with L. In a mathematical idealization, where onedisregards the discreteness of cells, this limit corresponds to a -+ O. In this limitthe spatial variation of the densities described by the discrete index m reducesto a functional dependence on the continuous variable x = rna.

Since the coarse graining corresponds in our case to integrating out themomentum variable, the spatial variation of the particle density Pm == bl2mfollows the one of the coarse-grained phase-space density 12m . In the large-systemlimit , the cell densities for the neighbors of cell m [i. e. , at positions x ± dx =(m±l)a] can thus be expressed through spatial derivatives of the particle-densitydistribution p(x) at x = rna. Up to second order in a one obtains

This yields for the temporal variation of the cell density ref. (44)]

p'(x) - p(x) _ b(!2'm - 12m)7 7

a a2

= -- [Tl - h + 2(T2 -l2)] oxP+ - [Tl + II + 4(T2 + l2)] O;p.7 27

In the limit 7 -+ 0, Eq. (50) goes over into an advection-diffusion equation

with a drift

(49)

(50)

(51)

a av == VI + V2 = - h - h] + -2h - l2],

7 7

and a diffusion coefficient

(52a)

(52b)a2 a2

D == D l +D2 = -h +hl + -2h +l2]'27 7

Here Vi (D i ) , i = 1,2, represent the 'part ial' drift (diffusion) coefficients,characterizing contributions from nearest and from next-to-nearest-neighbor


transitions, respectively. Eq.(51) is the desired transport equation describing(particle) transport in the context of multibaker maps . The particle current isfound to be

j = vg(x ) - Doxg(x). (53)

In order to have a well-defined transport equation with finite param eters v andD, the differences of the respect ive transition rates should scale as air , andtheir sums as a2 /r. The macroscopic limit required to derive (51) correspondsto a, r -+ 0 with v and D fixed.

4.3 The Macroscopic Limit for the Entropy Balance. The condit ionfor the existence of an advection-diffusion equat ion requires the finiteness of vand D, and thus of the combinations (52) only. These are two relations on thefour transition rat es and leave the four independent heights fully unconstrained .The existence of a thermodynamically consistent entropy balance heavily depends , however , on a prop er scaling of the heights.

Firs t, based on a physical constra int, we give a one-parameter scaling formof the transition rates. It is well-known in the theory of stochast ic processes[45,64] tha t v / D is proportional to the exte rnal field. Since the field is assumedto be constant along the chain , we impose as an addit ional condit ion on themicroscopic dynamics that the relation vi/D, is independent of i. This leads tothe condition

1 r2 - h----2 r2 + l2'

(54)

Therefore, one parameter , denoted in the following by a, can still freely bechosen when expressing rl, r2, h,l2 by a, r , v and D. A convenient form of therepresentat ion is

and

rD ( a V)r l = ~a 1 + 2D '

rD1- a( a v)r2 = ~ -4- 1 + D '

-t: ( a V)h = ~a 1- 2D ' (55a)

(55b)

Thi s form ensures the existence of a well-defined macroscopic t ransport equat ion, irrespective of the value of a . However, it remains to be seen whether alsomeaningful expressions for the irreversible ent ropy production are found.

While the transport equation is independent of the phase-space cont ract ionfactors Pi/Pi' these ratios do show up in the irreversible ent ropy production(48). After all, the irreversible ent ropy production is sensitive to dissipation inthe deterministic chaot ic dynamics describing the corresponding microscopicevolut ion. To parameterize these dependences, it is convenient to introducea representation for the Pi analogous to (55):

398

and

T. Tel and J . Vollmer

TD ( a V) - TD ( a V)r1 = - (30 1 +£1 - , it = - (30 1 - £ 1 -

a2 2D a2 2D

TD 1 - 0 ( a V) - TD 1 - 0 ( a V)r2 = -,-- 1 + £2- , 12 = - ,-- 1 -£2- .

a2 4 D a2 4 D

(56a)

(56b)

Here, (3 > 0, , > 0, £ 1 and £2 are parameters characterizing the dissipationin the system. The coefficient, can be expressed by 0 and (3, since due to (37)the sum of the Pi equals that of the Pi leading to 40(3 + (1 - 0), = 1 + 30.

When applying the condit ion of tim e reversibility (41) to the parameters,we find from (55) and (56) two condit ions: (i) (3 = , = 1, and (ii) £i = -l.This choice corresponds to a dissipative, time-reversible biased dynamics, similarto the one of the Lorentz gas subjected to an external field with a reversiblethermostat.

Since the parameters 0 , (3 , " £ 1 and £ 2 are also relevant for the macroscopicbehavior we define the macroscopic limit as:

a -+ 0, T -+ 0, with V, D , £i, 0 , (3 fixed. (57)

4.4 Entropy Production in the Macroscopic Limit. Substituting relation (50) into (48) yields for the irreversible ent ropy production in the largesystem limit a -+ 0

s.s; _ ( ) ( T 2) (Oxp)2 (_ 2 2 )--=ap -aa+-a_ OxP+ D --v - - +0 aaoxp .M 2 P

Here,

(58)

(59)

is the average phase-space contraction rate for a homogeneous phase-space density distribution. Furthermore,

(60)

is an average phase-space cont ract ion rate for the motion in the positive direction . a _is defined in an analogous way by replacing r via 1in the formula. Thefirst two terms of (58) are consequences of the phase-space contraction, whilethe third one arises from the mixing between neighboring cells and from thetime evolut ion. The fact that these equat ions are expressed solely by spatialderivatives of the particle-density distribution does not yet ensure the existenceof a well-defined macroscopic limit for the rat e of irreversible ent ropy production

Multibaker Maps and the Lorentz Gas

f::..iSm (irr)-- -+ (J

aT

399

(61)

and other quantities. The result obtained in the limit T -+ 0 strongly depends onhow the transition probabilities Pi behave . Using (55), (56), in the macroscopiclimit (57) one immediately finds

_ D [ 1 - a ] v2 [2 2]

(J = - a2 a In,8 + -4- lll"Y + 4D a(1 - cd + (1 - a)(1 - C2) ,

and

(62a)

(62b)

Note that the relation for 0' diverges in the limit a -+ 0 unless one requires,8 = 'Y = 1. This shows that the finiteness of the average phase-space contractioncannot be ensured in the model without the first necessary condition (i) on timereversibility. Observing this requirement when inserting the relations (62), aswell as (55) and (56), into Eq. (48) one obtains

Whether it coincides with the thermodynamic results still depends, however, onthe parameters Cl and C2 ' According to thermodynamics, the rate of irreversibleentropy production in a system with drift and diffusion is j2 j (pD) where j isthe particle-current density. Hence,

(64)

A coincidence can only be ensured if the dissipation parameters are properlychosen. One immediately sees that Cl = C2 = -1 is the only choice of parameterswhere agreement with thermodynamics is possible. The agreement is then foundindependently of the choice of the parameter a . Altogether we find that onlya time reversible dissipative dynamics is able to lead to the thermodynamicresults .

§5. Results Obtained with (nn) Multibaker Maps

(65)

(66)

Since the probability to jump to a next-to-nearest-neighbor cell in the Lorent zgas is rather low, the dynamics can faithfully be approximated by a nearestneighbor (nn) mult ibaker map . Research has mainly concent rated on thi s typeof multibaker maps , i.e., on the special case 12 = r2 = r2 = {2 = 0 of Eq. (38).Here we summarize results obtained for different problems related to irreversibility and transport . Doing this, we consider cases with a reversible dissipationmechanism only, i.e., with { = r and r = 1 (the subscript 1 of I , r , { and ris suppressed in the following). It is in general true that besides this case onecannot find agreement with Irr eversible Thermodynamics. Earlier result s onunbiased area-preserving mult ibakers (i.e., for the special case r = 1 = r = {)have been reviewed in [4].

5.1 Invariant Densities and Takagi Functions. The average densitiesin the cells of a multibaker chain evolve according to a master equat ion (44).The stationary density profile with fixed densities [lo and [IN+I at the left andright boundary of the chain , respectively, fulfills

_ (f)rn- 1[lrn - [lo + ([IN+I - [lo) (T) N+l .

T - 1

Starting with a density which is uniform in each cell and fulfills (65), thecoarse-grained density [lrn is station ary, while the exact density [l(x ,p) insidethe cells takes constant values in the horizontal (expanding) direction, and accumulate s more and more structure in the vertical (contracting) one (cf. Figs. 10and 11). In particular , afte r one iteration (Fig. 11 with 12 = T2 = T2 = I; = 0,rl = It = I , t. = r l = r , and s = s) one finds the uniform densit ies l[lrn+I / r ,[lrn , and T[lrn- I// , on three st rips with heights br , bs and bl from bottom totop , respectively. In the following iteration , the density in any of these regionis structured further in the vertical direction as shown in Fig. 12a. Subsequentiterations lead to a successive refining of the st ructure of the density such thatit approaches asymptotically a self-affine measure [4, 17, 18]. Gaspard thereforesuggested to consider instead the cumulat ive measure, which is in our model

Rrn(p) = la

dx lb

dp' [l(x ,p') .

Its t ime evolut ion is shown in Fig. 12b. One easily verifies that thi s functionfulfills the recursion relat ion

for 0< t <r

for r< t <r+ s

for r + s< t <b .

(67)


(a)

p(p)

.. .. .. ..ar~

00 b 0 b 0 b 0 b

(b)

:::~+~ .. lL .. L/,.. lLo bOb 0 b o b 0 bP P P P P

Figure 12: The invariant measure for cell m of a th ree-strip multibaker map with I =

0.15, s = 0.60 and r = 0.25 and with average densities ern - l = ~ern and ern+l = ~ernin cells m±1 fulfilling (44) with P-z = pz = O. (a) The first five levels of the hierarchicalconstruction of the invariant measure along the vertical direct ion. It approaches a selfaffine (multifractal) distribution. Regions entering in the first step of refinement fromthe right (left) are shaded in light (dark) grey. Subsequent ly, these colors are preserved.(b) The first five levels of the hierarchical construction of the cumulative measure whichapproaches a Takagi function [4, 5, 18]. The choice of shadings matches the one in part(a).

It rises strictly monotonously from 0 to fg rn , and approaches a nowhere differentiable function in the limit of infinite refinement .

Gaspard based his argument regarding entropy product ion in t he Hamiltonian multibaker map on the occurrence of t hese non-differentiabilities [4, 19]. Hesucceeded by this to derive t he first local entropy balance for a deterministicdynamical system. His original argument shows t hat irr eversible entropy pro duct ion cannot be avoided in steady states of nonequilibrium systems characterized by fractal invariant measures. Strictly speaking, however, fractal measuresare only enco untered in the limit where the number of cells te nds to infinity(N -+ 00) , which cannot be taken in a physica l system wit h a fixed gradientand cell size. Furthermore, thermodynamic ent ropy production is also expectedin transient states where t he densities are still (piecewise) smooth. A recentdiscussion of these questions can be found in Refs . [20,29].

T his ap parent contradiction can be reso lved by noting [23, 24] that the concept of a steady state in t hermodynamics must not be confused with t hat ofan invariant measure in a dynamical system. In fact , stationarity in t hermody-

namics only requires the stationarity of the average densities in regions of smalllateral extent, like for instance the unit cell of the lattice in the Lorentz gas, orcorrespondingly the cells of linear size a of the multi baker map. Structures inphase space below this finite scale are meaningless in thermodynamics. Theiroccurrence gives rise to entropy production. Thermodynamically meaningfulresults are expected to be found in the macroscopic limit, where the extent a ofthe cells is much smaller than the system size or typical length scales related tothe variation of the coarse-grained densities. Due to the exponentially refiningstructures in chaotic systems, one expects in this limit to encounter structures onmuch finer scales than a, such that the results of a thermodynamic descriptiontaking into account this cutoff turn out to be robust against details of theprescription for coarse graining. In particular this approach faithfully predictsentropy production also in finite systems (provided N » 1) and in transients.In the following we concentrate on results obtained in this spirit.

5.2 Particle Transport and Entropy Balance in Isothermal Systems. The macroscopic description of transport considers the evolution of(thermodynamic) averages in regions of small spatial extension. In this spiritparticle transport driven by an external field and a local density gradient isfaithfully described by the multi baker map of the previous Section. We sawthat the transport process is related to an advection diffusion equation on themacroscopic level, and that the irreversible entropy production is also consistentwith the thermodynamic form. These results remain valid when the next-tonearest-neighbor transition probabilities (r2' [2) vanish. One can show [22-24]that even a full entropy balance holds on the macroscopic level with

<Jl = -\7j(s) + <Jl(thermostat)

as the entropy flux. Here

(68)

(69)

is the entropy current in full consistency with thermodynamics. One finds,however, another contribution to the flux

<Jl(thermostat) = _ vjD '

(70)

which is not a divergence but is due to heat taken out locally by the thermostating procedure. Note that the full entropy balance is independent of the cellsize in the macroscopic limit, implying that the results of Sect. 2, when appliedlocally, can become independent of the coarse-graining size in this limit . The appearance of a finite <Jl(thermostat) is a deviation from traditional thermodynamicsbut its presence is unavoidable in an isothermal model. After all, such modelsare considered to mimic transport at a fixed temperature, such that they need


not have any temperature variable . A bulk system, however, is warmed up bythe Joule heat such that the assumption of a uniform stationary temperatureprofile is not valid. To let the system approach such an isothermal steady state,one has to remove the Joule heat leading locally to the heat flux .p(thermostat).

5.3 Green-Kubo Relation in the Isothermal Case. We consider longtrajectories of length t = nT which are consistent with a steady-state densitydistribution (}m . The Green-Kubo relation for multibaker maps should thenrelate the diffusion coefficient D of their dynamics to the variance of the displacements Iln of the trajectory segments around the mean a(r -l)n. The varianceof alln - a(r - l)n is to be taken with the probability density Pn(lln, /1) forfinding a trajectory segment of length t = nT centered at /1 with displacementalln. For the multi baker this probability takes the form

Pn(llni/1) = {}Jl;/~n/2 L ~(Jl)(nl ,nr) lnlsn srnr . (71)Lm=l (}m ru, n s , n r

nr +ns + nl = nnr - nl = Iln

The first factor of (71) accounts for the probability that the trajectory segmentstarts in cell mo = /1 - Iln/2 . The other factor is the probability to find

a segment with displacement Iln , where ~(Jl) (nl' nr) denotes the number oftrajectories which are centered at /1, make nl (nr) steps to the left (right),and never leave the chain . Consistently with the piecewise-linear character ofthe multibaker dynamics, these probabilities are taken to be independent. Inparticular, ~(Jl)(nl ,nr) = n!/(nl!ns!nr!) if the considered trajectories stayinside the chain , which we will assume in the following. 3

Using that Iln = nr - tu, and that Iln can take any value between -n andn , we have

XLnl ,ns ,nr

n r + n s + n l = n

(72)

Note that the density dependent factor cancels, and what is left is exactly thesame expression as for random walks. Thus,

3 This assumptions implies that the length of the system is sufficiently large suchthat the escape of trajectories from the chain can be neglected.

404 T. Tel and J. Vollmer

(a2(.6.n - (r -l)n )2) = 2DnT (73)

with D = a2(r + l)/ (2T).We can then express the diffusion coefficient as

2 n n - i

D = 2:T L L ( [mi - m i- I - (r - l)][mi+j - m i+j -I - (r - l)]). (74)i=1 j=l- i

Here m i stands for the index of the cell the tra jectory visited at time ir such that.6.n == mn - ma = 2:7=1m i - m i-I ' Taking into account that in a steady st at ethe average is expected to depend only on j due to time translation invariance,and that n is large, one obt ains

2 00

D ~ ~ ~ ([m l - ma - (r -l)][m + 1 - m · - (r - l)] )2T ~ J Jj=-oo

00

== ~ L ((va - v)(Vj - v)) .j =- oo

(75)

Here v = a(r - l)/T stands for the the drift velocity, and we have introducedthe instantaneous velocities

aVi == - (mi+1 - mi) (76)

T

which can take on the values ±a/T or zero. Eq . (75) const itutes thus a discreteversion of the Green-Kubo relation.

