statistical physics and dynamical systems: rigorous results
TRANSCRIPT
Progress in Physics Vol.lO
Edited by A. Jaffe, G. Parisi, and D. Ruelle
Springer Science+Business Media, LLC
Statistical Physics and Dynamical Systems Rigorous Results
J. Fritz, A. Jaffe, and D. Szasz, editors
Springer Science+Business Media, LLC 1985
Editors
J. Fritz MTA MKI Realtanoda u. 13-15 H-1053 Budapest (Hungary)
A. Jaffe Dept. of Physics Lyman Laboratory Harvard University Cambridge, MA 02138 (USA)
Library of Congress Cataloging in Publication Data Main entry under title:
Statistical physics and dynamical systems. (Progress in physics ; vol. 10) »Contains most of the invited papers of the
Second Colloquium and Workshop on >Random Fields: Rigorous Results in Statistical Mechanics< held in Koszeg, Hungary between August 26 and September 1, 1984« -- CIP pref.
Includes bibliographies. 1. Random fields -- Congresses. 2. Statistical
mechanics -- Congresses. 3. Quantum field theory --Congresses. I. Fritz, J. II. Jaffe, Arthur, 1937-111. Szasz, D. IV. Colloquium and Workshop on >>Random Fields : Rigorous Results in Statistical Mechanics« (2nd : 1984 : Koszeg, Hungary) V. Series: Progress in physics (Boston, Mass.) ; v. 10. QC174.85.R36S73 1985 530.1'3 85-1235
D. Szasz MTA MKI Realtanoda u. 13-15 H-1053 Budapest (Hungary)
ISBN 978-1-4899-6655-1 ISBN 978-1-4899-6653-7 (eBook) DOI 10.1007/978-1-4899-6653-7
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Statistical physics and dynamical systems : rigorous results I J. Fritz ... , ed.
(Progress in physics ; Vol. 10) ISBN 978-1-4899-6655-1
NE: Fritz, Jozsef [Hrsg.]; GT
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
© 1985 Springer Science+Business Media New York Originally published by Birkhäuser Boston, Inc. in 1985 Softcover reprint of the hardcover 1st edition
ISBN 978-1-4899-6655-1
C 0 N T E N T S
Preface vii Program of the Workshop ix List of Participants xv
v
R.A. MINLOS: The Algebra of Many-Particle Operators
H. ARAKI and T. MATSUI: C*-Algebra Approach to Ground States of
the XY-Model
M.Z. GUO and G. PAPANICOLAOU: Bulk Diffusion for Interacting
Brownian Particles
L.A. PASTUR: Spectral Properties of Random and Almost Periodic
Differential and Finite-Difference Operators
H. SPOHN: Equilibrium Fluctuations for Some Stochastic Particle
Systems
A.G. SHUHOV and YU.M. SUHOV: Linear and Related Models of Time
Evolution in Quantum Statistical Mechanics
P. COLLET: Systems with Random Couplings on Diamond Lattices
L.A. BUNIMOVICH: On the Diffusion in Dynamical Systems
A. KUPIAINEN: ~ 4 with Negative Coupling 4
C. BOLDRIGHINI, A. DE MAS!, A. NOGUERIA and E. PRESUTTI: The
Dynamics of a Particle Interacting with Semi-Infinite Ideal
Gas is a Bernoulli Flow
YA.B. PESIN: A Generalization of Carateodory's Construction for
Dimensional Characteristic of Dynamic Systems
M. MISIUREWICZ and A. ZDUNIK: Convergence of Images of Certain
Measures
1
17
41
49
67
83
105
127
137
153
191
203
vi
R.R. AKHMITZJANOV, V.A. MALYSHEV and E.N. PETROVA: Cluster
EXpansion for Unbounded Non-Finite Potential
C. BARDOS, R. CAFLISCH and B. NICOLAENKO: Thermal Layer Solutions
of the Boltzmann Equation
E. PRESUTTI, YA.G. SINAI and M.R. SOLOVIECHIK: Hyperbolicity and
221
235
Moller-Morphism for a Model of Classical Statistical Mechanics 253
L. ACCARDI: Quantum Stochastic Processes 285
P.M. BLECHER and M.V. JAKOBSON: Absolutely Continuous Invariant
Measures for Some Maps of the Circle
C. MARCHIORO: Some Problems in Vortex Theory
P.M. BLECHER: The Maxwell Rule and Phase Separation in the
Hierarchical Vector-Valued ~ 4-Model
R.L. DOBRUSHIN and S.B. SHLOSMAN: Constructive Criterion for the
Uniqueness of Gibbs Field
R.L. DOBRUSHIN and S.B. SHLOSMAN: Completely Analytical Gibbs
Fields
P.A. FERRARI, S. GOLDSTEIN and J.L. LEBOWITZ: Diffusion, Mobility
and the Einstein Relation
B. SOUILLARD: Transition from Pure Point to Continuous Spectrum
for Random Schrodinger Equations: Some Examples
M. AIZENMAN: Rigorous Studies of Critical Behavior II
303
317
329
347
371
405
443
453
vii
PREFACE
This volume contains most of the invited papers of the
Second Colloquium and Workshop on "Random Fields": Rigorous
Results in Statistical Mechanics" held in K8szeg, Hungary
between August 26 and September 1, 1984. Invited papers
whose authors could finally not attend the Colloquium are
also included.
The Colloquium was organized by the generous sponsor
ship of the International Union for Pure and Applied Phys
ics, the International Workshop Committee for Theoretical
Physics, the International Association of Mathematical Phys
ics, the Hungarian Academy of Sciences and the Janos Bolyai
Mathematical Society.
Members of the International Program Committee were
R. L. Dobrushin, A. Jaffe, J. L. Lebowitz, D. Ruelle, Ya. G.
Si'lai. The Organizing Committee consisted of J. Fritz, D.
Szasz (co-chairmen), D. Petz (secretary), A. Kramli, G.
Lippner, P. Lukacs, P. Major, A. Slit8, N. Simanyi, A. Vetier.
There were 112 participants from 21 countries represent
ing all the six continents. There were altogether 22 forty
five minute invited talks and 63 ten minute contributed
papers. The Workshop was organized on the last two days of
the Colloquium. It mainly concerned topics most interesting
for Hungarian physicists and, moreover, its program ensured
additional space for discussions.
We express our sincere gratitude to Denes Petz for his
careful work as the technical e~itor of this volume and to
Zsuzsa Er8 for her excellent and rapid retyping of several
manuscripts.
The Editors
Uonday
10.00
1 0. 20 - 11 • 05
11.10- 11.55
3.00- 3.45
3.50- 4.35
coffee break
5.00 - 5.45
5.50 - 6.35
7.30
Tuesday
9.00 - 9.45
9. 50 - 10.35
coffee break
11.00 - 11.45
10.50- 12.00
1 2 • 02 - 1 2 • 1 2
12.14- 12.24
ix
PROGRAM
Opening Ceremony
R. L. Dobrushin - s. B. Shlosman: Constructive criteria of uniqueness and analiticity in statistical mechanics
J. L. Lebowitz: Mathematical and physical ideas in nonequilibrium statistical n.:!chanics
E. Lieb: Various estimates for the eigenvalues of the Laplacian
H. Araki: C*-algebra approach to the ground states of the X-Y model
M. Aizenmann: Rigorous studies of critical behaviour
D. Kazhdan: c-representations
Welcome party
A. Kupiainen: Non-trivial continuum limit for negative coupling ~4
4
G. Papanicolau: Bulk diffusion and self diffusion for interacting Brownian motions
o. E. Lanford: Renormalization group methods for mappings with golden-ratio rotation number
H. Rost: Equilibrium fluctuations for a onedimensional nearest neighbour model
A. Verbeure: States stationary for the detailed balance of reversible processes
M. F. Chen: Jump Markov processes and interacting particle systems
12.26- 12.36
12.38 - 12.48
3.00-3.10
3.12- 3.22
3.24 - 3.34
3.36 - 3.46
3.48 - 3.58
4.00 - 4.10
4. 12 - 4.22
4.24 - 4.34
4.36 - 4.46
coffee break
5. 14 - 5.24
5.26 - 5.36
5.38 - 5.48
5.50- 6.00
6.02 - 6.12
6.14 - 6.24
6.26 - 6.36
6.38 - 6.48
6.50- 7.00
X
Ra. Siegmund-Schulze: On existence of nonequilibrium dynamics of multidimensional infinite particle systems: the translation invariant case
J. Fritz: Interacting Brownian particles: existence and self-adjointness
A. Kramli - D. Szasz: The problem of recurrence for Lorentz processes
D. Dawson: Ensemble and multilevel models of critical behaviour
K. Fleischman: Occupation time processes at a critical point
J. A. Galves - E. Olivieri - M. E. Vares: Metastable behaviour of stochastic systems: a pathwise approach
J. R. Klauder: Langevin equations for statistical computations
N. Ianiro: Stationary Boltzman-equation
G. Jetschke: On stochastic nonlinear parabolic differential equations
P. Calderoni: On the Smoluchowski limit for simple particle systems
H. Cranel: Stochastic systems on manifolds
C. A. Hurst: C*-algebra approach to the Pfaffian method for the Ising model
J. T. Lewis - J. V. Pule: Phase transitions and the weak law of large numbers
K. Kuroda: The Pirogov-Sinai theory of phase transitions for continuum systems
Y. Higuchi: A weak version of Russo-SeymourWelsh theorem for the two-dimensional Ising model
L. Laanait -A. Messager - J. Ruiz: Phases coexistence and surface tension for the Potts model
B. Toth: A lower bound for the critical probability of square-lattice site percolation
D. Merlini: On the Temperley conjecture for the two-dimensional Ising model
V. Warstat: A uniqueness theorem for systems of interacting polimers at low temperature
M. Arato: The distribution of stochastic integrals
Wednesday
9.00 - 9.45
9. 50 - 10.35
coffee break
11.00-11.45
11 . 50 - 1 2 . 00
1 2. 02 - 1 2 • 1 2
12.14 - 12.24
12.16- 12.36
12.38 - 12.48
Thursday
9.00 - 9.45
9. 50 - 10.35
coffee break
11.00 - 11.45
11 . 50 - 1 2 • 00
1 2. 02 - 1 2 • 1 2
12.14 12.24
12.26 - 12.36
12.38- 12.48
3.00-3.10
3.12- 3.22
3.24 - 3.34
xi
E. Presutti - Ya. G. Sinai - M. Soloviechik: Hyperbolicity and Moller-morphism for a model of classical statistical mechanics
Yu. M. Suhov - A. G. Shuhov: Linear and related models of time evolution in quantum statistical mechanics