In the macroscopic limit T -* 0, the sum can be approximated by an integral,and we find the common expression of the Green-Kubo relatio n

1 J ooD = - dt ( [v(O) - v][v(t) - vl).2 -00

(77)

5.4 Energy Transport. In order to gain further insight into the roleof heat and entropy fluxes in deterministic models for transport [65,66 ], weintroduce (d . [30-32]) a new field (}T, which plays the role of a kinetic-energydensity. This field is also driven by the baker dyn amics , but with a tim e evolut ionadapted according to physical intuition. In contrast to the numb er of particles,the kinetic energy per unit volume (}mTm is not a conserved quantity. Its newvalues (}'m ,jT:n,j on the st rips j = L , S , R (d. Figs. 10,11) contain in our modela contribut ion from the particle flow and, in addit ion, also a source term ofst rength qm (corresponding to local heatin g) per unit time, which is constant inevery cell:

I I r [ 1(}m,rTm,r = I (}m- I Tm- I 1+ Tqm ,

(}'m sT:" s = (}m Tm [1+ Tqml ,, ,

I I l [ ](}m I Tm I = - (}m+! Tm+! 1 +Tqm ., , r

(78)

Multibaker Maps and the Lorent z Gas 405

There is a unique choice for qm where the t ime evolut ion of the kinetic energybecomes consistent with thermodynamics in the macroscopic limit (cf. below).In other cases the source te rm leads to a non-vanishing ent ropy flux (ther most a t ) ,

and can be considered to characterize the thermostatin g procedure applied.An updat e of the average kinetic energy can be calculated similarly to the

upd at e of the density. In cell m the average value e'mT:" after one t ime step isobtained by averaging the different contribut ions Eq . (78) on the st rips, yielding

For a fixed set of transition prob abilities r, l and the choice qm = 0, Eq. (79)implies a passive advect ion of the T field by the deterministic (x,p) dynamics.The presence of the source te rms qm makes the advect ion non-passive. Thequantity Tm is considered to represent the (kinet ic) temperature of cell m .

In the macroscopic limit the particle current is still

j = pv - Doxp, (80)

where v is the drift due to an external field, and for the temperature equat ionwe find:

(81)

(82)

This clearly shows that the source distribut ion q(x) [the macroscopic limit ofqm] influences the temperature profile, and different choices of q(x) can leadto different steady states. In particular , states with spat ially and temporallyconstant T , as they are commonly considered in nonequilibrium molecular dynamics simulat ions based on dete rministic thermostats, require an identicallyvanishing q.

5.5 The Entropy Balance in the Presence of Temperature Gradient s. In the presence of a non-uniform spatial temperature profile, we adopta temperature dependent reference density e* (T) == T' , in close analogy witha classical ideal gas law. Here "I is a free parameter. The entropy 8m of cell m is

8m = -aem In /:'{r) .A calculation similar to the one carri ed out in Sects. 4.1 and 4.4 yields for theirreversible ent ropy production in the macroscopic limit [30, 31]

(T(irr ) = :~ + "IpD[O~T] 2 , (83)

in full harmony with the results of Irreversible Thermodynamics [2, 67] . Thecoefficient in front of the term (oxT/T )2 due to tempera ture gradients is thus

be identified with the heat conduct ivity A, i.e. , A= , pD . The ent ropy cur rentcan be expressed as

j(S) = -A j - AoxT/T,

with

For the flux into the thermostat we obtain

(ther mostat ) = , pq - vj / D.

(84a)

(84b)

(85)

It is the difference of a term proportional to the source q, and a term corresponding to the change of entropy associated with Joule heating vj / D due tothe drift v of the particles. In thermodynam ics of bulk syst ems (thermost a t ) = 0,and the source term q takes the form

q* = vj /A . (86)

All dissipative heat is then carr ied by the heat cur rent to the boundaries ofthe syste m. This can only be achieved if there is a temperature dynamics inthe syst em , and shows that a non-vanishing (t hermostat ) is unavoidable whenmodeling drift in a syste m with spat ially and temporally constant te mperature .

5.6 Thermoelectric Cross Effects. Cross effects appear in Irr eversibleThermodynamics if at least two independent driving forces cont ribute to thebasic cur rents . Our multibaker with the kinetic-energy field faithfully describesthermoelectric cross effects, where, besides the exte rnal field or density gradi ent ,also the te mperature gradient cont ributes to the particl e curre nt. Vice-versa , t hedensity gradient also gives rise to a heat cur rent in addition to the one generatedby the field or the density gradient.

In thermodynamics t he par ticle cur ren t is of the form

j = pv - (J~l (oxPc+ eaoxT ),e

(87)

(88)

where Pc denotes the chemical potential of the particles, and v is the driftvelocity du e to an exte rnal elect ric field E , which is related to the conduct ivity(Jel by

(Jel Ev = - -.

ep

The coefficient a in (87) is t he thermoelectric power (or Seebeck coefficient ).It is easy to see that the multibaker result (80) for the macroscopic particle

cur rent is consis tent with (87) . As indi cated by (82) , the equat ion of st ate of t he'mult ibaker gas ' is that of a classical ideal gas with , as its (constant volume)


specific heat measured in units of Boltzmann's constant. Therefore the chemicalpotential as a function of the density and temperature should be

Jic = T(l +,) + Tin (pT-"I). (89)

Hence, we can express the gradient of the chemical potential with that of thedensity and the temperature as

(90)

Substituting this into (87) we see that it is equivalent with (80) provided that

and

1a = --[1 + In (pT-"I)].

e

(91a)

(91b)

The first condition is Einstein's relation which is expected to be valid in a weaklyinteracting classical gas, and the second one is an expression of the Seebeckcoefficient with local thermodynamic state variables .

The thermodynamic form of the entropy current [2] in the presence of bothtemperature differences and an external field or density gradient is

(92)

with II as the Peltier coefficient. A comparison with the multibaker results (84a)and (84b) immediately yields the Peltier coefficient in the form of

This implies that

TII = -- [1 + In (pT- "I)] .

e

II= «t

(93)

(94)

which is nothing but Onsager 's reciprocity relation for thermoelectric effects. Bythis we have expressed all the kinetic coefficients (Jel- >., a and II with systemparameters.

The forms (87) and (92) of the currents express the presence of cross effects. The measurement of the (off-diagonal) Onsager coefficients is however notpossible in homogeneous systems. It requires a junction between two materials[68] . In analogy with typical experimental arrangements, this situation can alsobe modeled in the framework of multi bakers by considering two chains with

408 T. Te l and J. Vollmer

different transition probabilities (i.e. , different "material prop erti es") , which arejoined together [32].

Momentum and heat transport in viscous flows. Recently, also a consistent description of shear flow, the accompanied viscous heat ing, and the corresponding ent ropy balance has been given in the framework of a multibaker [71].The problem of the laminar motion of an incompressible viscous fluid shearedby the relative motion of two parallel walls has been treated. In this case themulti baker dynamics is area preserving and drives two fields: the velocity andthe temperature distributions. In the macroscopic limit , in which the kinematicviscosity is kept finite, the transport equat ions go over into the Navier-Stokesand the heat- conduction equation of viscous flows sheared in a given direction.Simultaneously, the entropy balance equat ion for thi s mult ibaker converges tothe well known thermodynamical form. T he inclusion of an arti ficial heat sinkcan stabilize steady states with constant temp eratures. This mimics again a thermostating algorit hm used in nonequilibriurn-molecular dynamics simulat ions.

5.7 The Irreversible Entropy Production as the Average GrowthRate of the Relative Phase-Space Density in Steady States. In steadyst at es of open systems with density gradients , the rate of irreversible entropyproduction <T( i r r ) / (} per par ticle is the (weighted) average of the growt h rate <TI}

of the relative local phase-space density (}t (x,p)/ (}*(x,p)

( ) _ 1 I (}r( x,p) / (}*(x,p)<TI} x,p = -:;. n (}o(x ,p)/(}* (x,p) ' (95)

where T is the tim e unit and (}* accounts for the temperature dependence of thenorm alization of the density introduced in Eq. (82). This can be seen by usingEq. (46) to work out (48) for the non-isothermal case, and setting (}o (x, p) = (}rn

and (}r(x,p) = (}'m i . This local expression holds irrespective of boundary condit ions. In general, it is different from zero even in macroscopically steady stat es(when th e coarse-grained density and other coarse-grai ned averages are t imeindependent) because of the never stopping time evolut ion of the phase-spacedensity (}. Besides phase-space cont ract ion, <TI} also character izes the mixingof regions with different macroscopic densities. Hence, it does not vanish inboundary-driven, volume-preservin g systems. Only in macroscopically homogeneous steady states <T1}(x,p) reduces to the local phase-space cont raction ratea(x,p) so that entropy production and phase-space contraction can be ident ifiedin such cases only.

In a steady state of a non-isothermal system the generalized form of theent ropy production (48) can be interpreted as a weighted sum of the logari thmofrat ios (}'m iT:" i - , /((}'mT:" -' ). This suggests as an alte rnative view to considerthe steady-stat e ent ropy product ion as the sum of cont ribut ions due to stepsalong trajectories, which contribute

Multib aker Maps and the Lorentz Gas 409

~ In [ rm (2mT

;;.'Y ] for a step from cell m to m + 1,T lm+l(2m+lT~~l (1 + Tqm+l )- 'Y

'l In [1 + Tqm] wheneve r it stays in cell m , (96)T

1 In [ lm(2mT

;;.'Y ] for a step from cell m to m - 1.-:;: rm-l(2m- lT~2.1 (1+ Tqm- d - 'Y

The density ra tios enter Eq. (96) becau se of the mix ing in cell m of differentcoarse-grained densities coming from the neighboring cells m - 1, m + 1. Thefact ors 1 + Tqm st em from the t ime-dependence of t he average local kineticenergy (i.e., of the temperature) as described by (78).

Due to the appearing of the source terms in Eq . (96), t he sum of cont ribut ionto the ent ropy production cannot be evaluated in general. For a thermostatedsystem, however , where qm = 0, the single-step cont ributions (96) add up likea telescoping sum along a t rajectory, and the entropy pr oduction changes signwhen following a trajectory backward in time. Consequent ly, the local irr eversibl e ent ropy production averaged over traject ories of length n, which makenl (nr) ste ps to the left (right) and are cente red at the half-integer cell index u ,

i.e ., which start in cell m a = J.l -/)"n /2 and end in J.l +/)"n /2 , where /)"n == nr - n/,is

It is indep endent of the details of t he sequence of steps . T he macroscopic limitof (97) is a (ir r ) of (83) divided by the density p.

5.8 Fluctuation Theorem for Entropy Production in Steady Stateswith Density Gradients. In this subsection we derive a large-deviationtheorem for ent ropy pr oducti on fluctuations of mul tibakers. First , we restrictto this end to syste ms with a spatia lly constant, stationary te mperature (i.e. ,Tm = T for all cells and qm vani shes identically). According to Eq . (97), t he rate

a~I-' ) ( nl ' n r ) then only dep ends on ti; and n r through /)"n . Hence, the probability

rr;l-')(ae) of finding a value a e == a~l-' ) (/)"n) along a traject ory segments of lengtht = nr centered at J.l is the same as the probability Pn (/)"n ; J.l) [Eq. (71)] forfinding a t raject ory segments of length t = nr cente red at J.l with displacementa/)" n :

(98)

We recall that ~(I-' ) (n/, n r) denotes the number of t rajectories which make ti;

(nr ) steps to the left (right) , and never leave t he chain. Comparing t rajectories

with their t ime- reversed cou nterparts, we find that ~(I-')(nl , n r) =~(I-')(nr , nd .


Moreover, time-reversed trajectory segments produce the same entropy up toa change of sign , aY')(b..n) = _a~L)( - b..n ). Thus,

II~IL) (a~L)(lm))

II~Jl) (abJl)(-b..n))

tu ;ns , nrnr + n s + ti; = n

nr - nl = ~n

nl , ns, nrnr + ns +nl = n

nr -nl = ~n

(99)

where we used in the last step that T is constant. After taking the logarithm

of both sides, the right-hand side is exactly t = nr times aY'\b..n). Eq. (99)constitutes therefore a local fluctuation theorem [27]

(100)

It applies to trajectories of given center J-l and of finite length t = nr, andstates that the logarithm of the ratio of the probabilities to find local entropyproduction rates aY') and -aY') over a time span t is the same as the total

entropy production taY') over this time interval.In order to obtain also a global fluctuation theorem one needs the probability

II t (a (]) of finding a value ae irrespective of the position of the trajectory. This is

obtained by summing up the contributions of all II~Jl\a(]), J-l = 1,~, 2,~, . .. N,and observing that J-l contributes to II t (a(]) if and only if there is a displacement

b..nlLwith a(] = a~L\b..nIJ . Consequently,

IIt(a(]) = LII~Jl) (a~)(b..nIJ)IL

= L (!Jl-t:>.nJL /2 (!:-) t:>.nJL II~Jl) (a~) (-b..nIL))Jl {!IL +t:>.nJL /2 l

= exp (ta (]) II t ( -0"(]),

where we used Eqs . (97,99) with b..nJl to obtain the second equality.This means that

(101)

(102)

Multibaker Maps and the Loren t z Gas 411

holds for the steady state {}m of the time-reversible system. It is worth emphasizing that the resul t does not depend on the parti cular choice of boundaryconditions. It generalizes the Gallavotti-Cohen fluctuation theorem [49,51] tost eady states which need no longer be homogeneous in density.

The calculat ion can be exte nded even to a class of st eady st ates with nontrivial temperature profiles stabilized by thermostating consistent with qm = O.Assuming t hat the number of traj ectory segments st arting in cell rna = J1 t:m/2 is proportional to {}m oT;;'ri , and using t he corres ponding modification ofEq . (71) , one can recover the fluctuation relation (102) also for non-isothermalcases. Wi th the original form of (71) on the other hand, the right hand sideof (102) is modifi ed such that a temperature dependent term appears besidesa I.! which removes the cont ribut ion of the heat current to a I.!' This indicatesthat the structure of fluctuation relations is more delicate in the presence oftemperature gradi ents than in isothermal cases, even with thermostating. Und erwhich condit ions fluctuation relations hold in non- thermostated systems , andwhich form they take in the presence of steady temperature gradients is anopen question at present .

§6. Discussion

We take up a few general issues here, which further illumin at e the relationbetween the modeling of transport by multibaker maps and other approachespursued in the lit erature [and in the other cont ributions to the present volume].

6 .1 Deviations from Dynamical Systems Theory. The multibakerapproach deviat es in several features from t hose based on dynami cal syste ms.The most essent ial one is the openness of the system, i.e., the consequences ofusing non-trivial boundary condit ions. These are prescribed for the fields {} andT speci fying the thermodynamic state, and not for the (x ,p) dy namics. Thephysical motivation of t his is that we consider the mul ti baker map as a mod elfor the dynam ics in the single-part icle phase space of a weakly interacting manyparticle system. In the spirit of kineti c theory, the state of the many-particleproblem is described by a distribution function in this phase space . If one isnot int ending to mod el the reservoirs , this dist ribution function is subjected toboundary conditio ns. It is this feature which leads (aft er averaging) to macroscopic transport or hydrodynami c equations subjec ted to non-trivial boundarycondit ions. We feel that it is unavoidabl e to go beyond the scope of dynami calsyst em theory in this point , in order to obtain a closer ana logy with transport phenomena. Multibaker models seem to represent the weakest necessarygeneralization. They st ill allow us to use well-est ablished tools of dynami calsystems theory. For instance, the Frobenius-Perron equa t ion of the map E( x ,p) ,which describes the time evolut ion of the phase-space density {}(x ,p) for x valuesin the multibaker chain, takes the tradi tional form except for an augment ation

412 T . Tel and J. Vollmer

with the boundary condit ions at t he two ends. A slight ly modifi ed version of theFrobenius-Perron equat ion also takes int o account the local source term, and canthen be applied to the kinetic-energy dens ity QT [31]. The invariant measuresare forced on the systems by the boundary condit ions and t hey differ fromthe natural (SRB) measures encounte red in the presence of periodic boundarycondit ions.