R. Caflisch: Thermal layers for the Boltzrnan equation
J. K. Percus: Evaluation of a class of functional integral
J. Bricmont - J. Frohlich: Random walks and the particle structure of lattice guage theories
D. Surgailis: On continuous contour-models and Arak fields
G. F. Lawler: Intersection properties of simple random walks
P. Major: Renormalization of Dyson's hierarchical vector-valued model at low temperatures
L. Accardi: Quantum probability
H. Spohn: Equilibrium fluctuations for some stochastic particle systems
C. Marchioro: Some problems in vortex theory
C. Kipnis: Asymptotics for the motion of a tagged particle in the simple exclusion model
J. R. Fontaine - Ph. A. Martin: Equilibrium equations and Ward identities for Coulomb systems
Ch. Gruber: On the invariance of charged systems with respect to external fields
J. Jedrzejewski: Phase transitions in models of itinerant electrons
G. Schlijper: Rigorous results for approxil'late variational principles
D. Dlirr: On Harris' collision model
H. Rodehausen: Diffusive behaviour for a class of Ornstein-Uhlenbeck processes
J. Gartner: On long-time fluctuations of weakly interacting diffusions
3.36 - 3.46
3.48 - 3.58
4.00-4.10
4.12 - 4.22
4.24 - 4.34
4.36 - 4.46
coffee break
5.14 - 5.24
5.26 - 5.36
5.38 - 5.48
5.50 - 6.00
6.02 - 6.12
6.14 - 6.24
6.26 - 6.36
6.38 - 6.48
6.50- 7.00
coffee break
7.20 7.30
7.32 - 7.42
xii
A. Kramli - N. Simanyi - D. Szasz: Transport phenomena and random walks with internal states
Y. Elskens -H. L. Frisch: Annihilation dynamics in one dimension
P. Ferrari - E. Presutti - M. E. Vares: Hydrodynamical properties of a zero-range model
D. Szasz - B. T6th: One-dimensional persistent random walks in random environment
w. A. Majewski: On ergodic properties of dynamical semigroups
A. Wakolbinger: Time-reversal co-dimensional diffusions
s. Pogosian: Cluster property of classical spin systems
B. Nahapetian: Limit theorems for weakly dependent random variables
D. Petz: Quasi entropies for finite quantum systems
H. Baumgartel: A class of nontrivial weakly local massive Wightman fields with interpolation properties
I. Daubechies - J. R. Klauder: Wiener measures for exp(-itH)
E. Bruning: On the construction of random probability measures of infinite dimensional spaces
K. H. Fichtner - G. Jetschke: A probabilistic model of a quantum mechanical infinite particle system
E. Orlandi - R. Figari: Gaussian approximation for the Green's functions of Laplacian in a domain with random holes
V. Schaffenberger: Borel summability in the disorder parameter of the averaged Green's function for Gaussian disorder
A. de Masi - P. Ferrari: Diffusion in per-celation regime I.
P. Ferrari - A. de Masi: Diffusion in per-celation regime II.
7,44 ~ 7.54
7.56 - 8,06
8.08- 8.18
Friday
9.00 - 9.45
9.50- 10.35
coffee break
11.00- 13.00
3.00 - 4.00
4.00- 4.10
4.12 - 4.22
4.24 - 4.34
coffee break
5.00 - 5.10
5.12 - 5.22
5.24 - 5.34
5.36 - 5.46
5.48 5.58
Saturday
9.00 - 9.45
9. 50 - 1 o. 35
10.40-11.25
11.30-12.00
xiii
R. Kotecky: On residual entropy models
s. Olla: Large deviations and variational principles
G. Royer: De Fortret-Mourier distance and log-concave functions
A. Katok: Random perturbations of dynamical systems motivation, conjectures, rigorous results
M. Misiurewicz: Convergence of images of certain measures
Discussions
Discussions
H. 0. Georgii: On the critical temperature of disordered ferromagnets near the percolation threshold
J. L. van Hemmen: Statistical mechanics of spin glasses
H. Englisch - M. Endrulis: Random alloys and special energies
F. Przytycki: Riemann maps and holomorphic dynamics
S. Pirogov: Automata systems with defects
A. Vetier: Ergodic properties of the Sinai billiard in an external field
J. Kotus: n-stability of vector fields
K. Ziemian: An almost sure invariance principle for some maps of an interval
P. Collet: Phase transitions on diamond lattices
L. Pastur: On the spectral theory of random and almost periodic operators
B. Souillard: Transitions from pure point to continuous spectrum for random Schrodinger operators. Some examples
Closing ceremony
XV
L I S T 0 F P A R T I C I P A N T S
L. Accardi Dip. di Matematica Universita di Roma II Via Orazio Raimondo I-00173 Roma (Italy)
A. Ag MTA KFKI PF. 49 H-1525 Budapest 114 (Hungary)
M. Aizenman Dept. of Mathematics Rutgers University Busch Campus New Brunswick, NJ 08903 (USA)
H. Araki 230-42 Iwakura-Nagatanicho Sakyoku Kyoto 506 (Japan)
M. Arata Fehervari ut 129 H-1119 Budapest (Hungary)
H. Baumgartel Inst. f. Mathematik Mohrenstrasse 39 1085 Berlin ( DDR)
G. Benfatto Via A. Fraccaroli, 7 I-00157 Roma (Italy)
C. Boldrighini Istituto Matematico Universita di Camerino I-52032 Camerino (Italy)
J. Bricmont FYMA 2 Chemin du Cyclotron B-1348 Louvain-La-Neuve (Belgium)
R. Caflisch Courant Institute 251 Mercer St. New York, NY 10012 (USA)
P. Calderoni ZIF Bielefeld Universitat Wellenberg 1 D-4800 Bielefeld (FRG)
M. Campanino Via Bisagno 14 I-00199 Roma (Italy)
S. Caprino Via Armando di Tullio 27 I-00136 Roma (Italy)
Mu Fa Chen Dept. of Mathematics Beijing Normal University Beijing (Peoples' Republic of China)
G.S. Chobanov Math. Inst. of the Bulg. Acad. of Sci. P.O.B. 373 1090 Sofia (Bulgaria)
J.S. Cohen Prinsengracht 1055 A-2 NL-1017 JE Amsterdam (The Netherlands)
P. Collet 26. rue Vergmiaud F-75013 Paris (France)
I. Daubechies Wolfshaegen 33 B-3053 Huldenberg (Belgium)
D. Dawson 2155 Delmar Dr. Ottawa KlH 5P6 (Canada)
R. L. Dobrushin Inst. of Problems of Transmission of Information Avia Motornaja 8 Moscow E - 24 (USSR)
xvi
D. Diirr Ruhr-Universitat Bochum Inst. fUr Mathematik Postfach 102148 D-4630 Bochum 1 (FRG)
Y. Elskens Fac. de Sci. Univ. Libre de Bruxelles C.P. 231• Bvd. du Triomphe B-1050 Bruxelles (Belgium)
H. English Steinstrasse 16 DDR-7030 Leipzig (DDR)
R. Eposito Piazzale Montesquieu 28 IA I-00137 Roma (Italy)
P.A. Ferrari IHES F-91440 Bures-Sur-Yvette (France)
A. Fialowski Villanyi ut 103 H-1118 Budapest (Hungary)
K. Fleischmann AdW d DDR Institut fUr Mathematik Mohrenstrasse 39 1086 Berlin (DDR)
J.-R. Fontaine 3A Ch. des Esserts CH-1024 Ecublens (Switzerland)
J. Fritz HTA MKI Realtanoda u. 13-15 H-1053 Budapest (Hungary)
J.A. Galves R. Riachuelo 296 13130 Sousas - S.P. (Brasil)
J. Gartner AdW d DDR Institut fUr Mathematik Mohrenstrasse 39 1086 Berlin (DDR)
H. -0. Georgii Sperlingweg 7 D-8031 Eichenau (FRG)
V. Gorini Dip. di Fisica Sez. Fis Teor. Via Celoria 15 I -20133 Milano (Italy)
Ch. Gruber Inst. Phys. Theor. EPL-Lausanne CH-1015 Lausanne (Switzerland)
Y. Higuchi Dept. of Mathematics Kobe University Rokko Kobe 657 (Japan)
C.A. Hurst 99 Fifth Avenue Joslin, South Australia 5070 (Australia)
xvii
N. Ianiro Via Urbana 143 I-00100 Roma (Italy)
K.R. Ito ZiF, Univ. Bielefeld Wellenberg l D-4800 Bielefeld l (FRG)
J. Jedrzejewski Budziszynska 135/5 Wroclaw (Poland)
G. Jetschke Friedrich-Schiller Universitat Sektion Mathematik DDR-6900 Jena (DDR)
A. Katok 1080 Spruce Str. Berkeley, Calif. (USA)
D. Kazhdan Harvard University Mathematics Department l Oxford Street Cambridge, MA 02138 (USA)
C. Kipnis 17 Rue Mathis F-75019 Paris (France)
J.R. Klauder AT and T Bell Labs Murray Hill, NJ 07974 (USA)
R. Kotecky Dept. Math. and Phys. V. Holesovickach 2 Praha 8 (Czechoslovakia)
J. Kotus Szcz~slinicka 29 m 29 P-02-353 Warsaw (Poland)
A. Kramli MTA SZTAKI Victor Hugo u. 18-22 H-1132 Budapest (Hungary)
R. Kuik Sloep 299 NL-9732 CT Groningen (The Netherlands)
A. Kupiainen Abrahamink. 17 C59 Helsinki 18 (Finland)
K. Kuroda Dept. of Math. Keio University Hiyoski 3-14-1 Kohoku-ku Yokohama 223 (Japan)
L. Laanait C.P.T. CNRS Lumniy Case 907 F-13288 Marseille Cedex 9 (France)
O.E. Lanford Inst. des Hautes Etudes Sci. 35, Route de Chartres F-91440 Bures-Sur-Yvette (France)
G.F. Lawler 311 S. Lasalle St. Apt. 30 A Durham, NC 27705 (USA)
xviii
J.L. Lebowitz 52 Locust Lane Princeton, NJ 08546 (USA)
G. !..eha Bavariastrasse 2 D-8551 Pinzberg (FRG)
J.T. Lewis San Clemente, Vico Rd. Dalkey, County Dublin (Ireland)
E. Lieb Jadwin Hall P.O.B. 708 Princeton, NJ 08544 (USA)
G. Lippner BME Villamosmernoki Kar Mat. Tsz. Muegyetem rkp. 3-9 H-1111 Budapest (Hungary)
P. Lukacs MTA SZTAKI Victor Hugo u. 18-22 H-1132 Budapest (Hungary)
W.A. Majewski Inst. Theor. Phys. Astrophys. Univ. Gdansk Wita Stwosza 75 P-80-952 Gdansk (Poland)
P. Major MTA MKI Realtanoda u. 13-15 H-1053 Budapest (Hungary)
C. Marchioro Dip. di Matematica Universita di Trento I-38050 Trento (Italy)
R. Marra Piazzale Montesquieu 28/A I-00137 Roma (Italy)
A. de Masi Ist. Matematica Univ. dell'Aquila Aquila (Italy)
D. Merlini Mathematisches Institut Ruhr-Universitat D-4630 Bochum (FRG)
M. Misiurewicz Asfaltowa 7/5 Warsaw
(Poland)
B.S. Nahapetian Inst. Math. Armenian SSR ul. Barekamutian 24 Yerevan (USSR)
S. Olla Via Bandello 52 I-09100 Cagliari (Italy)
E. Omerti Dip. di Matematica Univ. di Trento I-38050 Povo (Trento) (Italy)
ixx
E. Orlandi Dep. Mat. "G. Castelnuovo" Citta Universitaria Piazzale A. Moro I-00100 Roma (Italy)
G. Papanicolaou Currant Institute 251 Mercer Str. New York, NY 10012 (USA)
L. Pastur Pr. Lenina 47 Harkov (USSR)
J.K. Percus 340 Riverside Dr. New York, NY 10025 (USA)
D. Petz MTA MKI Realtanoda u. 13-15 H-l053 Budapest (Hungary)
S.A. Pirogov IPPI AN USSR Ermolovoi str. 19 101447 Moscow GSP-4 (USSR)
S.K. Pogosian Inst. Mat. Armenian SSR ul. Barekamutian 24 Yerevan (USSR)
E. Presutti Dip. di Matematica Univ. di Rorna I-00185 Roma (Italy)
F. Przytycki ul. Puszczyka 17 m 80 P-02777 Harszawa (Poland)
M. Pulvirenti Via A. di Tullio 27 I-00136 Roma {Italy)
M. Rattner Dept. of Math. University of California Berkeley, CA 94720 (USA)
M. Redei Rak6 ..~. 47/a H-1112 Budapest (Hungary)
H. Rodenhausen Schlierbacher Lanstr. 154 D-6900 Heidelberg (FRG)
H. Rost Seidenweg 7 D-6907 Nussloch (FRG)
G. Royer 13 rue Jules Cuillerier F-94140 Alfortville (France)
J. Ruiz CPT CNRS Lumniy Case 907 F-13288 Marseille Cedex 9 (France)
E. Scaciatelli Dip. Matematica Univ. di Roma Piazzale Aldo Moro 5 I-00100 Roma (Italy)
U. Scharfenberger Eckenheimer Landstr. 72 D-6000 Frankfurt (FRG)
XX
A.G. Schlijper Clavecimbellaan 399 NL-2287 VP Rijswijk (The Netherlands)
R. Siegmund-Schulze Elsa Brandstrom-Str. 38 1100 Berlin (DDR)
N. Simanyi Rozsa F. u. 5 H-2440 Szazhalombatta (Hungary)
B. Souillard 4 et 6 Rue Saint Nicolas F-75012 Paris (France)
H. Spohn Theoretische Physik Theresienstr. 37 D-8000 Munchen 2 (FRG)
A.K. Stepanov Inst. Fiz. AN SSSR Moskovskoe Obl. Cernogolovka (USSR)
Yu.M. Suchov IPPI AN SSSR Ermolovoi St. 19 101447 Moscow GSP-4 (USSR)
D. Surgailis Inst. of Math. and Cyb. 232600 Vilnius Pozelos 54 (USSR)
A. suto MTA KFKI Pf. 49 H-1525 Budapest 114 (Hungary)
D. Szasz MTA MKI Realtanoda u. 13-15 H-1053 Budapest (Hungary)
K. Szlachanyi MTA KFKI Pf. 49 H-1525 Budapest 114 (Hungary)
B. Toth MTA MKI Realtanoda u. 13-15 H-1053 Budapest (Hungary)
L. Triolo Univ. degli Studii di Roma Dip. Mat. Via A. Scarpa I-00161 Roma (Italy)
l~. Urbaiisk i ul. Gagarina 45 HA-2 P-87-100 Torun (Poland)
M. Vares Rua Sorocaba 484, Apt. 301 F Botafogo Rio de Janeiro (Brasil)
P. Vecsernyes MTA KFKI Pf. 49 H-1525 Budapest 114 (Hungary)
A. Verbeure Leuvense Straat 5 B-3050 Oud-Heverlee (Belgium)
xxi
A. Vetier BME Villamosmernoki Kar Matematika Tanszeke Muegyetem rkp. 3-9 H-1111 Budapest (Hungary)
A. Wakolbinger Reinprechtenstr. 2 A-4040 Puchenau (Austria)
J. Westwater Dept. of Mathematics Univ. of Washington Seattle, WA 98195 (USA)
V. Warstat Schonitzstr. 19 DDR-4020 Halle (DDR)
W.D. Wick Phys. Dept. Princeton Univ. Princeton, NJ 08540 (USA)
G. Winkler Alranstr. 17 D-8000 Munchen 70 (FRG)
M. Winnink Inst. for Theor. Physics P.O.B. 800 Groningen (The Netherlands)
K. Ziemian ul. Na Uboczu 24 m 64 P-02802 Warszawa (Poland)
THE ALGEBRA OF MANY-PARTICLE OPERATORS
R. A. Minlos
There exists a large class of operators which contains
practically all operators of many- (or infinite-) particle
physics. These operators were called in [1] cluster opera
tors. They were investigated from a different point of view
in the papers [1], [2], [3], [4]. Here we consider some pure
algebraic aspects of the theory of cluster operators. These
operators together with the space, where they act, naturally
form a category. It is convenient to describe some construc
tions used in the theory of cluster operators on this lan
guage.
1. The cluster categories
We introduce here a specific category of such kind. We
shall discuss the generalizations of this scheme at the end
of this section.
Let be a Hilbert space of the form
\! n Hn,h = .f_2 ( (Z ) ,h)
is the v -dimensional lattice, n is an integer,
a Hilbert space), i.e. the space of the h -valued { \! n
functions f = f (T), T= (t 1 , ..• ,tn) E (Z ) }
~~ f li = r , II f (T) ~~· 2) 1 I 2 11 h l L n , lhJ
TE(Zv)
( 1 )
(2)
The objects of our category (we denote it by N ) are
the direct sums of the spaces Hn,h
Hr = ED H h , (n,h) n,
2
where r = { (n,h)} is a finite our countable collection of
pairs (n,h) . We shall describe now the morphisms of the
category N which were called cluster operators. Evidently,
every bounded linear operator
where r = { (n,h)}, r 1 = { (n 1 ,h 1 )} are two collections of
pairs, generates a matrix of linear operators {B(nl,hl),(n,h)}
8 (n 1 ,h 1 ),(n,h): Hn,h _,. Hn 1 ,h 1 •
Furthermore, a linear operator
B: H h .... H I hi n, n ,
generates the matrix of operators
h .... h 1 •
We denote by Nn 1 ,n Nn 1 UNn the disjunctive union of the sets Nn={1, ... ,n} and Nn 1 = {1 1 , ••• ,n 1 } and by An 1 ,n the lattice of all partitions
of the set such that
It is convenient to assume that the blocks of the partition
y are enumerated so that the numbers of the smallest ele
ments of the sets ainNn i=1, ... ,s increase. The order
y~ Y1 in An 1 ,n means that every block of y 1 contains
some block of y • For each n~1 , n 1 ~1 , each partition
y = (a 1 , ... , as ) E An 1 n
E (Zv)s we denote by
and each collection ( T 1 , ... , T s) E
Tn E ( Z v) 4 and Th 1 E ( Z v) n the vee-· y y
{t1 , ... ,tn}
if jENnnai
T~ 1 ={t;, ... ,t~ 1 } so that tj=
, j E Nn I n ai
We say that the operator
A = A : H y n,h
3
_,. H I h' n ,
is connected with respect to Y = (a 1 , ... ,as) E An 'n if
its matrix blocks AT'T: h+h' satisfy the following condi
tions:
1) for every collection (T 1 , ••• ,Ts)
A I AT' T T'+Tn T+Tn ' y I y
2) the series
"/ T,T'
)I A p < li T' ,Tuh,h' ""
h 'i'Y converges. Here t e sum L
taken over T E: (Zv)n and T
means that the summation is
so that
t 0 . 1 .min .min= ' ~= , .•. ,s ' Ji Ji
<II ·llh,h, is the operator norm).
The operator
min j j E ai nNn
is called a cluster operator if there exists a representa
tion
B I yEA I n ,n
A y (3)
where every operator AY is connected with respect to the
partition yEA , • n ,n
Lemma 1. Cluster operators are bounded and their re
presentations (3) are unique.
Proof. The first assertion follows from the inequality
:: ., 1 r SUJ2 , I' II 'In II ] ll B :1 ~ 2[-T L : BT' T h h '+sup L II BT 'T h h'
T' I I T' T I
(4)
Conditions 1) and 2) imply that the right side of (4) is
finite. Again, 1), 2) and (4) imply for any yEA , that n ,n
4
lim B I = J ( Ay-) T I T T'+T~ ,T+T~ y~y I T.-T.-+00
~ J i;lj,i,j=1, ... ,s
(in the sense of uniform convergence of operators). Then
(Ay)T' T = J )JA (y,y) (Dy-)T' T I
I y >_y n I In I
where \lA (·,·) is the Mobius function of the lattice n' ,n
An' ,n . The representation is called the cluster expansion
of the operator B , and the operators Ay are its connec
ted components.
More generally, the operator
is called a cluster one if all its blocks
are cluster operators. B(n' ,h'J (n,h)
The set Mer (Hr, Hr,) := Mer ( r, r' ) of the morphisms from
Hr to Hr, in the category consists of the cluster opera
tors B: Hr 7 Hr, . The correctness of this definition fol-
lows from the lemma.
Lemma 2. 1) For every
is a cluster one. 2) Let
r the unit operator Er: Hr7Hr
B2 : Hr, 7 Hr II be
B = B2B1 :Hr 7 Hrll
B 1 : H r 7 H r , and
cluster operators. Then their composition
is also a cluster operator.
The first assertion of the lemma is evident. The second
one follows from the next lemma.
Lemma 3. Let the operators A( 1 ): H H y 1 n,h 7 n' ,h' and
(2) A : H ' h' 7 Hn II, h II y 2 n ,
be connected with respect to the par-
titions y 1EA , and y 2EA 11 , correspondingly. Then the n ,n n ,n
composition A (2 )A ( 1 ) =A:= A is connected with respect to y2 y1 y
the partition y E An 11 , n of set Nn 11 , n which is defined in
the following way. We denote by :Y 1 and :Y 2 the partitions
of the set NnUNn,lJNn 11 = Nn 11 ,n',n which are obtained from
y 1 and y 2 with the help of additional partitions of the
sets Nn and Nn 11 into one-point blocks. Then
5
y r1 v r ! 2 iN , n ,n
( 5)
where
and
y1vy2 is the greatest lower bound of
Y 1 v Y 2 11 N E A , is the partition of n",n n ,n
y1 and y2 Nn, , n gener-
a ted by the partition r1 vr 2 of Nn",n1,n
The proof follows easily from conditions 1) and 2).
Let s(y) be the number of the blocks of the parti
tion y . Then, as easily follows from the definition (5),
s (Y) ~, min ( s ( y 1 ) , s ( y 2 ) )
Let B be a cluster operator B: H h->- H 1 hI and n, n , AY its connected components. The largest value s(y) for
the partitions yEA 1 such that n ,n
is called the rang of the cluster operator B . If
B: Hr->- Hr 1 is a cluster operator we put
r(B) max r(B(n 1,h 1),(n,h))
where B(n 1,h 1),(n,h) are the matrix blocks of B .
Corollary of Lemma 3. Let Mor s be the set of nol:fhisms
of the category M which have rang r~s . Then Hors is a
two-sided ideal of the category N •
Let B*: Hr 1 -.. Hr be the operator conjugate to the
cluster operator B: Hr ->- Hr 1 • Then B* is a cluster opera
tor and the map *: B->- B* generates a contravariant func
tor from N to itself.
The set M(f,f) of cluster operators from Hr to it
self is an algebra with unity and involution.
Let {U~n,h), tEZv} be unitary representation of the
group z in the space Hn,h
(U(n,h)f) (T) = f(T+t) t
The direct sum of these representations give unitary re
presentation
@
(n,h) Ef
6
0 (n,h) t
in the space Hr . Evidently, every cluster operator
B: Hr +Hr 1 satisfies the relation
Remarks
BU (f) t
1. The previous constructions permit the following
generalizations:
(6)
a) instead of the group Zv we can consider another
infinite group G ;
b) instead of the spaces Hr we can consider a more
general class of spaces and representations of the groups n
(Zv) (or Gn which act in these spaces;
c) we can replace condition 2) of the definition of the
operator Ay connected with respect to the partition y
by some other condition providing "connectedness" of the
variables "b_elonging to the same block of the partition y "
2. We shall indicate certain subcategories of the cate
gory N
a) subcategory Nfin ; its objects are the direct suns
of spaces H n,n corresponding to finite-dimensional spaces
h ;
b) subcategory Np p>O ; connected components
morphisms of this category satisfy the condition:
for i 13 I , i 13 1 I s P
A y of
where S = (S 1 , ••• ,13n) , 13 1 = (13,, ... ,13~ 1 ) are multiindexes,
13 81 8n 8 1
ji3/ =i3 1+ ... +13n, [13 1 / =131~+ ... +13~~, T =t 1 ••• tn , T =
81 I 13~ 1
t 1 tn ; ;
c) subcategory Nh.f · c Nfin (hyperfinite), its objects
are finite direct sums of the spaces Hn,h , where h is
finite dimensional.