Another deviation from dynam ical-syst em approaches is the possibility of local investigations. Since spatial varian ce is ubiquitous even in the coarse-grainedfields of mul ti ba kers, it is worth concent rating on cell quantities. By this approachone can est ablish an analogy with the local tran sport equat ions and t he local ent ropy balan ce. Local invest igations can be carried out in periodic mul ti-cell models, too [32]. In order to see sustained inhomogeneit ies of the fields , one considersin that case a multibaker chain driven by a st rongly bias ed dynamics in a verysmall region of the chain. The steady-state field distributions are determined bythe difference of the bias in this region to the one in the remainder of the system .In steady st ates these models are equivalent with open multibakers [32] . However ,they differ in transient cases due to the correlat ion of the dynami cs in the vicinityof the two "ends" of periodic syste ms. Op en multibaker models are therefore atpresent the only analyt ically accessible tool to comprehensively st udy transportpro cesses driven by exte rnally prescribed boundary condit ions.

6 .2 Interpretation of Coarse Graining. Coars e graining plays an essenti al role in different bran ches of physics. Even the theory of classical , equilibriumstatist ical mechani cs describ ed by closed volume-preserving models, relies onthis concept [45] . After all, the concept of the number of states or of the par t iti onsum requ ires partitioning the phase space into small boxes. Before the rise ofquantum mechanics this par ti tioning was arbit rary, but later the Correspondence Principle together with Heisenb erg 's un certainty relation fixed the size ofpartitioning. It amounts to the smallest product of the location and momentumuncertain ty b.x b.p = h, where h is Pl anck's constant . In the not ation of Sect. 2of the present pap er , t his means that the dim ensionless linear size of the boxesfor coarse graining is e r-..; h1/ 2 . Since the size is given by a universal constant,in equilibrium statistical mechan ics we do not think of cha nging the box size. Itis illuminating to recognize, however , that there is this dependence. In fact , thewell known relation S(E) = In (r(E) jh3N ) between equilibrium entropy S andthe phase-space volume f(E) at fixed energy E in a closed syst em of N particl eseach moving in three dim ensions impli es that S(E) depends logari thmically onh. The coarse-grained ent ropy S€=h 1j2(t ) converges to the value of S(E) [69,70] .By a form al change of h to hi we find a shift by - 3N In (hi j h) corresponding tothe fact that the phase space is d = D I = 6N dim ensional. The coarse grainingdiscussed in Sect . 2 for general dynamical systems is similar in spirit since it isbased on coarse graining with the finest possible resolution in phase space.

The concept of coarse graining in Irreversibl e Thermodynami cs is markedlydifferent since it is used to define local thermodynami c vari abl es. This is done


by dividing the real space into boxes of linear size much smaller than the totalsize of the system, so that .the box is large enough to contain a yet macroscopicnumber of par ti cles. The coarse-grained fields are t hen the thermodynamic st atevariables of the box in accordance with the hypothesis of local equilibrium. Theyare obtained by integrating out all momentum-like vari abl es and averag ing overthe box in real space. The linear size a of t he mul ti baker map is the analog ofthe box size. Due to t he integration over p , the t ransport and the ent ropy flow isalong the x direction - a feature which is not present in the general dynam icalsystem approach. It is within this setting that the weak resolution depend enceof t he ent ropy can fully be removed after taking the appropria te macroscopiclimit. Only in this limit one can hop e for a full ana logy wit h t hermo dynamicrelations. Whether it is indeed found might depend on system par ameters. Forinst an ce, a non t ime-reversible dynamics never leads to results consiste nt withthermodynamics .

6.3 Interpretation of the Macroscopic Limit. The macroscopic limit ,which we bri efly denote as a, T -7 0, is to be understood as the limit with a clearseparat ion of length and t ime scales between t he dynam ical and macroscopicdegrees of freedom. More precisely, we consider systems of linear exte nsion Lmuch larger t han the cell size a, and observe the fields on the macr oscopicdiffusion ti me scale L2ID or on the drift t ime scale DIv2 . In this case thedim ensionless cell size aiL an d time uni t T DIL2 (or T V

2ID) are much smallerthan unity. The not ation a, T -7 0 should be interpreted as the commonly usedphenomenological conti nuum limit of hydrodynam ic theories . It is meaninglessto literally let t he cell size a to vanish , since the local averages and the conceptof local equilibrium would t hen become meaningless. As mentioned at the endof Section 3.1, the cell size and the time unit are related by the momentum pof particles. Therefore, when modeling the Lorentz gas, the limits a -7 0 andT -7 a should be carried out under observat ion of the additio nal conditio n thatthe momentum, i.e., t he ratio alT is fixed .

T he results obtained in the macroscopic limit are st ructurally stable. Carrying out coarse graining not only over one cell, but over any finit e numberme of cells or afte r every n t ime un its rather t han after just one, one findsa dependence of the relaxation ti me on m e an d n . Taking the macroscopic limitwith fixed me or n, however , means that the size of the coarse-graining regionor the ty pical ti me afte r which coarse graining is applied is still much smallerthan their macroscopic counterparts . Therefore, in the macroscopic limit theresulting transp ort equations become independent of me and n . Consequently,the physically relevant relaxation t imes do not depend on these details of theprescrip tion for coarse graining. Rather t hey are related to the spectrum ofthe time-evolu tion operator of the thermodynam ic densities, i.e., they are independent of the t ime to discussed in Sect . 2. The latter time characterizes theemergence of fract al structure s on the scale of E in the (t hermodynamically notimm ediately relevan t) exact densities.


6.4 Outlook and Open Problems. The modeling of t ransport by simple (i.e., analytically accessible) dynamical systems is presently a fruitful andrapidly expanding enterprise. Naturally, however , there is a number of openproblems still waiting for their solu t ion

• In our approach the fields () and T are considered as independent dist ribut ion functions in the single-particle phase space of a many-particleproblem . After all, there is no momentum conjugated to the spat ial var iable of multibaker maps which would relate the kineti c-energy density tothe density. A closer comparison with par ticle dynami cs would thereforebe useful.

• Weak interactions are needed in order to induce relaxations keeping up thelocal thermodynami c equilibrium in any small region in space. It wouldbe int eresting to implement their role explicit ly in order to clarify the rolet hey play for, e.g., the occurrence of the source te rm in the heat equat ion .

• A general reversible dynami cs for heat is missing. It is uncl ear how (oreven whether) t he source te rms in the kinet ic energy densi ty can be implement ed reversibly.

• The role of quantum-mechanical coarse graining in Irreversible Thermodynamics is open . At present there is not even a model in sight to addressthis question.

The multibaker approach can be considered as a sear ch for the simplestmathematical st ructures which allow for a consistent description of transpo rtphenomena. It is remarkable in how far consistency with Irreversible Thermodynami cs can be obtained even without referring to relations naturally usedin kinetic t heory, like for instan ce the relation between momentum and kinet icenergy. We are looking forward to explore other simple models to shed light ont he open questions in the field.

Acknowledgements. Illuminating discussions with E.G.D. Cohen , J.R. Dorfmann , D. Evans, P. Gaspard, J. Krug, G.P. Morriss, G. Nicolis and Z. Racz aregrate fully acknowledged. We thank W . Breym ann, L. Matyas, and L. Rondonifor enjoyable and frui tful collaborations, and grate fully acknowledge supportfrom the Hungarian Science Foundation (OTKA T17493, T19483, T032423) .

A. Traj ectories of the Thermostated Lorentz Gas

The kineti c energy of closed particle systems subjected to an exte rn al field growsquadratically in time. In order to avoid this undesired behavior in classical t ransport , such models are commonly coupled to noise t erms [64] or to deterministic

Mul tibaker Maps and t he Lor ent z Gas 415

thermostats [3]. The lat ter approach leads to deterministi c equations of moti on ,which are however no longer volume-preserving (i.e. , in par t icular they cannot beHamiltonian ). Here we explicit ly solve these equations to obtain the trajectoryof a single particle of charge e and mass m in the (x ,y)-plane of the Lorentzgas , when it is subjected to an external field E and a determinist ic thermostatfixing its kinet ic energy. The Newto n equations are in that case

. pq =-,

mp = eE -(p,

(104)

(105)

where q == (x, y) and p == (Px,Py), are the positi on and the momentum of thepar ticle, resp ectively. The dynami cal friction coefficient ( represents the actionof th e thermostat. It is fixed by the condit ion Otp2 = 0 tha t the kinetic energyof t he particle is constant in t ime, leading to

eE· p( = -2-·

P(106)

Since the modulus of p is constant , one convenient ly par ameterizes the mom entum by its angle with the y axis

p = P (- sin B, cos B). (107)

A.1 Time Reversibility and Phase-Space Contraction. For the choice(106) of (, the equations of motion (103) are invari ant under reversing thedirection of p and changing the sign of the t ime . Hence, the operation B H

B+ 7f is an involution, which gene rates the t ime reversa l also in the case of thethermostated Lorentz gas . As a consequence , the flow in the ph ase space (q, p )is st ill one-to-one, bu t it is no longer area preserving in the presence of the field .In that case the ph ase-space volume I' changes with the rate

(108)

This rate has no definite sign. The net ph ase-space cont raction a(t ) along a t rajectory segment from (xo ,Yo) at t ime t = 0 to (x ,y) at t ime t is proportionalto the displacement in the direction of the field. Taking (without restriction ofgenerality) E = (E,O), one finds tha t

i t meE lt meEa(t) = dt ( = - 2- dt i: = - 2- (x - xo).

o PoP(109)

(110)

A.2 The Form of Trajectories. We denote the initi al dir ecti on of thet ra jectory by Bo. After inserting Eq . (107) into Eq . (105) one then finds

. eEB = -- cos B.

P


(113)

(112)

With t hi s input Eq. (104) can b e integra ted t o yield

p2 10 p2Y - Yo = - -- dO = - -- (0 - ( 0 ) ,

m eE 00 m eE

x - Xo = L rdO t an 0 = _ L In [ cos0 ] .

m eE } eo m eE cos 00

Eliminating 0 from t hese rela tions and setting m = e = lone finally recovers(34) .

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Appendix

Boltzmann's Ergodic Hypothesis,a Conjecture for Centuries?"

D. Szasz2

Contents

§l. Boltzmann's Ergod ic Hypothesis . . . . . . . . . . . . . . . . 423§2. Finding a Mathematical Object , a Not ion and a Problem

(from Boltzmann to von Neumann, i.e. from 1870 until 1931) 424§3. Proving the First Relevant Theorem

(from Neumann to Sinai, from 1931 until 1970) . 426§4. Appear ance of Non-E rgodic Behaviour

(Cosequences of the KAM-Theory, 1954- 1974) 427§5. Sinai 's Setup. Billiards (1970) 428§6. N = 2 Balls (1970-1987).

Local Ergodicity of Semi-Di spersing Billiards 431§7. N ~ 3 Balls (1989- ).

Global Ergodicity of Semi-Dispersing Billiards 434§8. The Boltzmann-Sinai Ergodi c

Hypothesis in Pencase Typ e Mod els . . . . . . 435§9. Ergodicity of Systems with a Fixed Number of Degrees of Freedom 437§10.Ergodici ty of Systems with an Increasing Number

of Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . 440

1 Lectur e presented on International Symposium in Honor of Boltzmann's 150thBirthday, February 23-26 1994, Vienna.Reproduced from Studia Scientiarum Mathematicarum Hungarorum 31 (1996),266-3 22, with kind permission of Akadernia Kiad6, Budapest, Hungary.

2 Research supported by the Hungarian National Foundation for Scientific Research,grant No. 1902

422 D. Szasz

§11.Ergodicity of Systems with an Infinite Numberof Degrees of Freedom

§12.Concluding RemarksReferences . . . . . . . . .

441442443

Abstract. An overview of the history of Ludwi g Bolt zmann's more than onehundred year old ergodic hypothesis is given. The exist ing main results, themajority of which is connecte d with the theory of billiards , are surveyed , andsome perspectives of the theory and interest ing and realist ic problems are alsomentioned .

Boltzmann 's Ergodic Hypothesis, a Conject ure for Centuries? 423

In 1964 Werner Heisenberg was elected a honorary do ctor of Lorand EotvosUnivers ity, Budapest . In his inaugural lecture he made a point that soundedsome t hing like this: "A theore tic al physic ist feels best if there is no rigorouslydefined mathem atical object behind his cons iderat ions". Certainly, Heisenbergwas having the ea rly years of quantum mechanics in his mind but wh at he saidperfectly fit te d the work of Ludwig Boltzmann as well. One could choose severalareas of his int erest to illustrate t his st atement , out of whi ch the history of t heergodic hypothesis we are going to elabo rat e on is only one.3

As it was so nicely explained in P rofessor Gall avotti 's illuminating lectureat this conference, G (1994) , t hough the rigorously defined mathem a tical objectbehind Boltzmann's considerations around t he ergodic hypothesis was indeedmissing, Boltzmann was ingenious in inventing mathemati cal paradigmas andin mast ering mathem atical ca lculat ions on t hem to find out the t ruth and toobta in convinci ng power , and eve n without having the mathem ati cal ob ject heunderstood many things better than we do now.

§1. Boltzmann 's Ergodic Hypothesis

During the 1870s a nd 1880s , various forms of the ergodic hypothesis were usedby Boltzmann in his works on t he foundations of statist ica l mechani cs (seee. g. B(1871) and B(1 884) ; for a historic account also F(1989)). An advancedformulation of the hypothesis would sound as follows:

Hypothesis 1.1 (BOltzmann's Ergodic Hypothesis) . For large systems ofint eracting parti cles in equilibrium tim e averages are close to the ense mble, orequilibrium average.

(Rem ark: In this paper - with the exce pt ion of section 10 - equilibriumaverages always mean microcanonical ones , i. e. the Liouville measure on thesubmanifold of t he phase space specified by the trivial invar iants of the motion .)

3 Another striking example, perhaps not sufficient ly widely known, is t he case with theBoltzmann equation. He published it in 1872,13(1872), and the first mathematicallysat isfactory derivation of the equat ion was only obt ained more th an 100 years lat erin 1975 by Oscar Lanford , L(1975), t hough the pictu re is st ilI not complete. T husneedless to say that Boltzmann 's original argument was highly intu itive. At t hesame time, however, it was so much challenging for the great mathemati cian , DavidHilbert t hat he included among his celebrated collection of 23 problems presentedat th e International Mathematical Congress held at Paris in 1900 th e sixth one witht he title "Mat hematical Treatment of the Axioms of Physics" (see H(1900)) . In itsformulation, besides requ iring an axiomatic approach to the theory of probabilities,Hilbert also says: "it is therefore very desirable t hat t he discussion of the foundat ions of mechanics be taken up by mathematicians also. Thus Boltzmann 's workon the principles of mechanics suggests the problem of developing mathematicallyth e limiting processes, th ere merely indicated, which lead from the atomist ic viewto laws of mot ion of cont inua". Boltzmann 's law of motion of conti nua is, of course,his equat ion.

424 D. Szasz

More precisely, if f is a measurement (i.e. a function on t he phase spaceof the system), then as N , the size of the system (for instance, t he number ofparticles) tend s to infinity, then

(1)

where fL is t he equilibrium measure, and stx is the ti me evolut ion of the phasepoint x .

We immediately not e that if N varies, t hen f and fL also depend on N ,and thus, for a mathematically st rict statement one ought to specify the senseof the convergence in (1), too. Let us look at th e main steps of the history ofBoltzmann's hypothesis - without int ending to provide a complete accountthough I think such a study should be don e. One major incompleteness of oursurvey is that it does not go into the history of the quasi-ergodi c hypothesis atall; as to some recent resul ts about it see H(1991) and Y (1992).

§2. Finding a Math ematical Obj ect , a Notion and a Problem(from Boltzmann to von Neumann, i.e. from 1870 unti l 1931)

It took quite a time unti l the mathemati cal obj ect of the ergodic hyp othesis wasfound . Indeed , only in 1929, Koopman , K (1931), began to investigate groupsof measure-preserving transformations of a measure space or in ot her language,groups of unit ary operators in a Hilb ert space 4 . Koopman 's idea was apparent lyin the air , and several mathematicians, including among other G. Birkhoff, M.S. Stone and A. Weil, cont ributed to the birth of ergodic theory ; for a historicaccount see M(1990).

More precisely, let M be an abst ract space, the phase space of the system and fL be a probability measur e on (a a -a lgebr a of) M . The dynamicsis a one-par amet er group S 'R = {st : -00 < t < oo} of measure preserv ing transformations, i.e. for every measurabl e subset A c M , and for everyt E lR fL (S- tA) = fL (A ).

Here, of course, fL is t he equilibr ium measure of the system. Let finally,f : M -t lR be a measurement such that f E L2 (fL ). Thus the object [i. e.(M , SIR , dfL ) with the funct ions f l is defined.

4 This progress was preceded by the success of Lebesgue's theory of measure which,on another path, also led, in 1933, Kolmogorov to the laying down the axiomat icfoundations of probability theory.

Bolt zman n 's Ergodic Hyp othesis , a Conjecture for Cent uries? 425

In 1931, von Neumann proved t he first ergodic theorem, t he so called

Theorem 2.1 (Mean Ergod ic Theorem (N(1932)) ). As T --+ 00 ,

liT- f(St x) dt --+ f( x)T 0

in the Ls-sense.

(The exact story of the first ergodic t heore ms is explained in t he note ofBirkh off and Koopman , B-K (1932).) 5

T he proof of the mean ergodic t heorem is not difficult but it is worth notingthat - even more t han 20 years later - Neum ann very highly appreciatedexact ly this achievement among his various findin gs in the vast terr ito ry of hisint erest. In 1954, when answering a questionnaire of the American Mathemat icalSociety, his works on the ergodic theore ms were named by himself among hismost imp ortant discoveries (the ot her two were the mathematical foundat ionsof qu an tum mechani cs, and further operator-algebras , called to day Neumannalgebras) .

T he limiting function J (x ) satifies two fur ther imp or tan t propert ies:

• f( x) = E(J I I )where I is the IT-algebra of t he invariant sets , or in ot her words J is theprojecti on of f onto the subspace of functions invari ant with respect tothe dynami cs S'R ;

• JMfdp, = JMJP, whenever f E L2·An ext remely importan t consequence is the following: if t he only invari ant

funct ions are t he constant s or , in ot her words , there are only trivial invarian tsets (i.e. p, ((S- tA \ A) U (A \ S -tA)) = 0 implies p,(A ) = 0 or 1), then, first ofall, J is a constant for every f and , moreover, by (ii) , J = J[ du . Consequently,the ergodic theorem says t hat , then as T --+ 00,

(2)

in the L2-sense.This state ment is much remini scent to Boltzmann's hypothesis but here we

st ill have just one fixed system and not var ious ones for different values of N .Anyway, define t he system to be ergodic, if the only invari an t functions are theconstants . Then we know that , for ergodic systems, t he relation (2), i.e. a versionof the ergodic hypothesis holds.

5 T hough his name is not explicit ly me ntioned , Bolt zmann 's influence on von Neumann is also seen in t he t it le of his earlier work on qu an tum ergodic t heory, N[1929]:"Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik".

426 D. Szasz

Summarizing: we have a mathematical model (groups of measure preservingtransformations) , the notion of ergodicity and, finally, the problem of establishing the ergodicity of a system we are interested in from the mechanical point ofview.

We note that, a bit later in 1931, Birkhoff, B(1931) (and also Khintchine)could, moreover , prove that the convergence in (2) holds almost everywhere aswell.

This progress led to the birth of an independent branch of mathematics:ergodic theory. This theory then began his autonomous evolut ion within mathematics and several sub-branches were also born. Just to mention some , one ofthem studies various forms and generalizations of the ergodic theorems , anotherone stronger forms of stochasticity, a special bran ch - quite int eresting for ourpresent discussion - investigates the ergodicity of particular systems, amongthem those arising from mechanics, a further one the isomorphism problem ofvarious dynamical systems,etc .

§3. Proving the First Relevant Theorem(from Neumann to Sinai, from 1931 until 1970)

The methods for establishing the ergodicity of mechanic al systems came froma different though related domain, from the theory of dynamic al system. In 193839, Hedlund, He(1939) and Hopf, Ho(1939) found a method for demonstratingthe ergodicity of geodesic flows on compact manifolds of negative curvature.Their main conceptual discovery was that the so called hyp erbolic behaviour ofdynamical syste ms could imply and, in fact , did imply ergodicity in the aforementioned models .

Hyperbolicity means, in other words, instability, i.e. the exponent ial divergence of trajectories starting arbitrarily close to each other, or else sensit ivity tothe initial conditions. The simplest exa mple of a hyperbolic system is Arnold'sfamous cat , the linear automorphism of the torus (cat stands for a Continuous Automorphism of the Torus) . Ind eed , if we consider the map TA of the

2-torus 1R2 I 7i} onto itself defined by the (hyperbolic) matrix A = G ~),then we see that the image of the cat get s expanded in one direction and contracted in a transvers al one, with the expansion (contraction) being the st rongest(weakest) in the eigendirect ion of the matrix corr esponding to the eigenvalueAu > 1(A8 < 1).

In 1942, very soon afte r Hedlund's and Hopf''s fund amental results , the Russian physicist , N. S. Krylov discovered that systems of elast ic hard balls showan instability simil ar to the one observed at geodesic flows on manifolds withnegative curvature, cf. K(1942) . This finding and the progress of the ideas ofHedlund and Hopf in the theory of hyperbolic dynamical systems justified Sinai 's

Boltzmann 's Ergodic Hypothesis, a Conjecture for Centuries? 427

stronger version of Boltzmann 's ergodic hypothesis formulated in 1963 for theparticul ar system of elastic hard balls.

Hypothesis 3.1 (T he Boltzmann-Sinai Ergod ic Hyp othesis (S(1963))) .Th e system of N hard balls given on ']['2 or ']['3 is ergodic for any N 2 2.

Since mechanical sys te ms also have conserved quanti ti es, t his conject ur e isunderstood so that ergodicity is expec ted to hold 0 11 (connec ted components of)t he submanifold of the ph ase space spec ified by th e invari ants of motion.

The concept ua l surprise of this conjec ture compared to Bolt zmann 's originalformulation was that no large N was assumed. In fact , ergodicity (and furt herstronger mixing properties , like the K-property) was expec ted to hold for anyfixed N 2 2!

In 1970, Sinai, S( 1970) was able to verify this conj ecture in the case of N = 22-dimensional discs moving on the 2-torus ']['2.

Before giving an insight into Sinai 's approach, let us mention th e limitat ionsof this nice ergodic beh aviour for syste ms with a fixed number of degrees offreedom .

§4. App earance of Non-Ergodic Behaviour(Cosequences of the KAM-Theory, 1954-1974)

In nature, we have important exa mples of systems of inter acting particles (orbodies) that ar e st able and not unstabl e like sys te ms of hard balls.

The most strikin g example is - fortunately - the solar sys tem. The factthat it consists of bodies of differen t masses is not of great importan ce, moresignificant is the fact that here the interact ion is different .

The year 1954 brought two important discoveries. Kolmogorov's 1954 work,K(1954) and its later evolut ion - thanks first of all to the achievements ofArnold and Moser (in particul ar , A(1963) and M(1!J62)) in the 60's - indi catedthat we may well have a situation when invariant tori wit h dimension half of thatof th e phase space can fill a set of posi tive measure (we note that in completelyint egrabl e systems such invari ant tori do foliate the who le ph ase space). Another,not so explicit , warning came from the numerial work of Ferrni-Pasta-Ulam, FP- U(19 55) demonstrating that the asy mptot ic equipartit ion of th e energy ofmodes may fail. As to a detailed exposit ion of thi s experiment and its effects werefer to the survey H(1983).

In the 1974 work of Markus-Meyer , M-M (1974) summarizing the pr eviousprogress there were two importan t statement s out of which the first one is morerem arkab le for our discussion .

428 D. Szasz

Theorem 4.1 In the space of sm ooth Ham iltonians

• Th e nonergodic ones form a dense open subset;

• Th e nonintegrable ones form a dense open subset.

Without going into techni cal details we note that the state ments are formulated in the C OO-topology and a Hamiltonian is called ergodic if, for almost everyvalues of the energy, the system is ergodic on the corresponding submanifold ofthe phase space.

Thus, for generic Hamiltonians , we cannot expect ergodicity, and in thefinal sections of the pap er we will return to the question of what kind of ergodicbehaviour can then be expected for them. The forthcomin g discussion will befocused on the comparatively simple case of hard ball syst ems.

§5. Sinai 's Setup. Billiards (1970)

We start with a simple trick tradi tional both in mathematics and physics: instead of treating N particles we consider just one par ticle in a high dimensionalph ase space. More concretely: Let us assume, in general, that a syste m of N(? 2)balls of uni t mass and radii r > 0 are given on ']['V, the v-dimensional unit torus(v ? 2). Denote the phase point of the i 'th ball by (qi' Vi) E ']['v x lR v

• Theconfigurat ion space Q of the N balls is a subset of ']['N .v : from ']['N.v we cut out(~) cylindric scatterers:

c., = {Q = (ql , ''' ' qN) E ']['N .v :1qi - qj 1< z-} ,

1 :::; i < j :::; N . The energy H = ~ I:~ vf and the to tal momentum P = I:~ Viare first integrals of the motion. Thus, without loss of genera lity, we can assumethat H = ~ and P = 0 and , moreover , t hat the sum of spat ial components

B = I:~ qi = 0 (if P -# 0, then the cente r of mass has an addit ional conditionallyper iodic or periodic motion) . For th ese values of H ,P and B , the phase spaceof the system reduces to M := Q x S N .v-v-l where

Q ,~ { Q E Q \ Ul, « " NC' J '~ q, ~ 0}

with d := dim Q = N ·v - v , and where Sk denotes, in genera l, the k-dimensionalunit sphere. It is easy to see that the dynamics of the N balls , determined bytheir unifo rm motion with elast ic collisions on one hand, and the billiard flow{st : t E lR} on Q with specular reflections on 8Q on the other hand, areisomorphic and they conserve the Liouville measure dJ.l = const- dq -dv .

Boltzm ann 's Ergodic Hypothesis , a Conjecture for Centuries?

..,.. ..

> -....

Figure 1

429

We recall that a billiard is a dyn ami cal system describing the motion ofa point particl e in a connecte d, compact domain Q C ]Rd or Q C ']['d = Tord

,

d 2': 2 with a piecewise C 2-smooth boundary. Inside Q the motion is uniformwhile the reflection at th e boundary 8Q is elas t ic (the angl e of reflect ion equalsthe angle of incidence, cf. Figure 1). Since the absolute value of the velocity isa first int egral of motion, the ph ase space of our system can be identifi ed withthe uni t tangent bundle over Q . Nam ely, the configuration space is Q whilethe ph ase space is A1 = Q x Sd-l where Sd-l is the sur face of the unit d-ball.In other word s, every phase point x is of the form (q, v) where q E Q andv E S d-l ' The natural projections 7r : M ~ Q and p : M ~ S d-l are definedby 7r (q, v ) = q and by p(q, v) = v , respectively.

Suppose that 8Q = U~8Qi where 8Qi are the smoot h components of theboundary. Denote 8M = 8Q x Sd-l and let n(q) be the unit normal vectorof the boundary component 8Qi at q E 8Qi direct ed inwards Q . In billiards,isomorphic to hard ball systems , the scatterers are convex cylinders if N 2': 3,and are (strictl y convex) balls if N = 2. The observation of Krylov and Sinaiwas that a billiard with strictly convex scat te rers behaves like a hyp erbolicdynami cal syste m, whereas in one with just convex sca tte rers there is somepart ial hyp erbolicity. We will illustrate this observation after some definitions.

We say that a billiard is dispersing (a Sinai-billiard) if each 8Qi is strictlyconvex, and we say it is semi- dispersing if each 8Qi is convex. The billiards onFigures 2 and 3 are dispersing. Indeed, they correspond to the syste m of twodiscs on ']['2; the first one to the case R < 1/4 and the second one to the case1/4 < R < 1/2.

The third one is a semi-dispe rsing billiard given on ']['3 with two cylindricscatterers. This parad igm was the first semi-dispersing but not disp ersing billiard whose ergodicity was est ablished (d. K-S-Sz(1989)) .

The mechanism producing hyp erbolicity in a dispersing billiard can be seenthe best on Figure 5 borrowing the illustration from optics. Assume we havea st rict ly convex scatte rer on ']['d and imagine it is a mirror. Take, x = (Q, V) EM , and the codimension one hyperplane r t hrough Q in the configurat ion space

430 D. Szasz

Fig ure 2

Figure 3

... : :::':.t·.· ; .: : ..

.. ..";,.

.....; :.

Fig ure 4

perp endicular to the velocity V. By attaching to points of I' velocities identicalto V we obt ain a wavefront t in the phase space M . After one reflect ion from themirror sca tterer, our wavefront gets st rictly convex while the linear distancesmeasured on r get uniformly expanded. This mechanism is exactly t he oneproviding the (un iform) hyperbolicity of a dispe rsing billiard.

Sinai 's 1970 work used th e theory of uniforml y hype rbolic smoot h dynam icalsystems which had had an int ensive progress in th e 60s and culminated in th e1967 pap er of Anosov and Sinai , A-S(1967) . T he serious difficulty Sinai hadto cope with was t hat billiards were not smooth dynam ical syste ms. Indeed , if

Boltzmann 's Er godi c Hypothesis, a Conject ure for Cent ur ies? 431

Figure 5

a smooth wavefront gets reflected from a scat terer and it contains a tangency,then, though the reflect ed wavefront will be cont inuous, its second derivativewill have a jump at th e tangency. This circumstance causes serious technicaldifficulties: in smooth uniformly hyperbolic dynamical systems the st able andunstable invariant manifolds, the fund amental tools of th e theory are smoot hand unb ounded , whereas in billiard s th eir smooth components can be arbit rarilysmall.

§6. N = 2 Balls (1970-1987).Local Ergodicity of Semi-Dispersing Billiards

As mention ed earlier, Sinai , in 1970, in his celebra ted paper obt ained the firstrigorous result in relation to the Boltzmann-Sinai ergodic hyp oth esis: he couldshow th at N = 2 discs on the 2-torus 1['2 was a K-system.

In fact , his result was formulated for 2 - D dispersing billiards (Sinaibilliards) with a finite horizon. A billiard has finit e horizon if th ere is no collisionfree tr aj ectory in it. This condition is fulfilled by a two-billi ard if R > :t (cf.Figure 3). In this case the configurat ion space consist s of four connected component s, and, of cours e, ergodicity is claimed on each of th em. For the case ofR < t (d. Figure 2), a 2 - D billiard with infinite horizon, th e correspondingresul t was proved by Bunimovich and Sinai in 1973, B-S(1973). On th e basis oftheir work it was und erstood th at a 2 - D disp ersing billiard was ergodic.

A multidimensional genera lization of their th eorem was only obtained in1987. Indeed , Chernov and Sina i, S-Ch (1987) were, in genera l, investigatingsemi-dispersing billiards and introdu ced th e basic notion of suffici enc y of anorbit or equivalent ly of a phase point . The main consequence of sufficiency

432 D. SZ3sZ

is that , in a suitably small neighbourhood of a sufficient point , the system ishyperbolic, though not uniformly. Next we present this notion in its minimalform as suggested in K-S-Sz (1990) .

Our starting point is Figure 6, similar to Figure 5. It shows that , if a scattereris not strictly convex but just convex, like e. g. a cylinder , then the image ofthe hyperplanar wavefront r with parallel velocities will not be curved in thedire ctions parallel with the constituent subspace of the cylinder, but in thetransversal directions, only. However, the uncurved neutral directions can stilldie out after several reflections on differently oriented cylindric (or , in general,convex) scatterers .