7
2. The category N and the Fourier functor
The category N constructed above corresponds to the
"space"-representation ( x -representation) of states and
observables in quantum mechanics. We shall introduce now
another category N which corresponds to the "moment"-re
presentation ( p -representation) .
The objects of the category N are the spaces of the
form
Hr ~ Hn,h (n,h) H
where H n,h v n
L 2 ((T ) ,h) is the space of the h -valued
functions
v -dimensional
(dA=dA 1 ..• dAn , dA
The operator
torus) with the norm
( J ''-~'(A)~2dAl1/2 l II~ Rh (Tv) n j
the Haar measure on Tv ) .
is called cluster operator (or A -cluster) if it can be re
presented in the form
(Bf) (A I) f K(A 1 ,A)f(A)dA (Tv)n
(for a dense set of the functions f and the kernel
K(A 1 ,A) is the operator-valued distribution of the form
I yEA I n ,n
K(A 1 ,A) a (A 1 ,A)o y y
(7)
(8)
where
tion
A== (A 1 , ••• , AJ , A 1= (A; I , ••• , A~ I )
y= (a 1 , .•. ,as)
, and for the parti-
s a rr o(n.-n~l
y i= 1 1. 1.
8
1!. 1.
L A . jEainNn J
1!~ 1.
L A~ I with o(•) a 0 -func-j 1 Eai nNn 1 J
tion on the torus and continuous operator-valued
functions on the manifolds
D = {A A I. y I •
called the cluster functions of the operator B • The in
tegral in (7) is understood as Bochner integral. More gen
erally, the bounded operator
is called A -cluster operator if all its blocks - -
B(nl,hl) ,(n,h): Hn,h + Hn1,h 1 are the same. Assertions similar to Lemmas 1 and 2 are true.
Lemma 4. 1) An operator of the form (7), (8) is bounded.
2) The representation of the kernel K(A 1,A) in the
form (8) is unique.
3) The composition B = B2B1 of the A -cluster opera
tors B1 : Hr+Hr 1 and B2 : Hr 1 +Hr" is a A -cluster
operator.
4) The unit operator Er: Hr+Hr is a A -cluster
operator.
The set Mor(Hr,Hr) =.Mor(f,f 1) of the morphisms of
the category N consists of the cluster operators. As it
follows from Lemma 4 this definition is correct. We intro
duce two important subcategories of the category N simi
lar to the corresponding subcategories of N • -fin 1. Subcategory N : its objects are the direct sums
of the spaces Hn,h , where
spaces.
h are finite-dimensional
2. Subcategory N its morphisms are p
operators with p -smooth cluster functions
3. Hyperfinite category Nh ·f. c Nfin
finite sums of the spaces
sional.
H where h n,h
A -cluster
ay its objects are
is finite dimen-
9
Let F : H 7 H be the unitary transformation n 1h n 1h n 1h
(Fourier transformation)
(F hf) (A) = f (A) = 1 12 L exp{i}: (/-. 1t.) }f (T) n 1 (2n)nv J J
TE(Zv)n
where v n v n
A=0 11 ... 1/-n}E(T) 1 T=(t11 •.. 1tn)E(Z).
Let us assume also that
F = r L F h: Hr 7 Hr (n 1h)E:f n 1
The transformation Fr generates the covariant functor
F: N 7 N (the Fourier functor) from the category N to the
category N :
( 10)
and for B E Mor ( r I r I )
FB = B
Lemma 5. Definitions (10) 1 (10a) are correct i.e.
FB E Mor ( r 1 r 1 ) and in fact the map F is a functor.
Reraarks
1. The image F Mor(f 1f 1 ) cl Mor(f 1f 1 ) 1 i.e. the class
of A -cluster operators is wider than the class of Fourier
images (10a) of cluster operators.
2. The functor F transforms the subc~tegory Np in
to the subcategory N but F Mor (f 1f 1 ) #Hor (f 1f 1 ) p p p
where Hor (flf 1 ) and Hor (f 1f 1 ) denote the set of mor-p p
phisms of the categories N and N respectively. p p
3. The representation {U~ 1 tEZv} of the group zv
in the space Hr is transformed by the Fourier transform
F into the representation {U~ 1 tEZv} 1 which in the spaces
Hn 1h £ Hr has the form
(iJ~f) (A)~ exp{i(t 11- 1+ ... +An)}f(A)
A
10
3. The s -superstructures and s -functors
We give some very useful constructions for the inves
tigation of the cluster operators.
Let s and nls be integers and ~(s,n) the set of
pairs (b,0) where b is a partition of the set N con-n sisting of s blocks and 0E:II is a permutation s of the
set N s {1, .•• ,s} vle denote by Hs n,h the Hilbert space
s v n _ Hn, h = i 2 ( ( z ) , h ~ i 2 (:::: ( s, n) ) )
We can consider its elements as functions f {f (b,0),
(b,0)E~(s,n)} with the values in
r = { (n,h)} we call the space
v n .t 2 ( (Z ) ,h) • For every
H~ = ED H~,h
the s -superstructure of the Hr • Evidently, the spaces
are the objects of the category N • Every operator
generates a matrix of operators
B (b 1 , CJ 1 ) , (b, CJ) : Hn, h ~ Hn 1 , h 1 (b 1 , CJ 1 ) E ~ ( s, n 1 ) ,
(b,0) E ~ (s,n).
We consider the cluster operators B: Hs ~ Hs 1 h 1 which n,h n , satisfy the following conditions:
a) the operators B(b 1 101 ),(b,cr) have the form
B (b I I ) (b ' = M 1 ' 0 ' ' 01 b 1 ,b,0 1 ,0-
where ~I ,b is a cluster operator acting from Hn,h to
b) the cluster expansion of Mb 1,b,cr has the form
Mb 1,b,cr ( 11 )
where the partition y y(b 1,b,0) = (a 1 , ... ,as) is defined
in the follOWing Way: (Xl, = 8l. V 8 I i=1 1 • • • ,S 1 b = 0- 1 (i)
= (8 1 , .•• ,Ssl
blocks of b
b 1 = {81, ... ,8~} (the enumeration of the
and b 1 is similar to the enumeration of the
11
blocks of the partition Y E A ) . We denote by n' ,n
Mors(r,r') £Mor(H~,H~,) the set of cluster operators
Such that their blocks B · Hs ~ (n' ,h' ),(n,h) • (n, s)
s H(n' ,h')
satisfy conditions a) and b).
Lemma 6. The sets of the objects {Hs} and the morr phisms Mors(r,r') form a subcategory of the cate-
gory N we denote it by Ns and call it the s -super-
structure of N s
Proof. The unit operator Er:
every r to Mors(r,r) Further, (i)
the blocks B(b',a') (b,a) i~1,2
s s Hr ~ Hr belongs for
the expansions (11) of
of the operators
B (1) •. Hs ~ Hs d B(2) Hs ~ Hs nIh n I I h I an : n I I h I n II I h II
imply
again (with the help of Lemma 3) the expansion of the blocks
of operator B = B {2 )B (1 ) in the sums (11).
Similarly we can define the categories Ns s=1,2, ..•
as the superstructures of the category N The objects
of Ns are the direct sums of the spaces Hs h consisting n, of the functions a (b,a) I (b,a) E::: (s,n)} with values in
L2 ((Tv)n,h) • ~he morphisms of Ns , B: Hs ~ Hs, h' con-n,h n ,
sist of the blocks B defined by the kernels (b' ,a'), (b,a)
of the forms
K(b' ,a'), (b,a) (i\' ,1\)
(The partition y y(b',b,a',a-1 ) is defined above.)
For every space Hs we define a representation of the n,h
group (Zv)s by the formula
s -[Un,h (T) f] (b,a) (T) = f (b,a) (T+T),
where t. = T J a- 1 (i)
if
s fEH h n,
12
We define the representations s v s
{ U r ( T) , T ( z ) } of the s
group (Zv) as the direct sums of representations (12).
In addition for every B E Mors ( r, r 1)
From here it follows that the spaces
Us and the morphisms BE Mors ( r, r 1) r
H~ , the operators
can be decomposed in
the direct integrals:
s Hsr= J exp{ii(t.,n.)}Es{n 1 , ... ,n) IT dn.
s 1. 1. s '-1 1. (Tv) 1.-
s 8 J 8{n 1 , ... ,nsl.rr dni
(TV)S 1.=1
( 1 3)
Here {H~{n 1 , ••• ,ns) (n 1 , ••• ,ns) E (TV)s} is the family of
s spaces, Er{n 1 , ••. ,ns) is the unit operator in
Hr(n 1 , ••• ,ns) and 8(n 1 , ••• ,ns) is an operator from
s s Hr(n 1 , ... ,ns) into Hr1 (n 1 , ... ,ns) . In addition for every
s collection {n 1 , ••• ,ns) the spaces {Hr{n 1 , ... ,ns)} and
the operators {8(n 1 , ••• ,ns)} form the category
Ns{n 1 , ••• ,ns) . The decomposition (13) are called the
canonical ones of the category Ns . Similarly we can in
troduce the canonical decomposition of the category Ns ,
that has a very clear structure. Namely, for each parti
tion b of Nn we can introduce new variables on the n n
torus (Tv) with the help of the decomposition (TV) s
={TV) xLb:{\1, ... ,\n) + i=1, ... ,s,
JlELb , where Lb ~ (Tv) n is the subgroup of
Lb{(\ 1 , ... ,\ ): ff.= L A.=O (i=1, ... ,s), b~(S 1 , ... ,Ss) n 1 'f"B J J- i
Thus each function f(\ 1 , ... ,An) can be written as
f(n 1 , •.• ,ns,u) . The space H~,h(n~, ... ,n~) consists of
the functions {f (b,o) (U), (b,o) E::: (s,n), uc:Lb} and the
l3
decomposition (13) in this case means 0 0
A1l11•••11ls { f (b I 0) p, 1 I ••• I As I ]J) =f (b I 0) ( 1l 1 I ••• I 1l s I ]J) } ~ { f (b I 0) ( ]J) } I
where =f (b 10)
0 0 (1! -1 l•••lll -1 I]J). The
o ( 1) o (s) operator
kernels of
0 0 B(TI 11 ..• 11ls) is represented with the help of the
the form
0 0 1l11•••11ls
K (lJI Ill) (b 1
1 0 1 ) 1 (b 1 o)
-where a~(lJ 1 1 1J) are continuous functions on the manifolds
y = y (b 1 1b 1 o 1 01 ) and <'\ (lJ 1 1 lJ) is o -function concentrated
on this manifold. We define now the functor Ts: N ~ Ns ;
Ts: H ~ Hs and for r r
where
B = I y
A EMor(H h1H 1 h 1 ) y n1 n 1
s ( T B) (b I I 0 I ) I (b I 0) = I A
y~y y
- - 1 -1 y=y(b 1 1b 1o o ) is defined above. The morphism
TsB E Mors ( r 1 r 1 ) is defined by ( 14) for every morphism
BE Hor (f 1 r 1 ) of the category N
( 14)
Lemma 7. The map Ts N ~ Ns is a covariant functor
and Ker Ts coincides with the ideal Mor s c .t<!or N •
The proof is simple and we omit it. Similarly we can
introduce the functors Ts acting from N to Ns . The
decompositions (13) and its analogous counterpart generate
the decompositions of the functors Ts and fS into the
family of the functors {Ts(TI 11 ... 11ls)} and (j'-s(1l11 ... 11ls)}.
n4. The hyperfinite category Nh.f.
Theorem 8. Let B: Hr ~ Hr be an invertible cluster
operator acting in the finite sum of the spaces Hn 1h
14
Then the inverse operator B- 1 is a cluster operator.
We give a short outline of the proof of this theorem.
If we assume that B- 1 is a cluster operator with the
cluster function {b } we obtain equations for them after y
a substitution of the cluster expansions for B and B- 1
in the equalities
-1 B B = Er •
After the application of the functors we can
obtain the system of equations
s=c1,2, ••• , s = 0
max n (n,h) Er
s s where B (n 1 , ... ,ns) =T (n 1 , ... ,ns)B. This system of equa-
tions have a hierarchical structure: the first of them for
s s contains only the cluster functions {b } of B- 1 = 0 y
with s(y)=s 0 and for every fixed collection (n 1 , ..• ,n ) so
has the form of an algebraic linear equation. The matrix of
this equation is non-degenerate as it easily follows from
the existence of the inverse operator B- 1 Thus the clus-
ter functions
equation for
exist with s(y)=s 0 . Then, the next
is reduced with the help of the solu-
tion of first equation to an equation of Fredholm type. This
equation has the cluster functions by of B- 1 with
s(y) = s 0-1 , as its solution. By repeating this arguments
we get all cluster functions of B- 1 .
In the case when B has p -smooth cluster functions
the inverse operator B- 1 has the same property.
The notations and the constructions developed above
are very useful for an investigation of the spectrum and
the scattering of a self-adjoint cluster operator (see [4]).
References
[1] Abdulla-Zadeh F. H., Minlos R. A., Pogosian S. K. Multicomponent random systems. Ed. Dobrushin R. L., Sinai Ya. G. Harcel Dekker Inc. N.Y. -Basel, pp. 1-37.
15
[ 2] Malyshev V. A., Minlos R. A., I. J. Stat. Phys. 21, N 3, 231-242 (1979). II. Commun. Math. Phys. 82, 211-226 (1981).
[3] ManhlWeB B. A., MHHnoc P. A. Tpy~ ceMHHapa HM. lleTpOBCKOro. MrY, 9, 63-80 (1983).
[ 4] Minlos R. A., Mogilner A. I. (in preparation).
17
C*-ALGEBRA APPROACH TO GROUND STATES OF THE XY-MODEL
Huzihiro ARAKI and Taku MATSUI
Research Institute for Mathematical Sciences
Kyoto University, Kyoto 606, JAPAN
I. Introduction
C*-algebras are known to provide a proper mathematical framework
for quantum statistical mechanics of spin systems on an infinitely ex
tended lattice in the same way as Hilbert spaces for quantum mechanics.
In the present article, we try to show that theory of C*-algebras is
useful not only for a general formulation but also for a concrete com
putation by illustrating how all ground states of the one-dimensional
XY-model in an external transversal field can be determined through
C*-algebra methods.
The XY-model on a finite one-dimensional lattice has been ''exactly"
solved through the Jordan-Wigner transformation [6]. In the case of
equilibrium state at non-zero temperature, the infinite volume limit of
the finite volume Gibbs canonical ensemble provides the unique (KMS)
equilibrium state for the infinite system. On the other hand, a (kink)
soliton state for the Ising model on an infinite one-dimensional lattice
is a ground state in the sense that any local perturbation of the state
produces states of equal or higher energy (i.e. it is a state of the
lowest energy) and is not obtained as the limit of the ground state of
a finite system. We formulate the whole problem directly for an infinite
system and determine all such states, if they exist. In particular, so
liton states are shown not to exist as ground states for the XY-model,
except for the Ising model case (which corresponds to specific choices
of parameters of our model).
In the present approach, C*-algebra methods play a crucial role in
two different places. One is with the Jordan-Wigner transformation,
which changes Pauli spin observables to anticommuting Fermion creation
18
and annihilation operators. Each creation or annihilation operator at a
lattice site, in its definition in terms of Pauli spin operators, con
tains a product of spin operators on all sites to the left of the lat
tice point (a "string" connecting the left end of the lattice to the
lattice site of the "Fermion") and this product does not converge in an
infinite lattice. Our remedy taken from an earlier work [3] is to intro
duce this "string" operator as a new operator T outside of the given
C*-algebra A of Pauli spin operators on all lattice sites, define a
larger C*-algebra A generated by A and T and define Fermion ere-
ation and annihilation operators (by the Jordan-Wigner transformation
involving T) which generate a subalgebra ACAR of A which is dif-
ferent from A. As far as ACAR is concerned, we can follow usual
"exact" computation and find various information about it including its
ground state which is unique for all values of the parameter. We then
try to determine ground states of A from information about ACAR .
This is the second place where C*-algebra methods provide a powerful
tool. In particular, the criterion for equivalence or disjointness of
restrictions of two Fock states to the even part ACAR of ACAR (even +
relative to Fermion numbers) in terms of the Hilbert-Schmidt norm (of
the difference of proejection operators defining the two Fock states)
and the z2-index introduced in [2] plays a decisive role in finding the
number of pure ground states.