Now for the definition of sufficiency. Assume that s[a,b]x is a finite trajectorysegment, which is regular, i.e. it avoids singularities.

Let sa X = (Q,V) EM and consider the hyperplanar wavefront f(sa x) :=

{(Q + dQ,V) : dQ small E lRd and (dQ, V I = O} (by denoting 1l'(x) = Q forx = (Q,V) we see that , indeed, 1l'(i) is part of a hyperplane) .

We say that the traj ectory segment s[a,bl x is suffi cient if 1l'(Sbf) is st rict lyconvex (see Figure 7). (To obtain a geometric or optical feeling of this notion,the reader is again suggested to imagine mirror-surfaced scatterers.) A phasepoint x E M is sufficient if its trajectory is sufficient (i.e. it contains a sufficienttrajectory segment) . In physical terms, sufficiency of a trajectory segment meansthat , during the time int erval [a ,b], the trajectory of x encounters all degrees offreedom of the system.

Figure 6

Boltzm ann 's Ergod ic Hyp othesis, a Conjecture for Ce nt ur ies? 433

~ . . ~~y~~~~ ....~ CO LLIS IONS

F igure 7

If a traj ectory segment is not sufficient , then the curvature of 1l"(Sbt ) at1l"(Sbx) necessarily vanishes in certain directions forming the so-called neutralsubspace. Simple geometric considerations (cf.K-S-Sz(1990)) show that a sufficient traj ectory segment generates an expansion rate uniformly lar ger than 1 insome neighbourhood of the point sa x. Then, by Poincare recurrence and theergodic theorem, it is not hard to see that , in some neighb ourhood of sax, therelevant Lyapunov exponents of the system are not zero. In ot her words, in thisneighb ourhood , the syste m is hyp erb olic. This observation should motivate thenon- trivial

Theorem 6.1 (Fundamental theorem for semi-dispersing billiards (SCh( 1987)) Assume that a semi-dispersing billiard satisfies some geometric conditions and the Chernov - Sinai ansa tz, a conditi on strongly connected withthe singulari ties of the sys tem.

If x E M is a suffi cient point, the it has an open neighbourhood U in thephase space belonging to one ergonent (i.e. ergodic component).

(A sim plified and suitably generalized version of this theorem, the so-called't ransversal fund amental theorem' was given in K-S-Sz(1990). Moreover , a version of the fund ament al theorem formulated for symplect ic maps wit h singularit ies can be found in L-W (1994).) The property expressed in the statement isusually called local ergodici ty. If almost every phase point of a semi-dispersingbilliard is sufficient, then , of course, it may have at most a countable numberof ergonents . In some cases it is not hard then to derive the global ergodicity ofthe syste m, i.e. to show that there is just one ergonent in the phase space. Not ethat it also follows from the genera l theory that , on each ergonent, the systemis Kolmogorov mixing. A much imp ortant consequence is thus t he following

Coro llary 6.2 (S-Ch( 1987)). Every dispersing billiard is ergodic, and,moreover, is a K-fiow. In particular, the sys tem of N = 2 balls on the u-torusis a K-fiow if r < ~ .

(For det ails, see K-S-Sz(1990).)

434 D. SZ3.sZ

§7. N ;:::: 3 Balls (1989- ).Global Ergodicity of Semi-Dispersing Billiards

With the fundamental theorem for semi-dispersing billiards in mind , the proofof t heir globa l ergodicity boils down to

• first demonstrating the Chernov-Sinai Ansatz, an imp ortan t condit ion oft he fund amental t heorem, and

• to then showing that the subset of non- sufficient points is a to pologicallysmall set of measure zero; for instan ce, its topological codimension is notsmaller than two.

In Sz(1993), we gave a sket ch of the st rategy worked out in our pap ers with A.Kr arnli and N. Simanyi for the core par t , and here we will just list the mainresul ts obtained so far."

• in 1991, Kramli , Simanyi and the present au thor , (KSSz-91) demonstratedt he K-p rop erty of N = 3 balls on the v-torus whenever v 2:: 2;

• in 1992, agai n t he previous aut hors, (KSSz-92) improved their met hods toget the ergodicity of N = 4 ba lls on the v-torus whenever v 2:: 3;

• in 1992, Simanyi, 8 (1992) was able to est ablish the so far strongest resul tfor hard ball syst ems : t he system of N 2:: 2 ba lls is ergodic on the v-toruswhenever v 2:: N ; his method is based on his Connect ing Path Formulacharacterizing the neutral subspace of a t ra jectory segment.

T he configurat ion of the cylindric scatterers of a billiard isomorphic to a hardball system inherits the permut ation symmet ry of the balls. A natural generalizati on of hard ball systems is to investi gate cylindric billiards in general, i. e.billiards with solely cylinders as scatterers. To this end cons ider compact affinesubspaces L i

: 1 ::; i ::; N , N 2:: 1 in the d-torus T d (wit h dim L i ::; d - 2) ,and denote c' := {Q := (ql ," " qd) : dist(Q, L i ) ::; r' }, 1 ::; i ::; N whereeach r i is positive. The billiard in Q := T d \ (U~lC i

) is a billiard with cylindricscatte rers.

6 Most recently, in the Summer of 1994, Simanyi and SZMZ, were able to provea weaker form of Sinai's hypothesis in the general case N , t/ 2:: 2: typical hardball systems are hyperbolic, i. e. the relevant Lyapunov exponents are not zero almost everywhere. For details and further developments see Simanyi's survey in thisvolume.

Boltzmann's Ergodic Hyp othesis, a Conject ure for Cent ur ies? 435

For cylindric billiards the following results have been obtained:

• in 1989, Krarnli , Simanyi and the present author, K-S-Sz(1989) considereda 3-dimensiona l orthogonal cylindric billiard (cf. Figure 4) ; they obtainedits K-p rop erty and t hus this was the first semi-dispersing - but not dispersing - billiard whose ergodicity was shown.

• in 1993, motivat ed by a question of John Mather, the present authorstarted a systemat ic study of cylindric billiards and found a sufficientand necessary condition for th e ergodicity of a class of them : for orthogonal cylindric billiards, cf. Sz(1993), Sz(1994) . These are characte rizedby the propert y that the constituent subspace of any cylindric scatte reris spanned by some of the coordina te vectors adapted to the orthogonalcoordinat e syste m where T d is given;

• in 1994, Simanyi and the present au thor , S-Sz(1994) found necessary andsufficient condit ions for the K-property of a to ric billiard with two arbit rary cylindric scat te rers .

Since the class of cylindric billiards is relatively simple, one can hop e for general necessary and sufficient cond itions for the ergodicity (and the K-property)of these systems. Indeed, we next formul ate a conjecture containing a genera lsufficient condit ion.

Conjecture 7.1 (SZ8.SZ, 1992). Assume that the configuration doma in ofa cylindric billiard is connected, and no pairs of the scatterers are tangent . Ifthere is at least one sufficien t point, then the billiard is K.

§8. The Boltzmann-Sinai Ergodic Hypothesisin Pencase Type Models

In ord er to resolve some difficulties on the way to est abli shing the BoltzmannSinai ergodic hypothesis, Chernov and Sinai , S-Ch(1985) suggested the study ofa quasi-on e-dimensional model of hard balls. It is given on an elongated torusof the type (L1!'l) x 1!'v -l where L is a sufficiently large number compa red to R(see Figure 8) . The main assumption is

ensur ing that the ord er of balls (in the direction of L1!'l) is invari ant under thedynam ics. Thus the mod el, which was called by Chernov and Sinai a pencase, isrealizable if 2 ::::: v ::::: 4. If we number the balls in their order: 1,2" " ,N, then

436

.......~~ '

Figure 9

a particular feature of the model is that only the pair s of consecut ive balls (i.e.{1,2} ,{2, 3}, ·· · , {N, l }) can int eract.

T he first result for a pencase typ e model was reached in 1992 by BunimovichLiverani-Pellegrinot ti- Sukhov. Inst ead of a torus their model lives in a domainwith dispersing boundari es (see Figure 9) an d the sizes of the domain ensurethat

• each ball is rest ricted to a fundament al domain of the "pencase" (thethroats between them are smaller t han 2R);

• between consecut ive collisions of a particular ball , it should always hita dispersing boundary;

• the pairs of balls in neighbouring domains can, indeed , interact .

A billiard of this ty pe is reali zable in arbitrary dim ension and the results ofthe aforementioned authors was that the syst em was K . This par ti cular modelwas , in fact , the first one where the Bolt zmann-Sinai ergodic hypothesis gotsettled for any N and v ~ 2.

Theorem 8.1 (B-L-P-S(1992)) . Th e B-L-P-S pencase is a K-system forany N ,l/ ~ 2.

For some time it seemed so that t he proof of ergodicity for the originalChernov-Sinai pencase was not easier than that for general hard ball sys te ms.Never theless, - with Nandor Simanyi - we could recently demonstrate thefollowing

Theorem 8.2 (S-Sz (1994-B)) . Th e Chern ov-Sinai pencase is a K- systemfor any N ~ 2, l/ = 4. If l/ = 3, then the system has open ergodic components.

Boltzmann 's Ergodic Hypothesis, a Conjecture for Centuries? 437

The restriction v i- 2 seems, at present , import ant whereas t hat of v < 5only arises since the model, as invented by its aut hors, does not exist for v 2': 5.On e could, however, introduce less realisti c models that do exist for v 2': 5, too,and for them our proof would also work but we do not want to stay on t hem.

There is, however , another , more natural way to introduce models witha pen case-typ e interaction in any dimension. Consider, nam ely, N balls, numbered 1, 2, .. . , N on the unit torus '['1/ . The rest rict ion is that only pairs of ballswith neighbouring numbers, i.e. again only the pairs {I , 2}, {2, 3}, .. . , {N 1, N} , {N, I} interact while other pairs can go through each other . This billiardis, of course, again a cylindric one.

Theorem 8.3 (S-Sz (1994-B)). The system with pencase-type int eraction isa K-system uiheneuer N 2': 2, v 2': 4. If v = 3, then the system has open erqodiccomponents .

(Froeschle(1978) (ef. H(1983)) introduced the notion of connectivi ty as theratio of the number of particles a given particle can interact with and of thenumber of all particles. His experiments suggested that this ratio can be relatedto the good ergodic properties of a system; in particular, below a critical valueof the connect ivity, a significant fraction of the ph ase space is occupied withinvariant tori . Our theorem shows, however , that , for hard ball systems , theergodic behaviour already appears at a connect ivity arbitrarity close to zero.)

§9. Ergodicity of Systems with a Fixed Numberof Degrees of Freedom

From the work of Markus-Meyer mentioned in sect ion 4 we know that ergodicHamiltonian s are in a sense except ional. Nevertheless, it makes sense to look forpossibly more of them since the mechanism s occurring in these can also help tounderst and the onset of chaot ic behavior, for inst an ce, the appeara nce of a largeergodic component in nonergodic sys tems.

In sections 5-7 we discussed billiard sys te ms . Here we mention three classesof Hamiltonians, for which Donnay and Liveran i, D-L(1991) could , in 1991,demonst rate ergodicity. These are systems of N = 2 particles on '['2 interactingvia a rotation-invariant pair potential V(T). These system have the same conserved quantities as the system of two hard dis cs and we assume that vf + v~ =1, VI + V2 = 0, qi + q2 = O. We do not give here all the condit ions since we aremainly interested in the qualitative description of these interact ions.

Assume in all cases that for some R > 0

1. V(T) = 0 if T 2': R;

2. V(T) E C2(O, R) ;

438 D. Szasz

3. limr---+o r2V(r ) = 0;

4. for h(r ) = r 2 (1-2V (r )) , and for except one value of r E (0 , R ) h' (r) > O.

Potentials in the first class are repe lling ones (see Figure 10). The addit ionalcondition besides (1) - (4) is now

5. V (R- ) = 0 and V' (R- ) < o.

Then , under some more condit ions, t he system is K. As it is evide nt from thecondit ions, V is, t hough conti nuous, not C 1 at r = R (see Figure 11). Indeed , t hejump of V ' in R as requi red by (5) plays the same role as the effect of a reflectionin a disp ersing billiard . T his phenomenon was first observed by Kubo in 1976(K(1976) ), and he, an d later he and Murata, K-M (1981) could already establishthe K- and the B-property of such syst ems under more restr ict ive cond it ionsthan those of Donnay and Liverani. It is a natural question whether t he Kubotype singularity can also lead to ergodicity in the case of several particles. Infact , we recall t he following

Problem 9.1 (Liverani-Szasz , 1990). Let N = 3, v = 2. Is it possible to finda Kubo-tupe interaction (i.e. one satisf ying the conditions (1) - (5) formulatedbef ore) suc h that the sys tem is ergodic?

A simpler problem could be the generalization of the Kubo-Donnay-Liveraniresul t for the case N = 2,v :::: 3 though, as observed by Woj tkowski , W (1990- C),in the multidimensional case new, unpleasant phenomena may arise .

T he second class invest igated by Donnay and Liverani contains attractingpotenti als. In 1987, Kn auf, K (1987) showed that for attracting interact ions wit hsingularit ies at r = 0 of the ty pe - 2 (1~.!. ) ' n = 2,3, 4, . . . , the system was

r nergodic. Donnay and Liverani 's main achievement was that they could get ridof t he assumpt ion that n was an integer (see F igure 10). Their main condit ionbesides (1) - (4) is

6. V i(r ) :::: 0 if v E (0, R ) and V( R) = V ' (R) = o.

Figure 10

Boltzmann's Ergo dic Hyp othesis , a Co njecture for Centuries? 439

Figure 11

F igure 12

From t he conceptual point of view the most remarkable is their third classsince here the potenti al is everyw here smooth. The basic feature of interactionsin the third class is that , for some Tc < R, the circle of radius Tc is a closed orbi t(see Figure 12). Interest ingly enough thi s orbit plays the role of a singularity.

In all cases, the existe nce of potenti als satisfying the aforeliste d condit ionsis proved . For a given potenti al satisfying the appropriate requirements thenergodicity is fulfilled at sufficient ly high energy. It is worth notin g that havingproved first that the Lyapunov exponent is non-zero, the proof of ergodicitycan be obtained by a suitable ada ptat ion of the fund ament al theorem for semidispersing billiards (d. section 6).

An interesting class of models was introduced and studied by Wojtkowski ,W (1990-A) and W(1990-B). Here a one-dimensiona l system of N particles ofdifferent masses moves in an external field , and the int eraction is elast ic collision.T he non-vanishing of Lyapunov exponents has been proved in severa l cases , bu testablishing globa l ergodicity st ill seems to be difficult.

440 D. Szasz

§1O. Ergodicity of Systemswith an Increasing Number of Degrees of Freedom

The situat ion when the number of particles increases exac tly corresponds toBoltzmann 's original question which - in modern terminology - could soundas follows: find , for a generic Hamiltonian, the asy mptot ic behaviour in thethermody na mic limi t . T his question is still not formulated precisely. From t hevarious possible ways, t he right one should , of course, be selected as dict ated bythe main applications . At present , as it seems to me, a very important application should be in the field of t he derivation of hydrodyn ami c equation frommicroscop ic, Hamiltonian principles. It is clear t hat , t he so far st rongest methodworked out in the last decad e by Varadhan and his coworkers, O-V-Y(1993)would require a form related to Boltzmann 's hyp othesis but we can st ill notselect the right form (we note that the results ob tain ed until now are valid forstochastic syste ms and not for purely Hamiltonian ones).

The conceptua lly simplest and wide most known form of a hyp othesis is thefollowing: denote as before the number of par ticles by N , and by p(N) t herelati ve measure of the ph ase space occupied by invari an t to ri. For simplicity,the interaction is fixed and 1G- = canst (for defini teness, we ass ume that t he

system lives on the to rus v t ]'V). T hen t he first conjecture is that p(N ) -+ 0as N -+ 00. A st ronger conjecture would then requ ire that the complement tothe set of invari ant to ri contains a large ergodic component whose measure getsclose to one.

Henon (1983) and Galgani (1985) discussed in det ail the situation and theconnection of these conjectures to the one on the limiting equipartiton of energybetween the modes of t he syste m. The conclusion is that the picture is not clearat all. There are , on one hand, interactions when numer ical work of Froeschleand Scheidecker (1975) indicates that p(N ) -+ 0 as expected. They investi gateda one-dimensional model with t he Hamilton ian

On the ot her hand, t he famous Fermi - Pasta - Ulam (1955) experimentsupported doubts about t he conjectures by detecting the failure of t he limi ting equipartit ion of energy. This was also a one-dimensional model with theHamilton ian

N N-l

H = ~D; + LV(qi+l - qi )1 1

where V (q) = ~ q2 + o:q3.These works generated a vivid int erest in the problem . For the cont radictory

views about it , the reader is suggest ed to consult the aforementioned papers of

Boltzmann's Ergodic Hypothesis , a Conj ecture for Centuries? 441

Henon and Galgani and for a more recent review that of Galgani- GiorgiliMartinolli-Vanzini (1993).

As I learnt from Gregory Eyink , for establishing hydrodynamic equat ions inthe sense of the approach of Varadhan 's method, a weaker form of the conject urewould also be sufficient. It is not necessary to have one large ergodic component. It seems that a weakly increasing upper bound for the number of ergodiccomponents and, of course, a good upper bound on p(N) could be sufficient.The picture here , however , needs more elaboration and the problems seem verydifficult .

§11. Ergodicity of Systems with an Infinite Numberof Degrees of Freedom

Since the sit uation with large but finite syste ms is so complicated, I expect thatthe solution of equilibrium statist ical physics should be borrowed . Whereas evena rigorous definition of a phase transit ion in a finit e system - not speakingabout its demonstration - is not an easy task , the quest ion get s much simplerfor infinite syste ms. In my view, first the ergodici ty of infinite syste ms shouldbe understood.

The very first result for an infinite syste m was obtained in 1971 by Sinaiand Volkovysky for the ideal gas: it was shown to be a K-syst em (V-S(1971)) .(A weaker resul t was obtained by Dobrushin already in 1956, see D(1956) .)For the first glance this sounds as a surpris e since in the ideal gas there isno velocity mixing at all. Indeed, in the formul ation of ergodicity one shouldbe a bit caut ious. By denoting the phase space by M = {{(qi,Vi) : i E Z} :{q;} is locally finite}, the equilibrium measure is PA({qi : i E Z}) 0 Il F(dVi)where P), is a Poisson measure with density oX and F (dv ) is an arbit ra ry nondegenerate probability distribution in ]R"' , and ergodicity holds with respectto thi s invar iant measure. The proof reveals an apparent ly new mechan ism ofergodicity : mixing - understood, of course , in time - is the result of theinit ial spati al mixing. In other words: the equilibr ium measure is Poisson, i.e.a measure with independent increments. Now as time proceeds, in a fixed box ofour observat ion, particles start ing from more and more distant intervals appearand their numbers are, rou ghly speaking, ind epend ent. This phenomenon canbe proved to provide mixing in time.

The same observation was used , in more delicate arguments, for showing theK-property for different var iants of the Rayleigh-gas , among others, by Goldst ein-Lebowitz-Ravishankar (1982) , Boldrighini- Pellegrinotti-Presut ti-Sinai- Soloveychik (1984) and L. Erdos-Tuyen (1991) . In these models only one particleinteracts with all t he other ones and the equilibrium measure is still Poisson.A relat ed model is the Lorentz gas where - similarl y to the ideal one - th ereis no interaction between the particles, but the dynami cs of each particle obeys

442 D. Szasz

a strong mixing in space. Based upon this mixing, Sinai demonstrated the Kproperty in S(1979 ).

Now a problem which I find very interesting and quite realistic is the following one:

Problem 11.1 (Szasz, 1990). Consider- an infini te pencase obtain ed asN ---+ 00 of the fin it e ones was in troduced in section 8. Prone that the naturalGibbs m easur-e is er-godic. (Her-e, of cour-se, the possible values of the dim ensionar-e v = 2,3, 4.)

(Infinite models also ra ise t he quest ion of existe nce of the dynami cs, butfor thi s model it was answered affirmat ively by Alexander , A(1976). ) In theproof of ergodicity two mechanisms can be exploited : the hyp erbolic behaviourof the interaction as don e for the finit e pencase or the spatial mixing of theequilibrium distribution as in cases of the ideal gas or the Rayl eigh one . Atpresent , however , I do not see an easy way for any of these possibilities, inparticular , for the second one . For the first one it is a natural idea to startbuilding up the hyp erb olic theory of infinite-dimensional dynami cal syste msand trying to define, for instance , the stable and unstable invari an t manifoldsfirst , and then to prove their existe nce .

In sect ion 10 we already mentioned the problem of the derivation of hydrodynami c ty pe equa t ions. A sufficient condit ion in some cases for the method ofO-V-Y(1993) to work is the following ergodicity ty pe condit ion: ever-y "requlor"state, inv ar-iant with r-espect to both translations in the space and the dynamics,is a mixtur-e of canonical Gibbs m easur-es. This property is appar ently strongerth an ergodicity, bu t , as remarked in F-F-L(1994) , to prove such an ergodicity fordeterministic Hamiltonian systems is st ill a formiddable unsolved problem. (Infact , Fri tz, Fun aki and Lebowi tz verify thi s property for a random Hamiltoniansyste m, and their pap er is, moreover , also recommend ed for fur th er reference.)We add that regular ity above means that the state has finite relative ent ropy(per uni t volume) wit h respect to the Gibbs measur e. This assumption implies,in particular , that the condit iona l distribution in any finite volume A, given theconfigurat ion outside A, is absolutely cont inuous with respect to the Leb esguemeasure.


In this lecture I have been concent rating on the history of Bolt zmann 's ergo dichyp othesis. I think that the second half of the t it le of t he talk is already justifiedif we focus our interest to just t he question of ergodicity. After more t ha n onehundred years, ergodicity is still not established in the simplest mechani calmod el, in the syste m of elast ic hard balls though I expect we are not far froma solution. But as to generic interacti ons, even the questions ar e not clearly posed

Boltz mann 's Ergodic Hypothesis , a Conject ure for Centuries? 443

and it might well be that there will not be a final underst anding after t he nexthundred years eit her. And we have not to uched upon more delicate , physicallyfundamental pro perties for whose proofs one should refine the methods used inst udying ergodicity of t he system involved . Without aiming at completeness wejust mention the problems

• of the decay of correlations (d. Ch(1994) ; here and in t he forthcomingcases only the last reference, I am aware of, will be provided , where furtherones can also be found) ,

• of the convergence to equilibrium, K-Sz(19S3) ,

• of the calculation of and bounds on the entropy of mechanical systems,Ch( 1991),

• and , finally, of the recur rence propert ies of such systems, K-Sz(19S5).

Acknowledgements. Thanks are due to G. Eyink and J . Fritz for their remarksabout the relation of ergodicity and the hyd rod ynamic limit transition. I amalso much grateful to M. Herman and B. Weiss for their helpful remarks on t hehistory of the ergo dic hypothesis and some references .

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Boltzmann 's Ergod ic Hyp othesis , a Co nject ure for Cent uries? 445

[Ho(1 939)]

[K(19 87)]

[K(19 54)]

[K( 1931)]

[K-S-Sz(1989)]

[K-S-Sz(1990) ]

[K-S-Sz(1991)]

[K-S-Sz(1 992)]

[K-Sz(198 3)]

[K-Sz(1985)]

[K(1942)1

[K(1976)]

IK-M(1981)]

[L(1975)]

[M(1990)j

[L-W(1994)]

[M-M(1 978)]

[M(1962)]

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446

IN(1931)]

[O-V-Y(1993)]

[S(1992 )]

[S-Sz(1994-A)]

[S-Sz(1994-B)]

[S(1984)]

IS(1963)]

[S(1970)[

[S-Ch (1985)]

[S-Ch (1979)]

IS-Ch(1987)]

[Sz(1993)]

ISz(1994)j

[V-S(1971)]

[W(1990-A )]

[W(1990-B)]

[W(1990-C)]

[Y (1992)1

D. Szasz

Neumann, J . von (1929): Beweis des Er god ensatzes und des HT heorems in der neuen Mechanik. Zeitschrift fiir Physik 57 30- 70Olla , S., Varadhan, S.R.S ., Yau , H.T . (1993): HydrodynamicLimit for a Hamiltonian System with Weak Noise . Commun.Math. P hys. 155 523-560Sima nyi, N. (1992) : T he K-p roperty of N billiard bal ls I, Inven t .Math. 108 521- 548; II . ibide m 110 151-172 (1992)Sima nyi , N., Szasz , D. (1994) : T he K-property of 4-D Billiardswith Non-Ort hogona l Cy lindric Scatterers. J . St at. P hys . 76 587604Sima nyi, N., Szasz, D. (1994) : T he K-property of HamiltonianSystems with Restricted Hard Ball Interact ion. (in pr eparation)Simon, B. (1984): Fi fteen Problems in Mathematical Physics.Perspectives in Mahtematics, Anni versary of Oberwolfach,Birkha user, Bost on 423- 454Sin ai, Ya .G . (1963): On the Foundation of t he Ergod ic Hypothesisfor a Dynami cal Syste m of Statisti cal Mechanics. Dokl. Akad .Nauk SSSR 1531261-1264Sina i, Ya.G. (1970) : Dyn ami cal Systems with Elastic Reflecti ons.Usp. Mat. Nauk 25 141- 192Sina i, Ya.G ., Chernov , N.!. (1985): Ergod ic Properties of SomeSystems of 2- D Discs and 3- D Spheres . manuscriptSinai, Ya .G . (1979) : Ergodic P ropert ies of t he Lorent z Gas .Funkcionalny Analiz i P ri!. 13/ 3 46-59Sina i, Ya .G ., Chern ov, N.!. (Hl87): Ergod ic Properties of SomeSyst ems of 2-D Discs and 3- D Spheres. Usp. Mat . Nauk 42 153174Szasz , D. (1993) : Ergodicity of Class ical Hard Balls. P hysica A194 86- 92SZ[ISZ , D. (1994): T he K-p rop erty of 'Orthogonal' Cy lindric Billiards. Commun. Math. Phys. 160 581- 597Volkovissky, K.L ., Sinai, Ya. G. (1971) : Ergodic P roperti es of anIdeal Gas with an Infin it e Number of Degrees of Freedom (inRu ssian). Funkcina lny Ana!. i Prim. 5 19- 21Wojtkowski, M. (1990): A System of One-Dimensional Balls wit hGravity. Co mmun. Math . Phys. 126 425-432Wojtkowski, M. (1990) : The System of One-Dimensional Balls inan Ex ternal F ield . Corn mun. Math. Phys. 127 425-432Woj tkowski, M. (1990): Linearl y Stable Or bits in 3 Dimen sion alBilliards. Commun. Ma t h. Phys. 129 319- 327Yoccoz, J.- C ir. (1992): Travaux de M. Herm an sur les Tores Invaria nts . (Sem ina ire Bourbak i, Vol 1991/ 92, 44) Asterisque 206Ex p. No 754 311-344

Author Index

Anosov, D. 3, 94~96 , 98, 100, 101, 109,197, 317, 345, 430

Baldwin, P. R. 128, 197, 199Bernoulli 58, 60, 68, 78, 84, 93-95, 115,

150, 191-193, 196-199, 205, 208, 236Boltzmann, L. 4, 29, 31, 32, 34 , 66,

132-134, 137, 147, 154,218,233-237,246, 248, 251, 254, 255, 260, 271, 276,294, 296, 300, 301, 312, 322-328, 330,335, 336, 341, 345, 348, 358, 407 ,422-425, 427 , 431 , 435, 436, 440, 442

Burns, K. 189, 190

Cheng, J . 202Ch ernov, N. 1. 2, 3, 9, 13, 17, 52, 54,

56, 58, 64-66, 71, 72, 74, 89, 96, 97,127, 129, 137 , 140, 141, 147, 152, 153,190, 192, 360, 362, 369 , 431, 433-436

Cohen, E. G. D. 3, 29, 36-39, 95, 117,206, 317 ,353,362,411 ,414

Donnay, V. 180, 196, 198, 199, 437 , 438

Gallavotti , G. 95, 206, 312, 317, 411,423

Gerber , M. 189, 190Gibbs, J .W . 94,95, 136-141 ,208-210,

235, 321-323, 327-330, 344, 369,373-377, 379, 395, 442

Katok, A. 9, 13, 66, 70, 78, 188, 190Knauf, A. 197, 438Kolmogorov, A. N. 3, 4, 78, 121, 122,

209, 232, 236, 240, 286, 300, 301, 312,327, 356, 373, 424 , 427 , 433

Krylov, N. S. 2, 10, 52, 96 , 192, 426,429

Kubo, R. 3, 146, 151-153, 196, 286,332, 334, 335, 338, 339, 358, 359, 403,404 , 438

Liouville, J . 13, 53, 60, 64, 65, 68, 73,114, 122, 123, 136, 138, 149, 150, 163,208 , 209, 322, 329, 343 , 349, 351, 356,371, 379, 423, 428

Liverani , C. 3, 52, 62, 64, 65, 96, 180,196, 436~438

Maupertuis, P. de 192, 197Maxwell, J . C. 233, 245, 260, 264 , 275,

276, 358

alia, S. 208Oseledec, V.1. 185, 283

Rudolf, D. J. 193Ruelle , D. 4, 94, 95, 123, 206 , 292, 317,

344, 350, 351, 369

Sirnanyi , N. 2, 3, 51, 82, 202, 434~436Sin ai , Ya.G. 2-4, 10, 12, 13, 17, 19, 51,

52, 54, 56- 58, 61, 62, 64-66, 71, 72,74, 90, 94-96, 105, 109, 112, 113, 116,121-124, 127, 136, 137, 139-141 , 163,190, 192, 194, 196, 206, 217, 232, 236,238, 240, 283, 286, 288, 300, 301, 312,327, 335, 344, 350, 356, 373, 426~431 ,

433-436, 441, 442Strelcyn, J .-M . 13, 66, 70, 78, 188

Tasnadi, T . 206

Var adhan, S. R. S. 208, 440 , 441

448 Author Index

Wojtkowski, M. P. 51, 62-66, 135, 180 ,186, 187 , 201-203, 207, 438, 439

Yau, H. T . 208Young , L.-S. 3, 89, 96-98, 192

Subj ect Index

absolute cont inuity 70, 185advance (of a collision) 79- 82ad vect ion-diffusion equation 396, 397Alexandrov space 9, 14Anosov maps 109- smoo t h 3- wit h singular ities 3Anosov property see hyperbolicityAnosov systems 94-96, 101arcw ise connected set 55, 56att rac t ing par t icles in a line 204ax iom A 94, 95ax iom C 95

B-m ixing (Bernoulli mixi ng) 58, 84Baker maps 318, 321, 330, 344, 359-361BBGKY hierar chy 218, 220, 225, 248Bernoulli prop er ty 58, 68 , 78, 84,

93- 95, 115, 196, 197bill iard 2, 3, 9, 10, 12, 13, 15-1 8, 22,

51, 52, 54, 56, 59, 62 , 63, 65-68, 71-73,76, 90, 95-97, 105, 106, 109, 112, 113,116, 123, 126, 127, 146, 148, 149, 157,160, 162, 163, 165, 166, 168, 173, 174,17~ 180, 189, 192, 194, 197, 200, 201,205, 206, 210, 244, 247, 250, 264, 267,279, 280, 283- 288 , 291, 293, 326 , 334,335, 422, 428-431, 433-439

- dispe rsing 2, 10, 15, 96 , 97, 127, 146,157, 162, 163, 165, 166, 173, 192, 244,250 , 264, 267, 283, 429-431, 433 , 438