For large values of the external transversal field, the ground
states turn out to be unique, while for smaller values of the external
transversal field there exist two and only two ground states breaking
the symmetry of 180° degree rotation of all spins around the z-axis,
with exceptions of (i) the case of XY-symmetry and (ii) the case of
Ising model. At the critical value of the external transversal field
(i.e. at the boundary of the above two regions) as well as in the XY
symmetric case with subcritical external field strength, correlation
functions, which are otherwise real analytic in two paramters for exter
nal field and XY-asymmetry, become singular. Ground states are unique
also for those critical values of parameters. For the case of Ising
model, there exist two pure ground states, which analytically continue
to two pure ground states for nearby values of parameters, plus two sets
of infinitely many pure ground states corresponding to kink soliton
solutions which are not translation invariant (a reason for the infinite
number of them), the two sets being distinguished by non-equivalence of
associated representations of the algebra of spin operators, and again
19
breaking the symmetry under 180° rotation of all spins around the
z-axis.
The two phases with unique and non-unique ground states are also
distinguished by its stability under local perturbation. The unique
ground state will return to itself under a local perturbation but, in
the region of non-unique ground states, not only mixture but also pure
ground states do not return to itself under a local perturbation in
general.
While the main emphasis of this work is on the application of an
abstract technique to determination of number of pure ground states, the
present analysis yields also an explicit formula for correlation func
tions. However, we shall give only a brief sketch of C*-algebra argu
ments for determination of number of ground states and detailed ana
lysis is referred to forthcoming publication [4].
2. Model
At each site of a one-dimensional infinite lattice Z, ob-
servables for 1/2 spin are elements of the algebra of Pauli spin oper-
a tors
-1) 0 '
which commute with Pauli spin operators at other lattice sites. The
C*-algebra (of observables) generated by all Pauli spin operators at all
lattice sites is denoted by A below.
The Hamiltonian of an XY-model for a finite interval [-N,N]
(N 1,2, ... ) is as follows:
N-1 -J[ L:
j=-N
(2. 1)
Here J, y, and A are real parameters and we assume J > 0 . For the
infinite system, the above Hamiltonian defines the dynamics as follows:
iHNt -iHNt lim e A e (2.2) N->«>
Here A is any element of the observable algebra A, at (A)(- 00 < t< 00 )
20
is its time evolution and the limit in the defining equation (2.2) is
known to exist by a general theory on spin lattice systems [5].
Signs of each of three real constants J (total strength), y (x-y
asymmetry) and A (strength of transversal external magnetic field
relative to nearest neighbour interactions of x and y spins) can be
changed by the following symmetry transformations, which are automor
phisms of the C*-algebra A:
(i) The sign of J is reversed by 180° rotation of spins at all
even sites around the x-axis and 180° rotation of spins at all odd sites
around they-axis, which are obtained by the N-> oo limit of the unitary
transformation uAu * (A E A) with
N u = n
j=-N
(ii) The sign of y is reversed by 180° rotation of spins at all
even sites around the z-axis, which is obtained by the N-> oo limit of
the unitary transformation u Au* (A E A) with
u = N :L
j=-N
(iii) The sign of A is reversed by 180° rotation of spins at all
sites around the x-axis, which is obtained by the N-> oo limit of the
unitary transformation u Au* (A E A) with
u
N n
j =-N
The Hamiltonian (2.1) is invariant under 180° rotation of spins at
all sites around the z-axis given by
G(A) lim ( N 0 (j) \ A
( N 0(j)\ (2.3) - ' :L \ :L N-.oo \j =-N z ) 'j =-N z )
(More concretely, 8(0(j )) =-0(j) G(0(j)) =-0(j) 8(0(j )) 0 (j).) X X y y ' z z
Consequently, for all A E A
at(G(A)). (2.4)
21
3. Ground States
A state ~ of the C*-algebra A, as an expectation functional
~(A) for A E A, is given as a vector state
~(A) = (~ ,n (A)~ ) ~ (j) (j)
(3. 1)
by a unit vector ~ (j)
in a Hilbert space H lP'
which carries a represen-
tat ion nlP(A) (A E A) of the C*-algebra A and which is minimal in the
sense that the vectors nlP(A)~lP' A E A, are dense in H . (TheGNS-triplet.) (j)
If a state (j) is stationary in the sense that lP(at(A)) =
lP(A) for all A E A and for all real t then there is a continuous
one-parameter group of unitary operators UlP(t) determined by
U (t) n (A)~ = n (a (A))~ ~ (j) (j) (j) t (j)
(A E A) (3.2)
and they implement the time evolution automorphism at
U (t) n (A) U (t)* = n (ex (A)). (3.3) ~ ~ (j) (j) t
We shall denote the generator of
As a special case of (3.2) with
u (t) ~
A= 1,
by
~ (j)
HlP: UlP(t) = expitHlP
is invariant under U (t) (j)
and hence is an eigenvector of the
to an eigenvalue 0 .
selfadjoint operator HlP belonging
Interpreting HlP as an energy operator in the space HlP under
consideration, H~ ~ 0 is the same as saying that the state (j) given
by the vector ~lP has the lowest energy at least among states given by
vectors in HlP . (Since the vectors nlP(A)~lP with A a polynomial of
a finite number of Pauli spins are already dense in HlP, these states
may be regarded as a result of more or less local perturbation on 4).)
If this is the case (namely if (j) is stationary and
called a ground state.
H ~ O), (j)
lP is
Let f be a complex-valued function of class S of a real variable
and define
f(t) J e-ipt f(p)dp/(2n), (3 0 4)
A(f) (3.5)
22
Then
11\P(A(f) )!"liP f 11 IP(at (A) )!"liP f(t)dt
f UIP(t) f(t)dt 11\P(A)rliP
f(H ) 11 (A)rl IP IP IP
Therefore, if the support of f is contained in the open interval
(-co,O), then f(H ) = 0 IP
follows for a ground state \P, i.e.
o. (3.6)
Conversely, if ~P(A(f)*A(f)) vanishes whenever the support of f is in
~co,O), then IP is shown to be stationary and HIP~ 0, i.e. IP is
shown to be a ground state.
The characterization of a ground state by vanishing of IP(Bf) for
a certain set of positive Bf (= A(f)*,\(f)) E A (i.e. by (3.6)) immedi
ately implies that the set of all ground states is a "face" of the (com
pact) convex set of all states, namely
(I) any mixture
of ground states IPI and 1Pz is also a ground state;
(2) any limit
\P(A) = lim IPv(A) (A E A)
of ground states IPV is also a ground state;
(3.7)
(3) any decomposition (3.7) of a ground state IP into states IPI
and ~Pz (with 0 <A< I) yield ground states IPI and 1Pz due to
A(f)*A(f) ~ 0 with supp f c (-co,O).
If a ground state IP does not have a decomposition (3.7) into other
ground states IPI and IPz , then IP is said to be a pure ground state.
By (3), it is then a pure state, i.e. it does not have a decomposition
(3. 7) to any other states \PI and \P2 • It is known that a state IP is
23
pure if and only if the representation IT~ is irreducible, i.e. H~
does not have any non-trivial IT (A)-invariant subspaces. ~
It also follows from (I) and (2) that mixtures of pure ground
states are dense in the set of all ground states. Thus we shall mostly
discuss pure ground states.
Sometimes, H~ may have many eigenvectors of the eigenvalue 0.
In that case there are different pure ground states, for which the cor
responding (GNS) representations IT~ areequivalent. We call these states
equivalent. If the corresponding GNS representations are disjoint, then
states are called disjoint,,which is the same as not equivalent in the
case of pure states.
4. Main Results
We now describe our results about qualitative features of pure
ground states of the XY-model. The following 4 cases for values of para
meters A and y are to be distinguished. (Under the assumption
J > 0, the parameter J does not play any role for ground states.)
(I) 0-symmetry preserving region: lgl > 1.
(2) Critical points: (i) I A I 1, y * o. (ii) I A I < 1, y 0.
(iii) I A I 1, y 0.
(3) Region of broken 8-symmetry: IAI < 1, y * 0, (A,y) * (0,±1)
(4) Ising model: A = 0, lyl = ].
The number of pure ground states is as follows:
in cases (I) and (2).
2 in the case (3).
in the case (4).
The number of non-equivalent representations associated with
ground states is as follows:
in cases (I) and (2).
2 in the case (3).
4 in the case (4).
24
About the question of whether a pure ground state ~ returns to
itself under local perturbation in the sense that
lim ~(a (A)) =~(A) t->oo t
for all ~(A) ('l',rr~(A)'l'),
(A E A)
'¥ E H , 11'¥11 ~
yes in cases (I) and (2).
no incases (3) and (4).
(4. I)
I, the answer is
About the question of whether 0-symmtry is broken in the sense
that the state (~ o 8) defined by (~ o 0) (A) = ~(O(A)) is disjoint
from ~. the answer is
0-symmetric (~ = ~ o 0) in cases (I) and (2) ,
broken 0-symmetry for any pure ground state in cases (3)
and (4).
For a fixed polynomial A of Pauli spin operators (strictly local
observables), dependence of ~(A) on parameters A and y is
analytic in regions (I) as well as in the union of regions
(3) and (4)
singular
(for an appropriate choice of two pure ground
states ~±),
at points of (2) •
Pure ground states of the Ising model has been determined in [5]
and can be explicitly described as follows.
For A= 0 and y =I, the Hamiltonian is
-2J N-1
L: j=-N
(4. 2)
The pure ground states ~± , mentioned above in connection with analytic
dependence of ~(A) on (A,y), are the product of eigenstates ~(j) ±
of o(j) belonging to eigenvalues X
~± n ~(j) ±
±I at each lattice site:
(4. 3)
25
A class of kink ground states are given by
c n E; ~j)) c n E; (j)), k j<k j;:o:k +
0, ±I, ±2, .... (4.4)
They are associated with a single irreducible representation of A and
hence their superposition yields a pure ground state, too. Although the
eigenvalue of HN for (4.4) is larger than its eigenvalue for (4.3)
for all finite N, any local perturbation of (4.4) can not eliminate
the kink due to the different asymptotic behavior of the state at large
distance in the right and the left (asymptotically (j)+ for j -+ +oo and
(j)_ for j -+-oo) and yields states of the same or higher energy.
Another class of pure ground states are the image of (4.4) by
8-syrmnetry and are given by
0, ±I, ±2, ...
and their superpositions. These 4 classes exhaust all pure ground
states in case (4).
For A = 0 and y = -I , the Hamil toni an reduces to
- 2J N-1
2: o(j) j=-N y
(J (j+l) y
(4. 5)
and ground states are obtained by exchanging x and y in the above.
5. Introduction of the Operator T
We need an operator T which can play the role of the non-exist
ing limit of
TN o(-N) (J(-1)
z z
as N -+ +oo. Due to (o(j ))* z
adjoint unitary operator:
T* N I.
o(o) (5. I) z
(J(j) and (o(j))2 I, TN is a self-z z
(5.2)
26 iHNt
As in the case of Hamil toni an HN , for which N ->- oo limit of e
does not exist but (2.2) exists, we consider the N ->- oo limit of the
automorphism of A induced by the (selfadjoint) unitary operator TN:
8_ (A) "' lim N->-ao
(A E A). (5.3)
This limit exists and can be computed explicitly for Pauli spin oper
ators:
8 (O(j}} - X
8 (o(j)) - y
8 (o(j)) - z
s(j)o~j)'
s(j)o~j),
(i) a . z
Here s(j) = I for j ~ I and s(j) = -1 for j ~ 0 •
From the explicit form of 8 above, it immediately follows that
id. (i.e. 8 (8_(A)) A} (5.4)
first on Pauli spin operators and hence for all elements of A . We can
say that the group z2 (the additive group of integers n modulo 2)
is acting on A as a group of automorphisms en (= 8 for odd n and
= id. for even n). According to a general theory of crossed product,
there exists a C*-algebra A (written as A x8 z2) which consists of
elements
A + BT (5. 5)
with A and B arbitrary elements of A, satisfying
In effect, we introduce a new element T satisfying
I' (5.6)
27
T A T 8 (A) (5. 7)
and generate a new algebra A out of A and T . (The above formulas
follow from (5.6), (5.7) and general rules for algebraic manipulation.)
We note that no elements of A can satisfy (5.7), i.e. 8 is an
outer automorphism, which can be shown, for example, as follows: Let.
T be the lattice translation automorphism of A satisfying T (o(j))= n(j+n) n a aa (a = x,y,z) for all j . We have the following asymptotic
abelian property:
lim [Tn(B),A] = 0, (5.8) n->oo
This is immediately checked when A and B are Pauli spin operators
(they start commuting for large n ), which generates A, and hence (5.8)
holds for arbitrary A and B in A. In particular, if A is invert
ible, we have
lim Tn(B)A A.
If T were in A, then (5.9) should hold for
B-l = B because (5.7) with A= implies T2 = I
On the other hand, (5.7) implies
T 8 T (A) n - -n
B = T, where
i.e. T = T-l
(5. 9)
which tends to 8(A) as n -> + 00 and contradicts with (5.9). Therefore
8 is an outer automorphism.
6. CAR Algebra
We can now write down the Jordan-Wigner transformation for an in
finite system and introduce Fermion creation and annihilation operators
c! and c. (at the lattice site j) as follows: J J
c! J
(6. I)
c. J (6.2)
28
Here plays the role of the N -> +oo limit of 0 (j -I) z
and is given in terms of T by
if <: 2, (6. 3a)
T, (6.3b)
if j ~ 0. (6.3c)
For j < k, T(i)
anticommutes with
commutes 0 (j)
with T(k) 0(k) and
0 (j) ' X
0(k) , while y
and Therefore X y
(j * k)
where B. is either c'!' or c .. For = k, the operator T(j) dis-1 1 1 2 (c'!')2 (T (j)) 2 appears from cj ' c. c'!' and due to = I and the ma-
J J J trix representation of 0's in Section 2 can be used to find commuta-
tion relations. As a result, we obtain the following canonical anti
commutation relations (CARs):
[ c'!', ck*] J +
0, (6.4a)
[c.,ck*] J +
(6.4b)
where
by c.
[A,B]+ = AB + BA.
and c'!' , j E Z, J
We shall denote a C*-subalgebra of A generated
as ACAR. J
Pauli spin operators can be expressed in terms of T , cj
as follows: First we have
2c'!'c. - I (0(j) + i0(j))(0(j)- i0(j))/2 - I 0(j) J J X y X y z
and c'!' J
(6.Sa)
Now T(j) is expressed by T and 0 (j) '
j z
E;z by (6. 3). By (6.1)
and (6.2), we then obtain
T(j) i (c.- d). J J
(6.Sb)
The automorphism 8 of A defined by (2.3) can be extended to an
automorphism of A satisfying G(T) = T. Then
G(c'!') J
-c'!', G(c.) J J
-c .. J
(6.6)
29
Because of 8 2 = id. , any element A E A can be decomposed as a sum
of 8-even and 8-odd parts:
A± = (A+ 8(A))/2 E A± ,
A± 5 {A E A; 8(A) ±A}.
Since A and ACAR are setwise 8-invariant, we have
A = A + A +
(6. 7)
(6.8)
(6.9)
(6. I O)
From (6.6) and (6.7), it follows that A~AR is generated by even
polynomials of c's and c*'s. By (6. I) and (6.2), it is contained in
A+. (T cancels out.) On the other hand, A+ is generated by poly
nomials of a's which have even total degree in cr~s and cr;s. Such
polynomials are even polynomials of c's and c*'s by (6.5). Therefore,
A + ( 6. I I)
Similar argument leads to
A (6. 12)
By (5.5), we have
A A + A + A T + A T + +
(6. 13)
We define the time evolution at(A) for A E A also by (2.2).
To show the existence of the limit, it is enough to show it for A = T:
at(T) lim e iHNt
T -iHNt
(T • T) e N->oo
lim iHNt -iHNt
(6. 14) e 8(e )T = vt T, N->oo
iHNt -i8(H )t v - lim e e N (E A+), (6. 15) t
N->OO
30
where we have used T2 = I in the first equality, (5.7) in the second
equality and the automorphism property of 8 in writing Vt
To see the convergence of (6.15), we note that 8(HN) differs from ~
by an N-independent bounded operator:
HNt Therefore, e has an absolutely convergent perturbation ex-
pansion which converges as N + + to
v t n=o
0
Actually, we do not use an explicit expression (6.16) for Vt
Since H E A = ACAR at commutes with 8 (by 8(HN) CARN + .+
leaves A (and obv~ously A) setwise invariant.
7. Exact Solution for ACAR
(6. 16)
As far as ACAR is concerned, we can follow "exact" computation
for a finite system ([6]). We describe here only the result of such a
computation.
First, to describe the time evolution which turns out to be linear
transformations of c's and c*'s (mixed together), it is convenient
to introduce a notation which unifies c's and c*'s [1]. We define
B(h) (7. I)
where f = (fi) E i 2 (~). g = (gi) E i2 ~), h
to converge (in norm). Then
(f) and the sum is shown g
B(eitKh) ,
ZJ (U + U* - 2,\ \-Y(U - U*)
y(U - U*) ) -(u + u* -2f.)
Here U and u* are shift operators on i 2 (zz) defined by
(Uf). J f. I J+
(U*f). = f. . J J-]
(7.2)
(7. 3)
(7.4)
The merit of the Jordan-Wigner transformation (6.1) and (6.2) is that
at can be explicitly computed in the form of (7.2).
31
The operator K can be further analyzed by the Fourier series.
Let
f(8)
Then
K(8)
K(8)h(8),
( cose - J.. 4J
\iy sine
f n
(2n) - 1 J e -ine f(8)d8
-iysine\
)..- cose)
(7.5)
(7. 6)
(7. 7)
The eigenvalue of K(8) is ±4Jk(8;J..,y) where
(7. 8)
We immediately see that K has an absolutely continuous spectrum if
(J..,y) * (0,±1), while K is 4J times a selfadjoint unitary operator
with its spectrum concentrated on two points ±4J if (J..,y) = (0,±1)
(Ising model).
A slight digression on operators B's is in order. The CAR's for
c's are equivalent to the following relations for B's:
B(h)* = s cr h )
where ref) = (!:) and g
fACAR B's generate
acterize the algebra
of c's and c* 's .
(f) i
and the
of B's
(7.9)
(7. 10)
fi is the complex conjugate of fi .
relations (7.9) and (7. 10) completely char
just as CAR's characterize the algebra
Any unitary operator W (acting on h E K2 @ K2) commuting with r
preserves the relations (7.9) and (7. 10) and hence induces an automor
phism w of ACAR satisfying w(B(h)) = B(Wh). (It is called a BogoiKt liubov automorphism.) e appearing in (7.2) is such an operator.
The commutativity with f is pquivalent to
r K r = -K. ( 7. I I)
32
We now discuss the relation of c and c* to B . To go back
from B to c and c*, one has to specify a projection operator E
(E = E* E2) satisfying
fEf I - E. (7 0 12)
We call such a projection a basis projection. Whenever a basis projec
tion E is given, we may interprete B(h) as annihilation operator
(c's) if Eh 0. For a general h, we have h = ff + g with
g = (1-E)h, f = fEh= (1-E)fh (by (7.12)), and B(h) B(f)* + B(g).
The CAR's for B(g) follow from (7.9) and (7. 10). In particular we ob
tain original c's and c*'s by taking E = (b). Immediately below,
we will use a basis projection E for which annihilation operators B(g)
is a linear combination of the original c's and c*'s.
We now want to find ground states of ACAR (relative to the time
evolution at) , although this is not exactly the problem we are sol
ving. For this purpose, we need a digression on Fock states.
Let E be a basis projection and ~E be a state of ACAR satis
fying
~E(B(h)*B(h)) = (h,Eh). (7 0 13)
We can easily see that there is a unique state ~E satisfying (7. 13),
called a Fock state. The uniqueness is shown as follows:
By (7. 13) and the Schwarz inequality, we obtain
~E(A B(h)) = ~E(B(h)*A) = 0 (7 0 14)
whenever Eh = 0 i.e. B(h) is an (E-)annihilation operator. (We are
no longer talking about the original c's.) In any polynomial of B's,
a general B(h) is decomposed as a sum of B(f)* and B(g) with
Lf = Eg = 0, and then all B(f)* can be brought to the left of all
B(g) using anticommutation relations (possibly with additonal terms).
Then all terms containing B's (after such a reduction) contain either
B(f)* at the extreme left or B(g) at the extreme right and gives a
vanishing contribution for ~E due to (7. 14). Thus only the constant
term (a multiple of the identity) remains and the value of ~E on any
polynomial of B's is uniquely determined. Hence ~E is unique.
For any basis projection E, a Fock state can be explicitly given
(showing its existence) and is known to be a pure state.
33
Let E+ be the spectral projection of K for (0,+ ~):
(7. IS)
( 7. 16)
By (7. II), r E/ is the spectral projection of fK r = -K for (0, +""),
i.e. that of K for (- 00 ,0). Therefore
because K does not have an eigenvalue 0 . This shows that E is a CAR . +
basis projection. We now show that a ground state of A ~s necessar-
ily the Fock state ~E and hence is unique.
We choose A = B(h) in the characterization (3.6) of a ground
state. Then
A(f) J a (B(h)) f(f)dt t
= B( f eitK h f(t)dt) = B(f(K)h).
The set of all f(K)h with supp f c: (-oo,O) is dense in (I-E+) (!1. 2 Ell 9. 2 ),
i.e. in the set of all h satisfying E h +
0. Hence we obtain from
(3. 6)
~(B(h)* B(h)) = 0
whenever E+h = 0. This coincides with the characterizing equation of
a Fock state ~E (cf. (7.13) and (7.14)). +
8. Ground States of A ------------------------·+
Because 8 commutes with at' A+ is setwise
can talk about ground states of A+ (relative to
ground state of A, then (3.G) :s satisfied for all
a -invariant and we t
at). If ~ is a
A E A and hence
for A E A+ in particular. Therefore a restriction of any ground state
of A to A+ must be a ground state of A+. We shall first determine
all ground states of A+ A~AR and then find out its possible exten
sions to A.
34
We alredy have one ground state of A = + ACAR
+ ' which is the re-
striction of the unique ground state <.PE+ of ACAR to ACAR + In the
following, we shall see that this is a unique ground state of A if +
(A,Y) * (O, ±I), and we have some more ground states of A+ if (A,y)
(0,±1). First we shall describe these additional ground states.
If <.P is a 8-invariant state of ACAR, then <.P(A) = 0 for
A E ACAR Therefore the associated representation space H<.P splits
into a direct sum
H <.P
H ED H <,p+ r.p- (8. I)
(Note that (n (A )Q , TI (A )Q ) = <.P(A*A ) = 0 for A± E A± due to <.P + <.P <.P - <.P + -
A*A E A_.) The subspaces H are both TI (ACAR)-invariant. If <.P is + - <.P± <.P +
a pure state in addition, then one can show by a general argument the
following facts about the restrictions of TI (ACAR) to H + , which <.P + <.P-
we dante by TI<.P± (as a representation of A~AR):
(A) Both TI <,p+
and TI r.p-
(B) They are disjoint.
are irreducible.
We can apply this to <.PE which is a 8-invariant pure state. +
Since <.PE is at-invariant, we have U<.P(t) satisfying (3.3) for
<.P = <.PE+' Since+ 8 commutes with at' A± are setwise at-invariant and
hence (3.2) implies that H<.P± are both U<.P(t) invariant. Since TI<.P±
are irreducible, U (t) satisfying (3.3) for ~n replaced by TI . <.P CAR ~ <.P±
and A restr1cted to A is unique up to a multiplication of a com-itA +
plex number e , A E IR. Therefore its generator H<.P is also unique
up to an additive constant. This means that a vector ~ in H<.P± yields
a ground state of ACAR if and only if ~ is an eigenvector of H + <.P
and the eigenvalue is the infimum of the spectrum of H restricted to + <.P
H<.P±' respectively. In the case of TI , this infimum is 0 and Q<.P is
an eigenvector having this eigenvalue. We now discuss the spectrum of
H<.P and relevant eigenvectors.
The Hilbert space H for the Fock state <.P = <.P of ACAR is <.P E+
known to have the following structure
(8.2) n=o
35
where L is the range of E+ (as a Hilbert space). Asym denotes the
totally antisymmetric part (under permutation) of n-fold tensor product
of L with itself, and H~ is generated by n~(B(h 1 ) •.. B(hn))n~
with E h. = h. , which belongs to ~~ or ~"- according as n is + J J .... ... even or odd.
Relative to the above decomposition,
u (t) ~ n=o
iK t ®n (e + ) (8.3)
where K+ KE+ acting on L. Since K+ does not have eigenvalue 0
and is positive, n (corresponding to n = 0 in the above decomposi~
tion) is the only eigenvector of H~ belonging to 0 . Furthermore,
where H IH ~ ~
is the restriction of H~ to H~. If (,\,y) * (O,:tl),
then K + has an (absolutely) continuous spectrum, and hence H~IH~, whose spectrum is obtained by n-fold "convolution" of Spec K+, also has
an (absolutely) continuous spectrum. Hence there are no vector other
than (a multiple of) n which gives rise to a ground state. ~
On the other hand, if (.\,y) = (0,±1), then SpecK+= {4J} and
hence Spec(H~iH~) = 4nJ. Therefore all vectors in H~ 1 (and only
these vectors), which are in H~_, are eigenvectors of H~ belonging
to an eigenvalue 4J which is the infimum of Spec(H IH )={1,3,5, •.. }. CAR ~ lP"""
Hence they produce pure ground states of A+ These vectors are of
the form n~(B(h))n~ with
responding state by ~h .
E+h = h and llhll I. We denote the cor-
Finally we indicate how one can show that states described above
exhaust all ground states of A:AR. Suppose supp fj c (-oo,O) (j = 1,2).
It can be shown that
co
co
coincides with Ag whenever f g on (supp f 1) + (supp f 2) c (-oo,O),
where notation (3.4) is used. In particular, we may choose g satis
fying supp g c (-oo,O). By (3.6), a ground state of ~ of AGAR satis-+
fies
36
Since f is arbitrary, '~e may take the limit of f + 1 (for example, -<:p2
f(p) ~ with £ + +0) and obtain
<+J(A*A) 0
for A~ B(f,(K)h 1)B(f2 (K)h2) and hence for all A B(h 1)B(h2) with
E+hl E+h2 ~ 0.
Let $ be the 8-invariant extension of <+' to ACAR:
A E ACAR ± ±
Then ~A satisfies <+'
whenever E+hl ~ E+h2 ~ 0. This yields the following two alternatives:
Then <+' ACAR.
+
is the Fock state <+'E+
and is the restriction of <+'E +
to
case TIA(B(h2 ))~A. being annihilated <+' <+'
E+hl ~ 0, produces the Fock state <+JE after
proper normalization. This already shows that +
TIA contains the Fock reI+'
presentation TIE. By a general argument, one can show that rr$ is a
cyclic representation for a ground state of ACAR hence must coincide
with rr<+l , .E
sentat1on
E ~ E . Therefore +
rr of ACAR. This <+'E +
as determined above <+' = <+lh
9. Ground States of A
<+J mus be a vector state of the repre-
exists only if (J.,y) ~ (0,±1) and then
Let <+J be a ground state of A. Since 8 commutes with at , the
condition (3.6), being satisfied by <+J, will be satisfied also by <+' o 8.
This means that <+J o 8 and hence
(<+l + <+' 0 8)/2 (9. I)
37
are ground states of A. In addition \P is 8-invariant: (ji(A+ +A_) =
\ii(A+) if A± E A± Thus 4i is uniquely determined by its restriction
to A+ which is a ground state of A+ and hence is already known to
us. Therefore we can find all pure ground states of A by finding out
all pure state decompositions (9.1) of the 8-invariant extension \P of
A+ to A and check that \P is in fact a ground state of A. (Then \P
in (9.1) is a ground state of A as already discussed in section 3.)
It turns out that 8-invariant extension of ground states of A+ to A
is always a ground state of A in our model. If (A,y) t (0, ±1), \P
is unique by our uniqueness result about ground states of A+ Hence
the known existence of ground states of a spin lattice system (as accu
mulation point of S-KMS states as S ~ +00 , S-KMS states themselves as
accumulation points of finite interval Gibbs states as the interval ap
proaches Z, and the existence of accumulation points due to compactness
of the set of states) implies that the unique \P must be a ground state.
The representation space of A associated with \P splits as in
(8. I) and defines two representations of If \P is a 8-inva-
riant extension of a pure ground state \1)+ of which is necessarily
the case for the above scheme, then 11+ which is the cyclic representa
is irreducible. One can then show by tion of A +
associated with (j)+
1T quite a general arguemnt that is also irreducible and has one
of the following two alternative structures:
(a) 4i is pure, in which case it is a pure ground state of A. If
(A,Y) t (0,±1), this implies that the ground state (j) of A is unique
(and (j) = (j) o 8 = \ji).
(b) (j) is an average of two disjoint pure ground states of A. If
(A,y) t (0,±1), this implies that there are exactly two disjoint pure
ground states (j) = (j)± of A such that (j)_ +
In the rest of the paper, we concentrate on the case (A,y) t (0,±1)
and explain how one can find out which of (a) and (b) occurs for each
value of (A,y). An abstract criterion, derived by a general argument, is
as follows:
(a) occurs if and only if 1!+ and 11 are disjoint.
(b) occurs if and only if 1!+ and 1T are equivalent.
In order to compare 1T +
and 1T , we consider the state
(9.2)
38
of A , where A± B± E A;AR. (It is easily seen to be a state.)
The corresponding representation space splits as
H
The vector state by
n(T)rl is
ACAR on is
(9.3)
(9.4)
~E , while the vector state by +
(n(T)rl, n(A)n(T)Q) = ~E (TAT) = ~E (8_A) = ~F(A) + +
(A E ACAR)
where 8
ACAR) and
is extended to
F=8_E+8_, 8
A by 8_(T) = T (and hence defined on
being an operator in defined by
(8 h).= s(j)h., s(j) defined in Section 5. - J J
assocated with Fock states ~E and ~F (with
The cyclic representation
E = E+) are given on
H ED H and
/-AR :plits +
H+T ED H_T, respectively, and the restriction of each to
into a direct sum of disjoint irreducible represenations on
H ±
and H±T, respectively, by (A) and (B) of Section 8. Let us denote
these irreducible representations by TIE± and "F± , respectively.
The cyclic representation of A associated with ~ (a 8-invariant
extension of ~E lA+ to A) is the restriction of ~(A) to H+ ED H_T
(= n(A)rl). Hence+ TI = TI and TI_ = "F_ . A result in [2] then gives + E+ the following criterion for their equivalence:
"+ ~ TI if and only if
(a) E- F is in the Hilbert-Schmidt class.
(S) (-J)dim(EA(I-F)) =-I.
The rest is to check (a) and (S) using an explicit from of E
given by (7. IS) and (7. 16) and F = e E e - + - As long as k(8,1.,y)
for all e, we find liE- Fll < H.S.
00, This condition is satisfied
cases (I)' (3) and (4) of Section 4. In case (2) (i) and (ii)'
II E - Fll H. S • = oo
11.1 > 1, E+
while in case (2) (iii) as well as all cases y 0,
commutes with 8 and hence E - F = 0.
E
* 0
in
+
If liE- Fil < oo, then the quantity in (S) above, called z2-index, H.S
is continuous (i.e. constant) in E and F in the norm topology of E
39
and F . Since E and hence F = 8 E8 depend on (A,y) continuously in
any of connected components of case (1) and cases (3) and (4) together,
respectively, we can determine it at one point in each connected corn-
anent. For case (1), we take y 0, IAI > 1 (as well as y = 0, IAI =1
for case (2) (iii)) and we find out E~(1-F) = 0 due to E = F. Thus in
case (2) (i) and (ii), (o:) is not satisfied and in cases ( 1) and (2) (iii),
(S) is not satisfied while (o:) is satisfied. Hence in cases (1) and (2),
we have a unique ground state of A.
In cases (3) and (4), we evaluate the z2-index at (A,y) = (0,±1)
and, after some explicit computation, we find that dirn(E~(1-F)) = 1.
Hence (o:) and (S) is satisfied. Hence we have two pure ground states of
A for the case (3).
Discussion on case (4) as well as other information about ground
states such as ergodic properties and correlation functions will be gi
ven in a forthcoming full account of this work [4).
Acknowledgement
This work was completed while one of the authors (H.A.) was at the
Zentrurn flir interdisziplinare Forschung, Universitat Bielefeld, D-4800
Bielefeld 1, FR Germany. The financial and secretarial support of ZiF
is gratefully acknowledged.
References
[1] Araki, H.: On quasifree states of CAR and Bogoliubov autornorphisrns. Publ. RIMS, Kyoto Univ. ~. 384-442 (1970).
[2] Araki, H. and Evans, D.E.: On a C*-algebra approach to phase transition in the two dimensional Ising model. Cornrnun. Math. Phys. 2l· 489-503 (1983).
[3] Araki, H.: On the XY-rnodel on two-sided infinite chain. Publ. RIMS Kyoto Univ. 20, 277-296 (1984).
[4] Araki, H. and Matsui, T.: Ground states of the XY-rnodel, to be published.
[5] Bratteli, 0. and Robinson, D.W.: Operator algebras and quantum statistical mechanics II. Springer-Verlag, New York-BerlinHeidelberg, 1981.
[6] Lieb, E., Schultz, T. and Mattis, D.: Two soluble models of an antiferrornagnetic chain. Annals of Phys. ~. 407-466 (1961).
41
Bulk Diffusion for Interacting Brownian Particles
M, z. Guo and G, Papanicolaou*
Courant Institute, New York University
l, Introduction.
In a recent paper [1), H. Spohn analyzed the equilibrium
fluctuations in space time for a system of interacting Brownian
particles in the hydrodynamical limit.
The hydrodynamical limit is a rescaling of space and time that
leads to a reduced or collective description of the infinite particle
system, much in the spirit of passage from the Boltzmann equation to
the equations of gas dynamics. Rost [2) formulated the relevant
mathematical problem for interacting Brownian particles. Using the
fluctuation dissipation theorem, he showed how the bulk diffusion
coefficient can be identified within the framework of an equilibrium
theory. This is important because the study of the infinite particle
system at equilibrium is much simpler, There seems to be no
mathematical analysis in a nonequilibrium situation for interacting
Brownian particles,
In this note we will outline a proof of Spohn~s theorem that is
relatively simple, and therefore has wider applicability. Details of
our proof follow [3) and will be published separately. The basic ideas
are laid out in [4) for similar problems in the context of random
media.