- focus ing 66, 284- semi-dispersive 3, 51, 52, 56, 58, 68,

71, 72, 284Birkhoff ergodic t heo rem 91, 92, 123Boltz mann- ent ropy see ent ropy

- H-function 328- H-theorem 233, 324, 425Bolt zm ann constant 32, 137 , 246, 294,

300, 323, 407Boltz mann equat ion 29, 31, 32, 34, 133,

233-237, 248, 251, 253- 255, 260, 271,296, 301, 312, 323-327, 335, 341, 423

Boltzmann formu la 132, 134Bolt zm an n-Sinai ergodic hyp ot hesis 66,

427, 431, 435, 436box bill iards 59br an ch of a traj ectory 55, 171Browni an approxi mation 3Bunimovich stad ium 62, 71Burnet t coe fficient (see also transport

coefficient) 115, 337, 353

capped sys te m of par t icles in a line 203cent ral limit theorem 3, 61, 91-95, 104,

108, 115, 150, 155, 157, 160-1 62, 166,172, 176, 192

chao t ic 1, 3, 71, 93, 95, 97, 128, 157 ,206, 218, 232, 233 , 236, 238, 274-276,284,317,329,345,346,348-350,355,359, 361, 371, 375, 378, 380, 389, 390,397, 402, 437

chaot ic hyp othesis 206Chebys hev polynom ials 43, 44Chernov-Sinai ansatz 74, 434Cla usius entropy see entropyclock speed 252-257, 260-263, 276clock value 252- 255, 257, 259-263cluster ex pa ns ion 31-33, 248coa rse graining (see also entropy ) 330,

368 , 369 , 371, 372, 376, 379, 394, 396,402 , 412-414

collect ive modes 275, 279, 280, 312

450 Subject Index

collision frequency 245-247, 252, 254,259-262, 269, 290, 292 , 297 , 301, 312

collision graph 57, 59, 79, 80, 83collision map 183, 284 , 294 , 299- lineari zed 282, 285, 299collisions 1, 3, 9, 11, 13, 14, 17-22, 29,

31-37, 39- 47, 52, 53, 57, 60, 61, 64,73, 79-83, 90 , 96, 105, 111-1 14, 116,126, 127, 132, 133, 135, 157, 163, 181,183, 187, 188, 190-194, 196,201, 202,205-207,217-221 ,225, 233, 236-242,245 , 248, 249, 252, 253, 259, 261, 262 ,264-266, 268, 273, 276, 281, 282, 284 ,287-290, 294, 298 , 310- 312, 324 , 326,334, 335, 341-343, 345, 349, 351, 352,380, 385, 386, 388, 389, 428, 436

- advance of a collis ion see advance- cont r ibut ion of a collision

see cont ribut ion- noninteracting 31-33, 35,40- number of collisions 3, 9-11 , 15-22,

29, 34, 42-47, 112, 113, 116, 163, 190,221, 222, 268, 352

- - among hard rods 40, 43, 46, 47- - among hard sphe res 30, 34, 45, 46

among three hard rods 43, 44- - among t hree identical hard sphe res

30, 34- - cyclic collisions among three identical

hard spheres 35four collisions among three identica lhard spheres 30, 34, 35, 45-47recollision among t hree identicalhard spheres 35

color conduct ivity 310, 341colour diffu sion algorithm 341combinatoriall y rich (orbit segm ent) 54complet e hyp erbolicity see hyperbolic-

ity , completecomplet ely integr able 1, 195, 202-204,

208, 427computer simulation 1, 4, 235, 246 ,

275, 279 , 280, 297 , 299, 311 , 317, 330conduction 2, 319- electric cond ucti on 92 , 319- heat conduct ion 318, 321, 408configurat ion space 11, 12, 52-54, 57,

59, 60, 62, 67 , 68, 72, 126, 127, 133,135, 201, 203, 204, 244, 371, 428, 429 ,431

- factorizing 53, 83, 84conformally symplectic 206 , 348conjugate pairing 297 , 310 , 348, 349,

361conjuga te pairing theorems

see Lyapunov exponentsconj ugate vari ables 319 , 329 , 338connect ing path formula 67, 79, 82 , 434connectivity 437conserved qu antiti es 146, 187, 193, 312,

322 , 427 , 437cont ribut ion (of a collision ) 81convex boundary component 52, 63convex scattering property 62, 63, 71corre la t ion decay 2, 3, 91-93, 95 , 97 ,

103-106, 112, 116, 150, 165, 177, 286corre lat ion fun ction 91-97, 103, 105,

112-117, 218-224, 276 , 279, 286-288,291-293, 327, 335, 337 , 338

Coulomb gas 275cross effects 406 , 407cumulat ive 254, 359, 400 , 401cur vat ur e 1,9,11-14,16,19-22,60,63,

66, 94-96, 112, 124, 125, 148, 163, 197,243 , 266, 274, 284, 286, 353, 426 , 433

cyclic inte ract ion 57- 59cylindric billi ards 54, 66-68, 434, 435- t ransit ive 68, 69- tran sverse 68 , 69cylindric scat terer 53, 67, 428, 429, 432 ,

434 , 435cylindrical symmet ry 40, 45

D-interaction 58decay of correla tions

see long time tailsdensity 31- 33, 117, 132, 133, 136, 137,

139 , 141, 153, 154, 157, 161, 162, 210,218 , 220, 225, 233, 236 , 240 , 241 ,243-248, 250-253, 255, 257, 261-263,268-274, 276 , 290-293, 295- 305,307-309, 311, 312, 320, 325, 326, 332,344 , 353, 358, 359, 361, 372, 373, 376,382, 386, 395, 400, 402, 403 , 406-409,411 , 414, 441

Subject Index 451

~ cell density 393, 395, 396~ coar se-grain ed density 371-374, 379 ,

380, 395, 396, 400 , 402 , 408, 409- cur rent density 321, 399- energy density 153, 320, 404, 405,

412, 414- entropy density 370- exa ct densit y 371-373, 375, 379, 380,

396, 400 , 413- gradi ent 358, 379, 402 , 406-409- initial densi ty 373 , 382 , 395- invariant densit y 98 , 372, 385 , 400- - natural invariant density 386- particle density 153, 358, 361, 396 ,

398, 399- ph ase space density 311, 357- 359 ,

361 , 371, 373, 379, 393, 395, 396, 398 ,408 ,411

- - (average) growth rate of ph ase spacedensit y 408

- - contraction ra te of ph ase spacedensity 398, 408

- probability densit y 225, 226, 323,327, 329, 336, 343, 345, 350, 403

- reference density 374 , 375, 395, 405- st at ionary density 371, 386, 400density of gas 132, 133, 136, 137, 139,

141, 244development of a trajectory 14diamond bill iard 168, 174, 284, 286,

287 , 293diffusion 2, 3, 115, 146, 147, 151-153,

176,209,218, 317,318, 321 , 328,334,336, 337, 341, 343, 352, 353, 357, 358,360, 369, 380 , 389, 390, 396, 397, 399,402 , 413

- coe fficient see transport coefficient92, 146, 147, 151, 152, 286, 290, 292,319, 327, 328, 335-337, 341, 343,351-354, 357, 359, 360, 396, 403, 404

diffusion const ant 3, 352dimension 14, 15, 17, 22, 30, 34, 40, 45,

52- 57,59,62,64,73,83,96, 117, 127,131, 133, 147, 152, 153, 186, 188-190,208, 210, 232-234, 237, 240, 243, 251,255, 261, 264, 265, 267-272, 276, 279,280, 283-287, 294 , 296-298, 300-302,

310-312, 322-324, 326, 327, 331, 332,334 , 336, 341 , 342, 344, 355-357, 360,361, 375, 376, 412, 427, 436, 437, 442

- inform ation dimensionsee information dimension

- of the natural measure 376, 378- ph ase space dimens ion 311, 317, 346,

357, 371, 376, 378dimensionali ty dependence 45direct simulation Monte Carlo (DSMC)

246 , 260, 311disper sing 2, 10, 15, 57, 96, 97, 127,

146, 157, 162, 163, 165, 166, 168, 170,173, 192, 244, 250, 264, 265, 267,429- 431, 433, 436 , 438

distribut ion 61, 92, 93, 95, 98, 100,101, 105, 109, 110, 123, 136-138, 148,151-155, 205, 209, 217-220, 226, 233,235, 243, 245, 247, 248, 250, 251,254-257, 259, 260, 262-264, 269-271 ,274, 294, 305, 319 , 323-326, 328-331,336, 340, 341, 344 , 345, 348, 353,355- 360, 374, 398, 401 , 403, 408 , 411 ,414, 441 , 442

- asymptotic distribution 372, 379- coarse-grained distribution 375- fractal distribution 344, 369, 375- particle density distribution 396- phase space density distribution 398- source distribution 405- stat ionary distribution 254, 270, 372- st eady-state dist ribution 375, 376,

412- symbol-sequence distribution 376divergent t erms 33Donsker 's "invariance pr inciple" 61drift 334, 343, 360, 396, 399, 404-406,

413dyn amical sys te m 10, 11, 52, 53, 60, 62,

64, 66, 68, 70-73, 78, 84, 90-95, 97, 98,101-105, 116, 122, 145, 146, 148-150,153, 157, 163, 166, 172, 173, 182, 186,188 ,191 ,208,211 ,217,218,235,236,264 , 312, 317, 368-371 , 373, 375- 380,389,390,401 ,411-414,426, 429-431 ,442

ensembles 317, 319, 328, 338- 342

452 Subject Index

Enskog form ula 132ent ropy see coa rse graining, irr eversible

thermodynamics, 3, 4, 10, 13, 17,22,121-124, 127, 128, 130, 135, 137, 140,196, 209-211, 232, 236, 237, 240, 241 ,243-247, 250, 251, 266-268, 271, 273,274, 286, 300, 301 , 312, 318, 320-324,327 , 328, 330, 340, 344, 345, 353 , 356 ,361 , 362 , 369, 370, 373 , 375, 376, 378,379, 395, 402, 405-407 , 410, 412, 413,442 , 443

- and t hermostats 344-346balance 4, 321, 368-370, 377, 379,390, 397, 401, 402, 405, 408, 412Boltz mann 322, 323, 330, 348Clausius (thermodyn amic) 320, 321,330coarse-grai ned 369,373-380, 395 , 412dy namica l see Kolm ogorov-Sin aient ropyflux 321, 369 , 370, 377-379, 402 , 404,405Gibbs 327-330, 344, 369, 373-377,379, 395information- theoretic 373Kolmogorov-Sinai see ent ropy,measure-theoretic see KolmogorovSinai en tropymeasure-theo retic 122, 123, 127- 130,135, 137, 140, 141

- metric 17, 22, 53, 84, 293 , 373, 376,378

- pr odu ction of entropy see ent ropypr oducti on

- relative 209, 442- space-t ime 3, 137- thermod ynamic 4, 320, 368-370, 379,

401- to pologica l 16, 17, 22, 122ent ropy production 4, 295, 318, 321,

322, 328 , 330, 344-346, 348, 353-355,359- 361, 369-371, 375-380, 394,396-399, 401, 402 , 405, 408-410

- in dy namica l systems 371- vs loss of informat ion 376equilibrium 31- 33,60, 61, 94 , 95 , 105,

134, 146, 153, 154, 232-235, 240, 241,

245-247, 254, 268-271 , 273-275, 279 ,280 , 294-300, 302, 310, 316, 317,319-324, 328-330, 332-335, 338-341 ,344 , 353 , 355, 356, 358, 360-362, 378,412 , 423 , 424 , 441-443

- local 116, 146, 320 , 359, 413-- t he rmodynamic 414- relaxation towards 218, 378- t hermal 31, 153, 369 ,371,379ergo dic 10, 56, 60, 62, 63 , 66, 69, 91,

103, 123, 134, 140, 150, 166, 182, 191,192, 196- 200, 205, 208, 210, 211, 236,241, 284 , 287, 294 , 330, 340, 354 , 425 ,427, 428, 431, 433, 434, 438, 442

ergodic hyp othesis 1, 26, 66, 234- 236,422- 425, 427, 431, 435, 436, 442 , 443

ergo dic t heorem 91, 92, 123, 283, 425,426, 433

ergodic ity 1, 3, 4,10, 11, 51- 54, 56, 57,59, 62, 64-68, 71, 72, 74, 78, 79, 83,84 , 94, 96 , 97, 104, 108, 116, 147, 150,181 , 182, 188, 189, 191-193, 199, 200,202,205,207-211 ,280,317,322,323,330, 340 , 349 , 350, 354, 355 , 426-429,431, 433-443

escape rate formalism 356-358event ually strictly monotonemonotone

cocycle, 187

factorizing configuration spacesee configura t ion space, factorizing

falling balls 51,64, 65, 72,201,203, 205finit e comp uter precision 293finite hor izon see hori zon , finit eflow with collision 64, 183, 187, 188,

191-193, 201, 202, 205fluctuation t heo rem 206, 345, 346 ,

409- 411fluid 4,95, 132, 147, 148, 153-159 , 162,

163,165,168,172,175-177,233-236,276, 279, 280, 290, 300-302, 312, 317,318, 320 , 331, 408

fract al d ime nsion 276Fred holm determinan t 351front propagation 232, 255fundamental t heor em for semi-d ispersing

billiards 76, 433 , 434 , 439

Subject Index 453

Galton board 288Gauss distribution 61 , 137Gauss ' principle of least const raint 339Gaussian the rmostat see thermostats

147, 206, 294, 311, 339, 341 , 342, 346,347

generalized Sinai billiard 194geodesic 1,9-17,20,21 ,66,94-96,189,

197, 346, 426geodesic flow 1, 9, 10, 12, 13, 66, 94-96,

189, 197, 426Gibbs- ense mbles see ensembles- entropy see ent ropy- paradox 321, 328Gibbs measure 94, 95, 136-141 , 208,

210,442glass 276global ergodici ty 78, 79, 433, 434 , 439graph of int eractions 57, 58Green-Kubo formula 3, 146, 151-153,

332, 334, 359Green-Kubo relations see t ra nsport

coefficients

H-theor em see Boltzmann, H-theoremHenon-rnaps 3Hamiltonian syst em 1, 64, 123, 180,

203, 208, 240, 265, 286, 300, 322, 323,329, 348, 360, 371, 442

Hamiltonian s 1,64, 114, 181-183, 187,197, 200-208, 218, 234, 322 , 323, 329,346- 350, 360, 369, 378, 401 , 415, 428,437 , 440

hard ball gas 4, 11, 18, 19, 91, 123, 126,232, 233, 236, 237, 240, 242, 244, 245,250, 274, 275

hard ball gas model 18hard ball syste ms (HBS) 1-3, 17, 19,

22, 51-54, 56, 57, 59, 67, 68, 72, 78,79, 84, 97 , 121, 124, 140, 181, 182, 190,191, 202 , 236-238, 240, 241 , 243, 247,251,260,274,276,280,286,298,301 ,306, 310, 311, 317, 343, 428 , 429 , 434,436,437

hard chaos 205hard disk gas 279, 298, 302hard dumbbells 301 , 302, 310

hard rods 40hard sphere gas 165, 283, 298, 312Hau sdorff dimension 294Helfand moment 358Hopf chain 66, 78horizon- finite 60, 97, 105, 106, 114-116, 149,

150, 152, 153, 158-160, 163, 192, 205,290, 291, 332, 334, 335, 337, 343 , 350,354, 356, 358, 431

- infin ite 106, 113, 149, 157-160, 177,288, 291, 334-337, 343, 380, 431

hyp erbolicity 1, 3, 10, 11, 51, 52, 62 ,66,68,69, 79, 94, 97, 98 , 101, 104, 107,113, 116, 123, 147, 164, 165, 168, 178,182 , 186, 188, 189, 191, 196, 198, 200,205, 317,345, 346,350, 353 ,355-357,426, 429, 430