Our discussion below will be formal since we will be dealing with
differential operators in infinite dimensions and other objects that
must be defined carefully. The key steps make sense, however, as they
are described here.
*supported by the National Science Foundation.
42
2. The Process.
Let ~(x) be a smooth nonnegative function on Rd that is even and
consider the system of diffusions that satisfy
( 2. 1)
Here [wj(t)) are independent Brownian motions on Rd.
The system (2.1) of coupled diffusions has been analyzed by Lang
[5) and more recently by Fritz [6). We will deal with (2.1) formally
here. We denote by vt the interacting particle process regarded as a
Radon point measure with
( <P , v t) = L cp( xj ( t)) j
(2.2)
for any cp E CQ(Rd). The formal generator of the process is given by
(2.3)
wher 9 j is the gradient operator with respect to the coordinates of the
j th particle.
The process defined by (2.1) is reversible relative to the Gibbs
measure consuructed with the potential~. We will assume throughout
that the particle density is low so that this equilibrium Gibbs measure
is uniquely defined. We denote it by ~ and we have, at equilibrium,
(2. 4)
where p is the mean particle density.
It is useful to think of the Gibbs measure as one associated with
the formal density.
Then the generator has the selfadjoint form
i. = ~ L 9 f (a9 j) j
(2.5)
(2.6)
43
and the associated bilinear form is
B (f, g) - (f,Lg) (2. 7)
One can base the study of (2.1) on this bilinear form which is first
defined on a suitable class of test functions [3].
If f is a function of v, i.e.
coordinates, then we can write
a symmetric function of the
'VJ.f(v) = J - Of(v) 'Vo (x-x.) dx R cSv (x) J
( 'V cSf (v )~ ov (x) x=xj
With this operator notation the bilinear form (2.7) becomes, for f and
g test functions,
B ( f , g ) = Ell { ( 'V Of • 'V cS g , ~) } (2.8) ov ov
In this notation, for example, the weak form of the resolvent
equation
Au - Lu f , A > 0
becomes
The equation (2.9) is to hold for all v in a suitable class.
3. Scaling.
The scaled process is defined by
so that for any ~ £ cQ
(v~ .~ ) = L £ dH£ xj ( t /£ 2 )) j
The formal generator (2.6) in scaled variables has the form
(2.9)
(3.1)
(3. 2)
44
The scaled Gibbs equilibrium measure is denoted by uE. it as associated with the formal density
(3.3)
We may think of
(3. 4)
If at time 0, the process is distributed with the Gibbs
measure UE then at any time t > 0, v~ will have UE as its law. When p,
the mean particle density, is small enough we know that UE tends to a
point probability measure concentrated on the measure p dx. We
introduce the fluctuation process ~ by
v~ = p dx + E d/ 2 ~ • (3.5)
Let PE be the law of ~ at equilibrium which is simply the Gibbs
measure UE translated and scaled according to (3.5). One can show (for
example in [3]) that the measure PE converges weakly to a Gaussian
measure on s~(Rd), the generalized function on Rd. Let G denote this
Gaussian measure. Its covariance is given by
where
EG{ (f,N)(g,N)} = p 2 I d f(x) g(x) dx R
P 2 = P + I d w (z) dz R
(3.6)
(3.7)
and w(z) is the second cluster function [7] of the Gibbs measure u
(unsealed).
The central limit theorem for ~ at equilibrium amounts to the
statement
(3. 8)
4. Space-time Limit Theorem
We want to characterize the limit law of ~ as a process and not
at one time point. On the other hand we want to consider ~ only when
45
initially it has the equilibrium law ~E • We consider then the
resolvent equation for ~· If
= I 00 -),t UE (N) e E{ f(~) I lib N} dt 0
( 4. l)
then UE (N) satisfies
(4.2)
for any test function v(N) in a suitable class and each A > 0.
We want to show the following
Theorem. As E tends to zero, uE(N) converges to u(N) weakly, i.e.
E~E (uEv) + EG(uv) for all text functions v, and u(N) satisfies the weak
resolvent equation
(4.3)
From this theorem, the form of (4.3) in particular, we see that if
Nt denotes the Ornstein-Uhlenbeck process on S'(Rd) formally associated
with the stochastic equation
(4.4)
then ~ in fact converges weakly to Nt• In (4.4) wt is the Brownian
motion in S(Rd) with unit spatial covariance.
As Rost [2] noted, the fluctuation dissipation argument (explained
in [2]) leads to the identification of the bulk diffusion coefficient
(4.5)
The principal interest in the theorem is this identification.
We recall that throughout, and in particular in the theorem, the
density p is assumed small enough so the Gibbs measure ~ is uniquely
defined.
5. Outline of the Proof.
With A > 0 fixed, we may pass to the limit in (4.2) along
46
subsquences. We conclude that there are functions u(N) and w(N,x) such
that
(5.1)
At this stage we do not know that w = V 6u/6N • In fact this is the
main thing to be shown. Then (5.1) is the same as (4.3) and we are
finished.
To show that w = V 6u/6N, three steps are needed. First we reduce
the identification to a simpler problem by using special test functions -v. Then we use the ergodicity of the operator L in (2.6) in a suitable
form and finally we use the identity
p 1 f 2p 2 = 1 - '2 Rd z Vt (z) w (z) dz (5.2)
Here w is the second cluster function as in (3.7) and the integral on
the right is an isotropic tensor.
The identity (5.2) is proved directly by working with properties
of the Gibbs measure ~. We have not seen this identity in the
literature before, although it is implied in Rost's paper [2]. It is
not used there. We will not prove (5.2) here but we will use it.
Step one, the reduction, is accomplished by letting
v = (ljl ,N) g(N) (5.3)
with
1jl (x) = e• x cl>(x) (5.4)
Here e is a fixed vector in Rd and ~ € CQ(Rd) while g(N) is a test
function on S(Rd) (i.e. in S(Rd)). Using this v in (4.2) and in (5.1)
we see that the identification amounts ot showing that
-€ € limE~ {(V 6u • ~,v)g} = p EG{(V ~Nu •e, ~)g} €+0 "!N ON
(5.5)
Now we integrate by parts on both sides of (5.5). From the right
we get
47
From the left we get
+ - EG{ug(Vcj>.e,N)} + p EG{u(Vcj>•e, ~)} +lim JE
£-1-0
where, after symmetrization,
Expanding in a Taylor series, we see that
The main part of the proof is to show that
(5.6)
(5. 7)
- G lim JE = lim JE = H E { ug(Vcj>• e,N)} , (5.10) £+0 £-1-0
H = J z Vcfl (z) w (z) dz • ( 5. 11)
If (5.10) holds, then (5.6) and the limit in (5.7) are equal, in view
of the identity (5.2). This means that (5.5) holds and the theorem is
proven.
The proof that JE + H EG{ ug(Vcj>• e,N)} requires (i) the ergodicity
of the process (2.1) and (ii) the fact that
48
(5.12)
with C a constant independent of c. This follows from (4.2). The
ergodicity of the process (2.1) is used in order to construct a
suitable solution to the equation
LU - sU I (y j -yk) 17~ (y j -yk) - H ' s > 0 k=1
(5.13)
this is then used in (5.9) in much the same way as in [4] in pages
856-857.
References
[ 1] H. Spohn, Particles,
Equilibrium Fluctuations to appear.
for Interacting Brownian
[2] H. Rost, Hydrodynamik gekoppelter Diffusionen: Fluctuationen im Gleichgewicht, Lecture Notes Mathematics 1031, 1983.
[3] M. z. Guo, Dissertation, New York Univ., 1984.
[4] G. Papanicolaou and S.R.S. Varadhan, Boundary value problems with rapidly oscillating coefficients, in "Random Fields" Vol. II, North-Holland, 1981, pp. 835-873.
[5] R. Lang. Unendlichdimensionale Wiener prozesse in Wechselwizkung I, II, z. Wahr. Verw. Geb. ~ (1977) pp. 55-72; 277-299.
[6] J. Fritz, preprint, 1984.
[7] D. Ruelle, Statistical mechanics, rigorous results, New York, Benjamin, 1969.
[8] M.Z. Guo and G. Papanicolaou, Fluctuation theory and bulk diffusion for interacting Brownian particles, to appear.
49
SPECTRAL PROPERTIES OF RANDOM AND ALMOST PERIODIC
DIFFERENTIAL AND FINITE-DIFFERENCE OPEFA'IORS
L. A. Pastur
1. Introduction
In the last decade there has been a growing interest
in the spectral properties of the Schrodinger equation and
other differential and finite difference equations in d d L 2 ( JR ) , 1-t 2 ( lR ) respectively) with random metrically
transitive or almost periodic coefficients. In addition to
the obvious mathematical reasons for such interest caused
by the intuitive logics of the development of the spectral
theory of operators, probability theory and mathematical
physics, these questions have a considerable importance to
a large variety of problems in theoretical physics, first
of all in the physics of condensed matter. Theoretical phys
ics is the source of many problems and methods of the spec
tral theory of the discussed classes of operators and has,
up to now, a substantial influence on the development of
this region of mathematics.
2. General metrically transitive operators (r.1TO)
A rather general mathematical scheme describing the
class of the operators under discussion is the following
(Pastur [39, 40]).
Let (Q,F,P) be a probability space and H(w) be a
random operator, i.e. a measurable function with values in
the set of linear operators on a Hilbert space H • Let
T={T} be a group of measure preserving automorphisms of Q
and UT={UT} be a unitary irreducible representation of T in H . We call the random operator H(w) metrically tran-
50
sitive (1.1TO) if it satisfies the relation
UTH(w)u; = H(Tw) (mod P) .
Remarks. 1) The case of selfadjoint operators is of
primary interest. 2) In the case of unbounded operators we
suppose that there exists a dense linear manifold VcH
such that UTVcV , VTcT , H(w)VcV (mod P) and H(w) is
symmetric or essentially selfadjoint on V with probability
(resp. symmetric or selfadjoint MTO).
As an important, paradigmatic example of MTO let us
consider the one-dimensional Schrodinger operator H =
d2 =- -- + q in H = L 2 (JR) with a random potential q (x,w)
dx 2
= Q(Txw) , where Q(w) is a measurable function on ~ ,
i.e. q(x,w) is a stationary and metrically transitive
random process (one-dimensional random field) and
a) 1 dw (~,F,P) = (S ,B 1 , 2nl , where B1 is the Borel
o -algebra on the unit circle is the group
of rotations of S 1 : T w = w+x (mod 2n) , acting as q (x,w)"' X
=Q(x+w), Q(w) is a 2n -periodic function, i.e. it is a
periodic potential with random, uniformly distributed ori
gin. The shift of the origin on the x -axis in the spec
tral theory of the periodic Schrodinger operator proved to
be a rather useful device in the modern theory of completely
integrable systems (see Marchenko [34]).
(n) n -n b) (~,F,P) = (Tor ,B , (2n) dw 1 ... dwn) , (Txw)i =
wi+aix (mod 2n) , ai are rationally independent numbers,
and the corresponding potential is a quasi-periodic func
tion, whose module is generated by (a 1 , ... ,an)
c) q(x) is an ergodic Markov process.
Similar examples can be constructed easily in the mul
tidimensional and discrete cases (elliptic differential
operators in L2 (JRd) and matrix operators in t 2 (:Ed) ) . In
particular, the discrete analog of the Schrodinger operator
in which q(x) , x E:Ed are independent, identically dis
tributed random quantities is called the Anderson model.
51
These examples represent three main groups of MTO:
periodic operators, whose spectral properties are well
known, almost periodic operators and random operators.
In the framework of the described abstract scheme the
following facts were established:
1) If H(w) is a symmetric MTO then both of its de-
ficiency indexes equal 0 or with probability
(Figotin, Pastur [15]).
2) The spectrum a of H(w) as well as each of its
components (point, continuous, absolutely continuous etc.)
are almost surely nonrandom sets (Pastur [39], Kunz,
Souillard [28], Kirsch, Martinelli [25]).
3) The event: any fixed point A is an eigenvalue of
finite multiplicity has zero probability (Pastur [39]).
4) The spectral multiplicity of every fixed interval
6co is not random (Figotin, Pastur [17]).
5) The point spectrum, if it exists, is a dense set.
3. Integrated density of states (IDS)
For differential and finite difference MTO the inte
grated density of states can be defined. This is the sim
plest nontrivial spectral characteristics of this class of
MTO and was extensively studied because of its relevance
both from mathematical and physical points of view. Namely,
let {II} be a family of bounded domains in lRd (Zd) which
sufficiently regularly exhausts lRd (Zd) and let HII be
the operator defined by the same operator inside II and
some selfadjoint boundary conditions on 3II . Denote by
NII(A) the number of eigenvalues of HII, which are less
than A , divided by the measure IIII of II . Then under
some mild conditions on the coefficients of the matrix MTO
and elliptic HTO of arbitrary order, there exists a non
random, nondecreasing function N(A) such that with proba-
bility at every continuity point of N(A)
1) lim NII(A) = N(A)
Il->-lRd
(Benderski, Pastur [6], Figotin, Pastur [15], Shubin [45]).
52
2) N(A) = M{E(O,O,f.)}
where E(x,y,>.) is the kernel of the resolution of iden
tity EA of the selfadjoint MTO H(w) , and M{ ... } de
notes the expectation with respect to the probability meas
ure P (Pastur [38], Figotin, Pastur [15]).
3) The spectrum o of H(w) is the set of points of
nonconstancy of N(f.) (Pastur [39]).
The last property illustrates the role of IDS in the
spectral analysis of MTO. Namely, it answers one of the
main questions: what is the spectrum of MTO as a set on the
real axis.
A large group of results concerns the asymptotic be
haviour of N(f.) near the bottom of the spectrum. Here are
some of them.
1) For the Schrodinger operator
a) y (x) = l:n (x-x.) , where n (x) J
smooth integrable function and {xi}
Poisson process in lRd with density
H = -L'l+q (x) and
is a sufficiently
are the points of a
v •
---"--ln--"-- 0 n ( o) n ( o) ' n~ , .\._--+00
ln N (A) (1+0 (1 ))
(Pastur [39-41]). c1 (d) , c2 (d,a) are known constants.
b) The "Poisson" potential from the previous case can
be represented in the form
q (x) = f n (x-y)m (dy) d p
lR
where m(dy) is the random, shift invariant measure, such
that m(A 1 ) and m(A2 ) are independent random variables
of A1 n A2 = ~ and
Prob{m(A)=m} = e-viAI (viAl) m!
If instead of this Poisson distribution for m(A) we take
the so called r -distribution
53
!AI -1 IAI-1 -vm Prob{m(A) E (m,m+dm)}= v f (!AI )m e
then if n<:O and -d-2 n(x) =o(lxl )
2 and if m(dy)=g (y)dy, where g(x) is the Gaussian proc-
ess in JRd with correlation function exp{-2:~ !xi!} , then d
ln N(A) = -c4 (d)A-2A- 1 iln A! 2 (d- 1 )(1+o(1)), A+O
(see Figotin, [13], [14] for these and more general cases,
which were studied using a generalized form of known re
sults of Donsker and Varadhan).
2) A more general but weaker result was obtained by
Kirsch and Hartinelli [23] in the continuous case (Schrodin-
ger equation) and by Simon [ 46] in the corresponding dis-
crete case. These authors assumed that the nonnegative po
tential has sufficiently good mixing properties and the
probability to take small values is not too small, and
proved that
lim ln 1 ln N (A ) : d A+O ln A -2
The results for 1b) show that this asymptotic formula cand not be sharpened in general. The number - 2 in the r.h.s.
is the so called Lifschitz exponent.
3) For one-dimensional Schrodinger operators with po
tential
where
¢(z)
k 0 >0 , x. 1-x. := v .~0 , yJ. = ¢ (jS) , S is irrational, J + J ~ J is monotonic, unbounded at z=1 and has period 1 ;
we have
a N(A) = c5 (v,a)A (1+o(1)) , :\+0
if -1/a -1 1
¢ (z) cv (1-z) , z+1 , a>O , v = !¢dz (Gredeskul, 0
Pastur [22]).
All these asymptotics are called now Lifschitz tails
54
and correspond to the so called fluctuation region of the
spectrum, where it exists due to the large deviation part
of the potential. These fluctuations have the form of rather
deep and wide wells with large distance between them. Due
to the random nature of the potential, these wells have dif
ferent forms and therefore the resonance tunneling between
them is extremely weak. The asyrnptotics of N(A) is deter
mined by the largest well for which the given small A is
the lowest eigenvalue. This fluctuation ideology proposed
by I. Lifschitz [30] gives the possibility to obtain a large
variety of asyrnptotics'for N(A) and related quantities
and is widely used by physicists (see Lifschitz et al. [31]
and references therein) . It also shows fairly well the qual
itative structure of the eigenfunctions, which should be
localized in essence in the neighbourhood of the correspond
ing well and suggests the point nature of the spectrum in
the fluctuation region.
4) Another type of spectral boundaries is the stable
one (Lifschitz et al. [31]). Here the spectrum exists due
to all typical parts of the random potential. In the vicin
ity of such a boundary, N(A) behaves as the same function
of some equation with constant coefficients (which corres
ponds to the so called effective medium). Here are some
examples.
a) Schrodinger equation. Stable boundary is the point
A=oo and its vicinity,
N (A ) = N 0 ( A ) ( 1+o ( 1 ) ) ,
where N0 (A) =C(d)Ad/ 2 is the IDS for H0=-Ll..
1 d d b) H = -rn(x) dx(kdx·) , where rn(x) and k(x) are
strictly positive metrically transitive processes. Here the
stable boundary is the point A=O . In the simplest case
rn(x)=m , k(x)=k , rn and k are constants; the correspond-1 IDA 1 I 2
ing IDS is N0 (m,k,A) =-:;r(k) In the general case it
turns out that
N (A) = N0 ()J,a>.,A) (1+o (1 l l , A-+0
55
where JJ=~~{m(x)}, ce=JJ- 1{k- 1 (x)} (Lifschitz et al. [31]).
4. The nature of the spectrum of random operators
As it was mentioned in the previous section, N(\)
shows where does the spectrum lay. But it does not tell any
thing about the nature of the spectrum, i.e. about the
structure of eigenfunctions. The answer to this important
question of spectral analysis as well as other interesting
features of MTO depend on a more detailed structure of the
coefficients, such as whether they are almost periodic, or
have good mixing properties, etc.
The main body of known results concerns the one-dimen
sional case. In this case the Liapunov index plays an im
portant role. It is defined as
y(\) = limi2xJ- 1 ln(u 2 (x)+u' 2 (x)) lxl->=
where u(x) is the Cauchy solution of the Schrodinger equa
tion, i.e. -u"+qu=\u, u(o) =sina, u'(o) =cos a,
aE[O,n) • The following facts are true:
1) The limit in the r.h.s. exists with probability 1
for every pair (\,a) and is nonrandom in the general case
of a metrically transitive q(x) (Oseledets [37]).
2)
y(\) = y 0 (\)+!lni\-JJ!d(N(JJ)-N 0 (JJ))
,.----, -1 h where Yo(,\)=+"-\ , N0 (\) = n +"\ are the Liapunov expo-
nent and the IDS for the Schrodinger equation with q=O
(Thouless [47], Avron, Simon [2], Figotin, Pastur [18],
Craig, Simon [10]).
3) y (A) = 0 for almost every A from a Lebesgue meas
urable set 6co if and only if 6no ae 1 0 , where o ae de
notes the absolutely continuous part of the spectrum (if
part, Casher, Lebowitz [7], Ishii [24], Pastur [39]; only
if part: Kotani [27]).
4) For random MTO (i.e. if q(x) is a Markov process)
y(\)>0 , V\Eo (Ishii [24], Pastur [39], Molchanov [35]).
This property is opposite to the property of periodic Schro-
56
dinger operators, for which a = {A: y (A) =0} .
From properties 3) and 4) it follows that the spectrum
of the random Schrodinger operator (and the random Jacobi
matrix as well) has no absolutely continuous part. But in
the random case much more is known. Namely, the spectrum
of the one-dimensional Schrodinger equation with a poten
tial of the form q (X) = Q (t;X) , where f:;X is an ergodic
Markov process on a smooth Riemann compact manifold and
Q(t;) is a smooth and nondegenerate function, is pure point
(Goldsheidt et al. [21], Carmona [8]) and all eigenfunc
tions decrease exponentially (Molchanov [35], Carmona [8]). The same assertions hold in essence for any one-dimensional
second order operator (see Kunz, Souillard [28], Delyon et
al. [11]) and for some one-dimensional operators of higher
order (Goldsheidt [20], Lacroix [32, 33]). The main condi
tions are the good mixing property of coefficients (e.g.
Markov in continuous case, i.i.d. ~n discrete one) and
smoothness of their distributions (the last condition is
needed to exclude the so called quantum resonance effects
(resonance tunneling).
we see that the spectral theory of one-dimensional
random MTO is at the present time in a state of complete
ness not too far from the same theory of periodic operators.
These two cases represent the "end" (extremal) points, the
pure cases of a large variety of possible spectral situa
tions which is possible for MTO and can be described sym
bolically as the points of segment [0,1]
periodic
p.rre abs. cont. sr:ect.J:um
of nul tipl. 2
mixing property of pot.
resarance effects
random
p.rre point sr:ect.J:um of I!Ultipl. 1
Concluding this short and fragmentary review of the
spectral theory of MTO, I want to make some remarks.
1. It is widely believed and sufficiently well argu
mented by physicists that at the vicinity of the fluctua
tion boundary (deep fluctuation region) the spectrum should
be pure point and the eigenfunctions are exponentially lo-
57
calized in any dimension. In contrast to this, in the vicin
ity of the stable boundary, in particular in the high-energy
region of the Schrodinger equation, the spectrum of random
operators should be continuous or even absolutely continuous (at least for d~3 ). At present there exists only a
few rigorous results, supporting these believes (more ex
actly facts of theoretical physics). These are the results
on the absence of diffusion in the vicinity of the fluctua
tion region in the Anderson model (Frohlich, Spencer [19]),
the announcement of Goldsheidt and Frohlich, Spencer on the
existence of point spectrum and the results by Kunz,
Souillard (Kunz, Souillard [29]) on the Anderson model on
the Cayley tree (Bethe lattice) •
2. The two-dimensional case is of considerable inter
est and rather intriguing. Here, according to the just as
beautiful as nonrigorous arguments, the spectrum may be sin
gular in the high energy region with a power law fall off
of the generalized eigenfunctions.
3. Even in the one-dimensional case, one has to answer
(if one has a theoretical physicist's taste) not only the
traditional questions of the spectral theory, but also to
analyse a variety of quantities constructed from the eigen
elements of random Schrodinger or Jacobi operators. These
quantities have a sufficiently complicated structure and
therefore such calculations, as a rule, are possible only as
asymptotic or approximate ones. An important example of such
quantities is the zero-temperature conductivity in the homo
geneous electric field of frequency v :
a (v) =M{f..i_ e(x,y,"A+v)f e(y,x,"A) I } >.. ay Y x=O
where
E ("A)
dE (A) e(x,y,"A) is the kernel of the operator ~ and
is the resolution of identity of the Schrodinger oper-
ator. Physicists have worked out some approximation methods
for handling this and similar quantities, at last for some
range of the parameters >.., v and for some characteristics
of random potential, which is called the semiclassical re
gion (see our book [31] and the references therein). But
these methods require substantial foundations. Besides, the
58
similar methods are badly needed for treating many other
relations between the parameters.
4. It is of great interest to analyze the picture of
the spectrum in the case, when in addition to the random
potential, an electric or (and) a magnetic field are pre
sent. The latter case demonstrates a very interesting inter
play of randomness and almost periodicity in the spectral
and related questions.
5. Almost-periodic operators
In recent years the "hot point" of the MTO spectral
theory has been shifted to the almost periodic case. Such
operators model the so called incommensurable systems and
some other physical situations. Besides they are filling
partly in the gap between periodic operators and random
ones, demonstrating a wide variety of types of spectral
behaviour. As a consequence, the study of this class of r.no turns out to be very useful for the spectral theory of MTO
as a whole.
To my opinion the main discovery here is the essen
tially non-algebraic nature of the spectral picture in con
trast to the periodic case, where a mere commutativity of
the corresponding operator with the translation operator
gives readily the main features of the spectral picture.
This discovery was made and can be demonstrated upon the
rather simple case of the so called almost-Mathieu equation
l/J (x+1) +l/J (x-1) +2g cos (2n o:x+w) l/J (x) =ill/! (x), x E Z
where g>O and a: is an irrational number. A very impor
tant and surprising observation was made by Aubry and Andre
[1). Namely, they argued that for g>1 the Liapunov expo
nent (in the discrete case it is defined as
lim :2x[- 1ln(u 2 (x)+u 2 (x+1)), where u(x) is the solution lxl-+oo
with u(1) =sin a: , u(O) =cos a: ) of this equation should
be strictly positive, moreover that the following estimate
holds
y (A) ~ ln g .
59
As a consequence, for g>1 , oae is empty. The arguments
of Aubry and Andre were made rigorous by Avron, Simon [2]
and Figotin, Pastur [18]. Hermann [23] gave a general proof,
based on ergodic theory of smooth dynamical systems, accord
ing to which the estimate y ~ lnJa I holds for potential n ik
q(x)=p(2nxa:+w) , where p(¢) = I a.e ¢ a* =a . From lkl~mK k -k
the other side, J. Bellisard et al. [4], generalizing the
results by Dinaburg, Sinai [12] and Belokolos [5] showed
that if a: is not well approximated by rationals then for
g>>1 the spectrum of almost-Mathieu operators has a point
component whose closure has positive Lebesgue measure, and
for g<<1 the spectrum has a massive absolutely continuous
component. At last, Avron, Simon [2], based on Gordon's re
sult proved that if a: is a Liouville number pn' qn are
I Pn' -q integers and a:- qn I ~ C n n , n-+oo ) and g>1 , then the
spectrum is singular continuous. We see that the almost
Mathieu operator demonstrates practically all possible types
of spectral behaviour depending only on the arithmetic prop
erties of the parameter a: (quasiperiod) but also on the
magnitude of the potential, i.e. parameter g
Now I report on the results of the study of two more
classes of almost-periodic operators. These results develop
and add new features to the outlined picture.
A) Limit periodic operators
Recall that an almost-periodic function q (x), x E JR
is called limit periodic if there exists a sequence of peri-
odic functions
growing periods
qn(x) 'qn(x+an) =qn(x) a such that
n
sup Jq(x)-q (xJI = ~q-q I ->- 0 xEJR n n C
with infinitely
The class of the Schrodinger operators with such potentials
was studied first by Moser [36], Avron, Simon [2] , Chula
evsky [9]. These authors showed that
i) for a dense G0 set of limit-periodic potentials
(they form a complete metric space) the spectrum is a (no
where dense) Cantor set of positive Lebesgue measure;
60
ii) for a dense set of potentials the spectrum is a
Cantor set and pure absolutely continuous.
The proofs of these results use the natural idea of
the approximations of limit-periodic operators by peri
odic ones, i.e. "closing" the spectral theory of periodic
Schrodinger operators. But the above mentioned authors used
the classical part of this theory. We (Pastur, Tkachenko
[42]) used its more modern and refined form, especially the
theory of Marchenko, Ostrovsky (see Marchenko [34]) contain
ing in particular the complete solution of the inverse spec
tral problem.
According to this theory, every square integrable peri
odic potential determines the Nevanlina function 8(~) ,
~ 2=A (this is the quasimomentum as a function of the square
root of energy) and the sequence of complex numbers {8k}~ ,
Im 80 =0, 18 0 12+ I k 2 i8k[ 2 <"",and in their turn, 8(~) k~1
and {8k} determine the potential uniquely. One can say
that they form a complete set of the spectral data. In par-
ticular, the function 8(~)
the upper half-plane c+ =
II+ = C+' U {kn ,kn+i [ 8 I kl! } kiO
gives the conformal mapping of
{~: Im ~ .:ot O} onto the domain
(comb) , and for every such func-
tion there exists a periodic operator whose spectrum is the
set 0: Im 8 (IA-8 0 )=0} •
Consider now the space of the Besikovitch limit-
periodic functions which is generated by the norm
Denote by
a)
II II - 1' r -1 Tf I r2 1112 lq 11 2 = 1.m ·lT 0 iq (x) 1 dxJ
B T+oo
A = {a }"" a sequence of positive numbers such that n 1
b) Q00 (A) the class of Besikovitch limit-periodic
potentials q(x) such that for some sequence of an -peri
odic functions qn(x)
lim ilq-q ii n+"" n B
0 , vc > 0
61
(we call such potentials superexponentially approximated); c) R the set of real numbers of the form r = 'ITk
n an k E Z and by R (A) = U R ;
n~1 n
d) K00 (A) the set of complex-valued
{Kr}rER (functions on R ) such that
sequences
{ 2 2}1/2 Ca +1 lim L r lK I e n n~oo rER(A)-Rn r
0 , Y C>O •
Then the following facts are true.
1) Every potential qEQ00 (A) determines uniquely an element of K00 (A) and vice versa, i.e. the set K00 (A) consists of the elements K which are the complete sets of spectral data for the potentials from Q00 (A) •
2) A closed set aE~ is the spectrum of some Schrodinger operator with potential qEQ00 (A) if and only if
a= {A: A~AO , Im K(/A-A 0 )=0} , where the function K(~)
is a conformal mapping of C + onto rr: = C ' U [r ,r+i I K 1 1 I] + rER(A) r
(dense comb) specified by the conditions: K (0) = 0 , lim (iy) - 1 K (iy) ='IT •
This statement gives the possibility to construct limit periodic operators with qEQ00 (A) which have a prescribed spectrum, i.e. it solves the inverse spectral problem for this class of potentials.
-1 ~ 3) N(A) ='IT Re K(vA-A 0 ) , y(A) =Im K(IA-A 0 ) .
4) The spectrum of the Schrodinger operator with
qEQ00 (A) is absolutely continuous and has multiplicity 2. For almost every AEa there exists two linearly independent
± ±iKX + ± 2 solutions 1jJ (A,X) =e u-(~,x) , where u (~,·) E B and + for some sequence of an -periodic functions u~(~,x) and
every Borel set 1:!. E ~+ and C>O
+ + p Can+1 lim J !u-(~,·)-u~(~,·)~ 2 d~·e 0 , n~oo 1:!. B
Yp E (1, 2)
We see that this class of almost-periodic operators is maximally close to the periodic one. The only novel feature is the possibility to have nowhere dense spectrum which is al-
62
ways absolutely continuous, because this Cantor set has al
ways a positive Lebesgue measure. This picture is a "typi
cal" one in the sense that the corresponding potentials are + dense. One should only use the dense comb rroo for it, i.e.
a function K E Koo (A) which has a dense support in lR .
B) Quasi-periodic operators
This class of operators is, on the contrary to the pre
vious one, rather close to the random t-!:TO. Recall that an
almost-periodic function f(x) is called quasi-periodic if
it has a form f(x) =F(2Tia 1x, ..• ,2Tianx) where F(x 1 , •.. ,xn)
is a 21T -periodic function for every variable xi E lR ,
i = 1 , ••. , n and a 1 , .•. , an are rationally independent num
bers. In the discrete case the simplest quasi-periodic func
tion has a form f (x) = F (2Tiax) , x E Z where F (x) is a
2TI -periodic function, F(x) =F(x+1) (e.g. F(x) =g cos x
as in the almost-Mathieu equation) .
Let us consider now the matrix operators Hd which
act in t 2 (Zd) according to the rule:
'i' vl 1jJ(x) +g tg ax·1jJ(x) , xEZd L x-y
yE2td
where wx is a sequence of complex numbers satisfying the
conditions
and
a X
w -K
lw I ~ Ce-piKI c P > 0 I K I I
1T (a , x) +w , a E lR d , w E [ 0 , 1T) , d
(a,x) = L:a.x. 1 ~ ~
This class of operators, whose elements are obviously ?-1TO
for any a E lRd and are almost-periodic for rationally in
dependent "frequencies" a 1 , ... ,ad, is the natural general
ization of the remarkable class of one-dimensional operators
proposed by Prange et al. [44]. These authors proved that if
a is approximated by rationals sufficiently badly, then H2 has a dense set of eigenvalues on lR and the corresponding
eigenfunctions are exponentially localized. However their
method was somewhat indirect from the point of view of the
63
spectral theory. We (Figotin, Pastur [16]) found, using
rather simple and essentially algebraic arguments, a con
venient representation for the resolvent of the operator
Hd and proved the following properties.
1) The IDS of Hd i.e. N(:\) , is an absolutely con
tinuous and even real analytic function and its density
p(:\) has the form
p ( :\)
W(k)
It is rather surprising that this N(:\) coincides with the
same function for the random operator of type Hd such
that a , x E :ltd are independent random variables with a X
common uniform distribution on [O,n) . Thus we have here
two examples of rather different operators (almost periodic
and random) which have the same IDS.
2) If the frequencies a 1 , ••• ,ad satisfy the condi-
tion
I I ' '-8 i (a , x ) -m 1 ;:: C 1 x 1 , C , 8 > 0 , x t- 0
then the spectrum of Hd is the set of solutions of the
equation
N(\) = ~ + * + (a,r) , IJ rEZd
and as a consequence of continuity and strict monotonicity
of N (A) , the spectrum is dense in JR • d 3) To every eigenvalue Ar , r E Z there is a unique
eigenfunction ~r(x) , r,x€Zd It has the form
X (x, A) (2TI)-d f
(Tor)d
exp[-2ni(k,x)-it(k,A)] dk W(k)-A-ig
where t(z,k) is the unique and (z,k) holomorphic solu-
64
tion of the equation
f(z,k)
f (k)
f 0 (z)+t(z,k)-t(z,k+a) , z EC ,
(2n)-d J f(z,k)dk,
(Tor) d
f(z,k) ::_W(k)-z+~g W(k)-Z-l.g
4) These eigenfunctions form a complete set in t 2 (~d) Thus the spectrum of Hd is simple pure point and the cor
responding eigenfunctions are exponentially decreasing as
lxl+oo .