- complete 53, 66- 68, 71, 72, 202, 204

infinite horizon see hori zon , infiniteinfinitesimal Lyapunov exponent see

Lyapunov exponent , infinitesimalinform ation 10, 12, 13, 35, 97, 98,

101-104, 106, 164, 191,218,276,294,311 , 322, 329, 344, 355- 357, 360, 372,374-376, 378, 384, 387

- loss of information and irreversibility376

- theory 329, 373, 376information dimension 294, 311, 344,

355, 357, 375, 376, 378integral of motion 53, 72, 114, 140, 187,

188, 428, 429invariance principle 61, 91-93, 116invari ant cone field 62 , 64, 66, 69 , 72invari ant tori 1, 199, 211, 427 , 437, 440irrational mass ratio 83irreversibility 4, 311 , 318, 320, 322, 330,

362, 372, 400irreversible thermodyn amics (see also

ent ropy, time reversibility) , 318-322,329, 330, 340, 368, 369, 400 , 405 , 406 ,412,414

isoenergetic (constant int ernal energy )dynamics see t hermostats

isokineti c (const atn kin etic energy)dynamics see thermostats

454 Subject Index

K system 58, 191, 286, 431, 436, 437,441

K-flow 58, 191- 193,284,433K-mixing (Kolmogorov mixing) 52, 58,

61, 78, 84 , 85, 93, 94, 96K-part iti on 61KAM-theory 1, 181, 182, 200 , 207 , 427Kaplan-Yorke relation 344Katok-Strelcyn t heory 78Kawasaki d istri bu tion 345Kolm ogorov mixing 433Kolm ogorov-Sin ai entropy 3, 4, 121,

122, 141, 232, 236, 237, 240, 241,243- 247,250, 251, 265- 268, 271, 273,274, 276, 286 , 300, 301, 312, 327, 356,357, 373

KS entropy see Kolmogorov-Sinaient ropy

Lagra ngian subspac e 70, 186-1 88, 201length space 9, 10, 14Lennard-J ones (LJ) potent ial 198, 331limit 3,9, 12, 17, 61, 91- 95, 98, 104,

108, 115, 128, 130, 134, 136, 138, 139,150, 151, 155, 157, 158, 160-1 62, 166,172, 176, 192, 200, 217, 218, 220, 222,238, 240, 241, 253, 261 , 276, 295, 301,305, 317, 319, 320, 324, 326, 336-338,340, 341 , 343, 345, 348, 350-352,357-360, 372, 373, 375 , 380 , 389, 396 ,398, 399 , 401, 402 , 413, 440 , 443

- continuum limit 389 , 413- la rge-syst em limi t 396- macroscopic lim it 336, 380, 396-399,

402, 404, 405, 408, 409, 413linear respo nse theory see t ra nsport

coefficientsLiouville measure 53, 60, 64, 65, 68, 73,

114, 122, 123, 136, 138, 149, 150, 163,371, 423, 428

local conservation laws 153local currents 153local ergodicity 54, 62, 64-66, 71, 72,

74, 431, 433logist ic transformations 3long time tai l 33 , 276, 327 , 337Lorentz gas 3,4, 11, 17, 60, 61, 89- 91,

95-97, 105, 112-116, 121, 123, 124,

126, 128, 130 , 131, 145, 147- 153, 165,176, 192, 210, 232 , 236-238, 244 , 250,264-267, 270-272, 274, 275, 280, 286,288, 290, 291, 294 , 297, 316-320,326-328, 332, 334, 335, 337, 341-345,347 , 351, 353-355, 358, 360-362, 368,370, 380 , 382, 385, 386 , 389 , 390, 393 ,398, 400, 402 , 413, 415, 441

- driven 206 , 289, 295-297, 299 , 370- - three di mensiona l 296- nonequilibrium 341-345, 347, 350,

353- 355, 360- op en 356-360- periodi c 96, 105, 113-116, 128-131 ,

146-150, 152, 153, 157, 160, 163,218, 288, 289, 291-293, 318, 326, 32~334-337, 341, 352

- random 4, 130, 274, 275 , 295-299,318, 324-327, 334, 336 , 337, 341

- t hree di mensiona l 270, 296 , 298 , 299,348 , 355, 361

Lorent z gas model 11, 126, 147, 148,264

Loren t z process 2, 3, 60-62Lorent z-Boltzmann euqa t ion (ext ended)

271,273Lyapunov exponents 4, 51, 53, 58 , 59,

64,68, 71,97, 122-1 24, 126, 128, 130,135 , 137-140, 166, 184, 186- 188, 194,196-1 98, 201 , 202, 205, 232 , 236-238,240-242, 251- 253, 255 , 260, 261, 263,265-268, 270-272, 274-276, 279, 280,283, 284, 286-288, 290-293, 295-306 ,312, 348-35 1, 355-357, 360, 361, 371,373, 433, 434 , 439

- and nonlinear response 355- calculatio n of 4, 232, 260, 265, 267 ,

275, 276, 327, 342- conjugate pairing of Lyapunov

ex po nents 348-350, 361- infini tesm al 71- of per iodi c orb its 350-353- related to transport 344 , 356 , 360- sum rul e 344, 348 , 353Lyapunov modes 301,310Lyapun ov spectrum 3, 137- 139, 232,

265 , 270, 274, 275, 280, 281, 283, 286,

Subject Ind ex 455

287, 290, 295, 300- 305, 308 , 310, 311,348

~ for billiards 279- for fluids 280, 298

Markov partition 2, 10, 61, 94- 97, 146,173, 317, 350

Markov returns 3, 98, 99Markov sieves 3, 173mass metric 53maximum principle 256Maxwell molecules 275mean free t ime 132, 134, 244, 250, 266,

267, 273, 274, 326measure 1, 53- 56, 59-62, 64, 65, 68, 70,

73, 74, 77, 78, 84, 85 , 91, 93, 95, 96,98-101 , 103, 105, 108-110, 114-116,122, 123, 127, 136-141 , 148-1 50, 152,153, 155, 162, 163, 173, 175, 183-185,188- 190, 195, 202, 208, 210, 211,234-236, 263, 287, 292- 294, 306 , 322,323, 350, 355, 356, 358, 360, 368, 369,371, 372, 375, 400 , 401, 423, 424,426-428, 434, 440-442

- "forced" st at ionary measure 368- conditionally invariant measure 371- cumulat ive measure 400, 401- invari ant measure 61, 94, 95, 101,

103, 115, 122, 123, 132, 151, 158, 206,208-210, 235, 323, 349 , 350, 369, 371,401, 412, 441

- Liouville measuresee Liouville measure

- natural invariant measure 60, 372,375

- natural measure 185, 322, 368, 371,372, 374-376, 378

- SRB measure 95-98, 101, 103, 115,123, 344, 350, 368, 369, 371, 412

- stat iona ry measure 210, 368, 375metric ent ropy see ent ropy, met ricmicrocanonical ensemble 234, 319mixing 52, 58, 61, 67, 78, 84, 85, 93-96,

103, 110, 150, 155, 157, 191, 218, 235,236, 317, 321 , 328,329, 369,370,382,398, 408, 409, 427, 433, 441, 442

monotone 186, 187, 202

- st rict ly 71, 186-188, 190, 194, 202,203, 205

multibaker 4, 368-370, 380, 390,392- 397 , 400-403, 406- 409, 411-414

multibaker chains 4, 394, 400, 411, 412multifractal attractor 280, 294, 295,

311mul tifracti onal measures see dimensionmultiple points 107, 110, 111, 164, 166,

167

neutral space 54, 55, 79, 80, 83neutral subspace 433no-slip collision 207non-degeneracy condi t ion 18, 19non-integrable 1, 150, 428non- slip collision 207noncontraction 189non equilibrium stationa ry st ates 4,

316-318, 320, 321, 323, 327, 330, 338,361

non equilibrium steady st ates 4, 123,275, 279, 280, 294, 295, 298, 31~ 311 ,318, 319, 321, 323, 334, 338, 342, 359,379

nonquilibrium st ead y state 310Nose(-Hoover) see Hamiltonians,

thermostat s

n (fun ction) 196-198orbit segment 54, 63, 83orthogonal cylind ric bill iards 435or thogon al non-splitt ing property 69outer geometric parameters 59, 72

partially focusing 199, 200pencase 52, 57, 58, 435-437, 442periodic disks fluid 147, 148, 153, 158,

159, 162, 163, 168, 172, 175-177periodic orbits 65, 175, 199, 200, 207,

317,331, 344, 345, 350-357Pesin formula 123, 139Pesin t heory 66, 78ph ase space 1, 11, 17, 34, 35, 45, 54, 57,

59- 61, 64, 68, 72, 73, 133-136, 138,140, 146, 149, 150, 153, 155, 160, 162,163, 170, 172, 173, 184, 185, 188, 193,195, 196, 201, 203, 208, 211, 234-238,

456 Subject Index

255, 265, 280-283, 287, 294, 295, 300,301, 311, 317, 319, 322, 323 , 326-329,341, 343-346, 350, 352, 355-359, 361,370-373, 376, 378, 379 , 386, 390 , 393 ,395, 396 , 411, 412, 414, 415, 423, 424,427-430, 433, 437, 440, 441

phas e space cont raction 343-348, 356,359, 360, 369 , 370, 372, 374, 377, 386,396-399, 408, 415

phase spac e density see density, ph asespace

- (average) growt h rate of see density,phase space

- cont rac t ion rate of see density, phasespace

ph ase tran sition 286, 290, 291, 301,332, 340, 441

piece of submanifold 184, 185, 189Pinsker par tition 84, 85piston 4, 217- 222, 224- 226Poincare map 182, 184, 188, 191, 192,

194, 196, 197, 201, 294, 344, 351, 354Poincar e rec urrence see recurrence,

Poincar epo int- particle billiar d systems 2Poisson brackets 187Poisson measure 60, 137, 441Po isson process 60poten tial 4, 13, 32, 33, 57, 65, 72, 114,

135, 136, 154, 157, 159, 181, 182,192- 203, 205, 206, 208, 210, 218, 233,236, 275, 300, 301, 329, 331, 332, 334,341, 347, 406, 407, 437- 439

proper alignme nt 65, 189pull ed front 255

Q-form 187qu asi- ergodic hyp othesis 424

radius of cur vat ure (ROC) matrix 243,248-252, 266, 267, 269, 271

ran dom matrices 274, 276, 310Rayleigh-Brillouin spectrum 312recur rence 61, 62, 98, 99, 443- Poincar e 234, 433regular 11, 12, 14, 22, 26, 75-77, 111,

126,128,163,166,171 ,183-1 86, 264,267, 271, 432, 442

regul ar cover ings- family of regular coverings 75-77regul arity of degenerate tangencies 74regularity of do uble singularities 75regularity of singu larity sets 186rela t ive entropy see ent ropy, re lativerelaxation t ime 413Reshetnyak 's gluing t heorem 14, 21reversibili ty 345 , 354 , 382, 394 , 398-400rotation functi on 194, 195, 197-199

scaling regime 217second viri al coefficient 31, 32sector 111, 186, 188, 189semi-disp ersin g 2, 3, 9-12, 15-19, 22,

65, 76, 126, 127, 163, 177, 189, 244,247, 250, 260, 264, 267, 429 , 431,433- 435, 439

semi-dis pe rsing billiard table 11, 16,18, 19

semi-dispe rsive billiard see billiard,semi-d ispersive

shortes t path 14, 16, 20-22simple wedge 203simulation 1, 4, 201, 233, 235, 236,

246, 247, 251, 260-264, 270, 271, 275,279, 280, 283, 287, 297, 299-303, 305,307-312, 317, 330-333, 338 , 339 , 348,355 , 358 , 405, 408

Sina i billi ard 2, 3, 10, 90, 96, 105, 109,113, 116, 163, 192, 194, 206, 288, 335,429,431

Sinai 's "pencase" mod el 52, 57, 58Sinai- Ch ernov ansatz 190Sina i-Ruelle-Bowen (SRB) measure

see measure, SRB 95, 123, 344, 350singular points of the boundary 162,

163, 166, 167slim set 55, 56, 73, 83smoot h dyn ami cal sys te ms with

singularit ies 188smooth dy na micaldyn amical sys te m

sys te ms with singular it ies, 182smooth sym plectic dynamical systems

with sing ularit ies 186, 191spec ular reflect ion 2, 68, 133, 237, 284,

428sphere of velocities 54, 60

Subject Index 457

stable foliation 65, 70, 185, 192, 197stadion billiard 284standard (v, k, r )-model 60state- asymptotic state 371 , 376- equilibrium state 94, 146, 274,

319-323, 328, 335, 356, 378- steady state 4, 123, 206, 275, 279,

280, 294, 295, 298, 310, 311, 318, 319 ,321-323, 334 , 338, 342, 344, 348, 356,358-360, 369, 375-379, 401, 403-405,408, 409, 411, 412

- thermodynamic state 340, 407, 411,413

- transient state 401Stosszahlansatz 254, 324strictly unbounded 190strong ball avoiding theorem 83sufficiency 54-56, 76, 431, 432symplectic 64, 182-187, 189, 191, 192,

194, 203, 206, 283, 287, 298, 302 , 345,346, 348, 433

symplectic box 185symplectic flow 13, 183, 302symplectic manifold 69, 184symplectomorphism 64, 65, 70, 71, 185- monotone 71

Takagi functions 359, 400, 401tangent space dynamics 280, 281, 300,

301temperature 32, 93, 132, 137, 139, 154,

161, 235, 236, 246, 257, 295, 300,319-323, 328, 332, 338, 340, 341 , 348,358,402,403,405-409,411

theorem on local ergodicity 56, 64-66,71,72,74

thermal conductivity see transportcoefficients

thermodynamicdescription of transport 369, 389thermodynamic density 413thermodynamic entropy

see entropy, thermodynamicthermodynamic entropy balance 369 ,

370thermodynamic entropy production 401thermodynamic field 370

thermodynamic limit 136, 138, 139,301, 305, 319, 320, 338, 440

- for Lyapunov spectrum 301thermodynami c steady state 401thermostat 147,206,294, 295, 310 , 311,

323 , 333, 338-342, 344-349, 356, 359 ,360 , 362 , 385, 398, 402, 403, 405, 406,415

- Gaussian see Gaussian thermostatthermostatting 340, 361 , 369 , 370, 402,

405, 408, 411time reversal 40, 154, 234, 294, 382,

386, 394, 415time reversibility

(see also irreversible thermodynamics)322, 329, 343, 345 , 353-355, 360, 382,394 , 398, 399 , 415

topological dimension 56, 73, 83, 434topological pressure 276tower 97,101-103trace formulas 351 , 352trajectory branches 55, 171transitive cylindric billiards

see cylindric billiards, transitivetransport- energy transport 404- heat transport 370, 408- particle transport 370, 402transport and entropy production 394transport coefficients

see Burnett coefficients 3, 33, 92,145-148, 150, 151, 153, 154, 165, 166,176, 177, 233, 276, 295, 311, 316 , 321,332 , 333 , 335 , 337, 339 , 340, 348, 358,360 , 369, 370

- Boltzmann equation derivation 326,327

- defined 319- dynamical expressions

for transport coefficients 344, 352,357 , 358, 360

- entropy,relation to 321- Gr een-Kubo expressions for transport

coefficients, 332-338- nonlinear 353-355- thermostat methods 338-341transport equation 233, 397

458 Subject Index

tree 57,58turbulenc e 317,319

unstable foliation 65, 70, 185, 192, 197,375

variational principle 94, 257velocity correlat ion dec ay 286velocity correla tion fun ction 279,

286-288, 291-293velocity distribution 233 , 254 , 257, 260 ,

262-264, 274viri al expansion 31

viscosity see transport coefficients 3,92, 146, 147, 154, 156, 162, 177, 233,311 , 319, 334, 337, 341, 353, 358 , 408

walls of a billi ard table 12, 18wandering set 61, 62Weeks-Chandler-Anderson (WCA)

potential 332W iener process 61Wojtkowski 's conj ecture 64

zet a fun ctions 351, 352zig-zag theor em 77, 78

hard ball systems and the lorentz gas

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