5) If d=1 then for any irrational a the spectrum
of H1 has no absolutely continuous component.
6) If d>1 and a is a Liouville number then the
spectrum of H1 is singular continuous.
7) Statement 2) remains valid for the operators
where
(V1/J) (x) = V(exp(2ni(a,x))+2iw)lji(x) , c:«1
and V ( z) , z E C is an analytic function in the vicinity
of the unit circle lzl=1 , mapping it into the real line.
The listed results were independently and approximately
in the same time discovered by Simon [46].
Properties 1)-4) seem rather natural in the one-dimen
sional case (d=1) because if a is not too well approxi
mated by rationals, then the potential g tg ( nxa+w) , x r:: ~ takes on arbitrary large values on a sequence of points
going to infinity. It is, however, interesting that these
peaks are also sufficiently thickly and irregularly placed
even in the multidimensional case, so that for any energy
a particle still becomes "entangled" between them and can
not escape to infinity. Due to this property of the poten
tial g tg (n (a,x) +w) , they are in essence supressed and
thus the spectrum in this case looks very similar to the
deep fluctuation spectrum for random MTO. This similarity is
supported by the analogous asymptotic behaviour of conduc
tivity in the low-frequency regime. Namely, according to
65
Figotin, Pastur [17] and Prange et al. [44]
exp {-rv 0l 1 /S} v+O • o(v)~C 1 (vJ ,
As it is well known (see for example Lifschitz et al. [31]),
the low frequency conductivity in the random case has a sim
ilar behaviour:
v+O .
References
[1] Aubry S., Andre G. Ann. Isr. Phys. Soc. 3, 139 (1980).
[2] Avron J., Simon B. Comm. Hath. Phys. 82, 101 (1982).
[3] Avron J., Simon B. Duke Math. J. 50, 369 (1983).
[4] Bellisard J., Lima R., Testard D. Comm. Math. Phys. 88, 207 (1983).
[5] Belokolos E. Theor. Math. Phys. 25, 344 (1975).
[6] Bendersky M., Pastur L. Mat. Shorn. (1971).
[7] Casher A., Lebowitz J. J. l.fath. Phys. 12, 1701 (1970).
[8] Carmona R. Duke Math. J. 49, 191 (1982).
[9] Chulaevsky v. Usp. l1at. Nauk 36, 143 (1981).
[10] Craig W., Simon B. Duke Math. J. 50, 551 (1983).
[11] Delyon F., Kunz H., Souillard B. J. Phys. A16, 25 ( 1983) .
[12] Dinaburg E., Sinai Ja. Func. Anal. Appl. 9, 279 (1975).
[13] Figotin A. Funct. Theory Appl. (Kharkov) 30, 132 (1980).
[14] Figotin A. Doklady Ukr. Acad. Sci. N6, 27 (1981).
[15] Figotin A., Pastur L. In Dif. eq. and Funct. Anal., Kiev, Naukova Dumka, 117 (1978).
[16] Figotin A., Pastur L. JETP Letters 37, 575 (1983), and to be published in Comm. Math. Phys. (1984).
[17] Figotin A., Pastur L. (unpublished) (1983).
[18] Figotin A., Pastur L. J. Math. Phys. (to be published) (1984).
[19] Frohlich J., Spencer T. Comm. Math. Phys. 88, 151 ( 1983) .
[20] Goldsheidt Ja. Doklady Acad. Sci. USSR (1979).
[21] Goldsheidt Ja., Molchanov s., Pastur L. Funct. Anal. Appl. 11 , 1 ( 1 9 77) .
[22] Gredeskul S., Pastur L. Theor. Math. Phys. (to be published) (1984).
66
[23] Herman ~1. Preprint Ecole Polytichnique (1982).
[24] Ishii K. Progress Theor. Phys. Suppl. 53, 77 (1973).
[25] Kirsch W., Martinelli F. Comm. Math. Phys. 85, 329 (1982).
[26] Kirsch w., Martinelli F. Comm. Math. Phys. (to be pub-lished) ( 1983).
[27] Kotani s. Proc. Kyoto Conf. on Stech. Proc. (1982).
[28] Kunz H., Souillard B. Cornm. Math. Phys. 78, 201 (1980).
[29] Kunz H., Souillard B. J. Phys. (Paris) Letters 44, L411 (1983).
[30] Lifschitz I. Adv. Phys. 13, 485 (1964).
[31] Lifschitz I., Gredeskul S., Pastur L. Introduction to the Theory of Disordered Systems. Moscow, Nauka (1982) and to be published by Wiley and Sons in 1985.
[32] Lacroix J. Ann. Inst. E. Costan N 7 (1983).
[33] Lacroix J. Ann. Inst. H. Poincare A38, N 4 (1983).
[34] Marchenko V. Sturm-Liouville operators and their application. Kiev, Naukova Dumka (1977), and to be published by Riedel in 1985.
[35] Molchanov S. Math. USSR. Izv. 12, 69 (1978).
[36] MoserJ. Cornm. Math. Helv. 56,198 (1981).
[37] Oseledec V. Trans. Mosc. Math. Soc. 19, 197 (1968).
[38] Pastur L. Math. USSR Uspekhi 28, 3 (1973).
[39] Pastur L. Preprint Low Temp. Inst. Kharkov (1974).
[40] Pastur L. Comm. Math. Phys. 75, 179 (1978).
[41] Pastur L. Theor. Math. Phys. (1978).
[42] Pastur L., Tkachenko V. Doklady Acad. Sci. USSR (to be published) (1984).
[43] PrangeR., Grempel D., Fishman S. Phys. Rev. Lett. 49, 833 (1982).
[44] PrangeR., Grempel D., Fishman s. Preprint of Maryland University (1984).
[45] Shubin M. Trudy Sem. Petrovskii 3, 243 (1978).
[46] Simon B. Caltech. Preprint and to be published in J. Stat. Phys. (1984).
[47] Thouless D. J. Phys. C5, 511 (1972).
67
EQUILIBRIUM FLUCTUATIONS FOR SOME STOCHASTIC PARTICLE SYSTEMS
Herbert Spohn
1. Introduction
Our knowledge about the microscopic structure of macroscopic sy
stems ln thermal equilibrium is encoded as equilibrium time correlations.
These objects are measured in scattering experiments : a purely angular
resolution of the scattered beam translates into static correlations and
its frequency (energy) resolution into time correlations. Despite their
central physical importance our mathematical understanding of equilibri
um time correlations lS very modest, quite in contrast to their static
counterpart on which we have a wealth of qualitative information.
If one insists on Hamiltonian dynamics, the problem of understan
ding the structure of equilibrium time correlations is indeed a diffi
cult one. This s!J>uld come as no surprise, since even in statics very few
results pertain to physically realistic Hamiltonians. Our beloved Ising
model is certainly a crude approximation to the "real life" ferromagnet
Tb(OH) 3 , say, although it catches a great deal of essential physics.
Following the philosophy of approximation in static equilibrium models
leads to particle or spin models with stochastic dynamics. This is a
well accepted procedure within statistical physics and has been applied
with great success to critical dynamics, to the dynamics of phase segre
gation and to many other problems. I will follow this practice here and
consider only models with a. stochastic time evolution.
I started to become interested in equilibrium time correlations for
stochastic particle models about three years ago. To my surprise the on
ly av,ailable result at that time was the one of Holley and Strook [ ·1 ,2]
who prove an exponential decay of correlations in space-time for the sto
chastic Ising model at high temperatures. They use an in essence pertur
bative argument exploiting that the stochastic Ising model bas no con
servation law. In this case the model resembles an Ising model ln d+1
dimensions and exponential decay is the well-known high temperature be
havior. To my knowledge the only extension is the one by Wick [3] to the
68
stochastic Heisenberg model. Further generalizations should be feasible.
The natural question to be asked next concerns models with one or
several conservation laws. This is the topic of the present article. To
summarize very briefly : we have now a technique which enables us to
prove the asymptotic behavior of correlations in space-time for suffi
ciently small interaction. The technique is in no sense perfected and
there are many open problems some of which will be listed at the end.
The work to be described here developed in close cooperation with
T. Brox, A. Der.lasi, A. Galves, C. Kipnis, E. Presutti, H. Host and D.
Wick. Without their contributions none of the results would have materi
alized. I would like to draw the attention of the reader to [4], which
is of a more programmatic nature and where some aspects are mentioned
which I will not repeat here, and to the very extensive and most recom
mendable expository article [5] which however predates the results to be
described here.
2. Some Models
I describe three classes of models of physical interest. For fi
nite volume the existence of the dynamics is standard Markov process
theory, for infinite volume cf. remarks at the end of this section.
A) Stochastic lattice gases (exclusion processes with speed change,
spin exchange dynamics [6,7]).
In a stochastic lattice gas particles jump on a bounded set A c Zd,
the d-dimensional hypercubic lattice. There is at most one particle per
lattice site. The space of configurations is then {0,1}A. A configura
tion is denoted by nand nx is its value at site x. nx 0(1) ·corre
sponds to site x being empty (occupied). The dynamics is specified
through the jump rates c(x,y,n) ~ 0 which give the rate of exchange of
the occupations at sites x and y when the configuration 1s n. Clearly
c(x,y,n) = c(y,x,n). c is assumed to be short ranged and translation in
variant. To avoid degeneracies we also assume c(x,y,n)>O for lx-yl = and nx * ny. The generator of the jump process is
L1/J(n) (2.1)
where nxy denotes the configuration n with occupations at sites x and y
exchanged. The jump process for L is denoted by nt•
Let
H(n) <!>(A) n nx xEA
69
( 2.2)
be the energy constructed from the short range and translation invari
ant potential <1>. We impose then the condition of detailed balance as
c(x,y,n) c(x,y,nxy) exp{ - (H(nxy) - H(n) ) } ( 2. 3)
Then nt with initial measm;-e
~ exp { - H(n) + ~ Ln Z X X
(2.4)
1s reversible. In particular, the Gibbs measures (2.4) are time Invari
ant. The conservation of the total number of particles is reflected by
the fact that nt has a one parameter family of stationary measures.
One may remove the restriction of single site occupancy. This case
has not been studied systematically. The only model investigated 1n con
siderable detail is the zero range process [8]. There a particle at a
given site x jumps with equal probability to neighboring sites with a
rate, c(nx)' depending only on the number nx of particles at x. Again
the process is reversible and admits a one parameter family of statio
nary measures which happen to be product measures.
B) Interacting Brownian particles.
This is a reasonable model for dilute suspensions, e.g. small poly
styrene balls in water [9,10]. One has N particles in a bounded volume
~cRd. Their time evolution is described by the diffusion process
grad V(x.(t)-x.(t)) + dw.(t) J l J
( 2. 5)
j=1 , •.. ,N, plus reflecting boundary conditions. Here x.(t) denotes the J
position of the j -th particle and {w. ( t)} are independent Brownian mo-J
tions. V 1s a short range, even potent-ial. Reversibility is ensured
through the drift being given as the gradient of a potential. The sta
tionary measure is the Gibbs measure
70
1 1 N 1 - exp{-- L V(x.-x.)} -N, dx 1 .•• dxN z 2i*j=1 l J •
(2.6)
If we think of the particle number as free to vary (grand canonical
prescription), then the diffusion process has a one parameter family
of stationary measures.
There are other versions of this model which however have not been
studied in any detail. In (2.5) the diffusion matrix may be process de
pendent. Under some further conditions this case should be handable by
our method. Sometimes one would like to give a finer dynamical descrip
tion by including also velocities in (2.5). This leads to the interac
ting Ornstein-Uhlenbeck processes,
dx.(t) J
dv.(t) J
v. ( t )dt J
N L F(x.(t)- x.(t))dt -yv.(t)dt + dw.(t)
J l J J i*j=1
(2.7)
j=1, ••• ,N, plus reflecting boundary conditions. Now, time reversal in
cludes velocity reversal and, as it stands, our method fails.
C) Time dependent Ginzburg-Landau models (Cahn-Hilliard theory f11],
model B of critical dynamics [12]).
These are "field theory" type models of very wide use. We describe
a lattice version which for our purposes is good enough since only the
large distance behavior of correlations is of interest. To each lattice
site x E A c Zd we associate a real random variable ¢x(spin). A spin
configuration is denoted by ¢. To each nearest neighbor bond (x,y) we
associate a random current from x toy, dj (x,y,t). {j (x,x+e,tlllel=1,e r r
positively oriented} are independent Brownian motions and we define
jr(x+e,x,t)= -jr(x,x+e,t). Then the dynamics is specified by the system
of stochastic differential equations
d¢ + t,x E dj(x,y,t) x ,yEJ\., I x-y I = 1
0 (2.8)
(()H ()H \
dj(x,y,t) = - <l¢ - <l¢ ) (¢t)dt + y X
dj (x,y,t) for lx-yl=1 r
(2.9)
where the (free) energy H lS given by
H(¢) .l E ( ¢ -¢ ) 2 + E V( ¢ ) 2 x-,yt.J\., I x-y I = 1 x Y xEJ\. x
(2.10)
71
with V(q) ~ aq2+b, a > 0. (2.8) and (2.9) are written in a form to make
their meaning transparent. (2.8) is the local conservation law. In (2.9)
the current from x to y is the sum of a random, white noise part and a
systematic part which is minus the difference in chemical potentials at,
x and y - the quantity the system tries to equilibrate. One readily
checks that the process ¢t is reversible and admits the one parameter
family of stationary measures
1 z exp{-H(¢) + h L ¢X} n d¢x xEll. xEll.
(2.11)
The model may be extended to several components, but this lS not the
place to go into details.
Let me summarize briefly the essential features of the stochastic
dynamical models discussed so far. The interaction is short ranged and
translation invariant. The basic field is locally conserved and this is
the only conservation law. The dynamics is reversible and admits a one
parameter family of stationary measures. These are the canonical, with
respect to the particle number or the magnetization, Gibbs measures.
The potential corresponding to the Gibbs measures is short ranged and
translation invariant.
As a first step, we have to establish the existence of the infinite
volume dynamics. Since we focus on equilibrium time correlations, the
existence of the equilibrium dynamics suffices, i.e. the existence al
most surely with respect to the initial Gibbs measure. For stochastic
lattice gases the complete theory is developed in [7]. The equilibrium
dynamics for interacting Brownian particles was constructed by R. Lang
[13,14]. For Ginzburg-Landau models there are only partial results [15]
wh<ic:b are extendable however.
3. The problem and its Expected Solution
The essential physics is contained in the equilibrium two-point
function in space-time
( 3. 1)
(and similarly for the other models), in particular its behavior at large
distances, I x-y I and It-s I large, which should be independent of the
fine details of the interaction. At such generality we have no control
72
over (3.1) and we therefore refrain o)lrselve~ to the more modest goal
to obtain the asymptotic behavior of the two point function for small
interaction. We fix once and for all the density (chemical potential,
magnetic field) of the initial, time invariant Gibbs measure and set
<nx> = p (similarly for the other models). By standard high temperature
expansion and by Dobrushin uniqueness, the Gibbs measure is then unique,
translation invariant and has good cluster properties, <•>p always re
fers to expectations of the process with this initial measure.
We rewrite (3.1) as <nt (n - p)> which we read as the aver-,x o,o p age density at x at time t for the initial signed measure (n0 - p))lp ,
the Gibbs measure somewhat modified at the origin. Because of the con
servation law we expect that this local perturbation spreads out accor
ding to the diffusion equation. Therefore for large separation in
space-time
<n n > - p2 ~ x(p) eD(p)~[t[/2 (x,O) t,x o,o p
( 3.2)
Here ~ is the (lattice) Laplacian and exp(t~) (x,O) the corresponding
transition probability. x(p) = L (<n n > - p2 ) is the static compressi-x X 0 p bility which is seen to be the correct normalization upon summing both
sides of (3.2) over x. The speed at which the local disturbance spreads
out is determined by the bulk diffusion coefficient D(p) , where, for
simplicity, we assumed isotropy. D(p) may be identified by considering
L x2(<nt n > - p2 ) • As a general result one finds the Green-Kubo X ,X 0,0 p
formula
( 3.3)
d,f3=1, ••• ,d, where <J (t)JS(O)> is the total current-current correla-a P
tion, cf.[4,5] for details.
We notice that in the Ginzburg-Landau model for the total current
the systematic part drops out and J (t) = [A[- 112 L djr(x,x+ea,t) . . a . . . xEA
wlth ea the unlt vector along the posltlve a-axls. Therefore
DG (m) = 1/2x(m) with x(m) = L (<¢ ¢ > - m2 ) and m <¢ > • Similarly L x xom om one finds that for interacting Brownian particles DB(p) = p/2x( p) ·
For stochastic lattice gases the situation is more complicated. The ana
logue of (2.9) is not valid in general. In (2.9) the average current gi
ven the configuration ¢ is the gradient of something else and this is
why the analogue of (2.9) has been baptized "gradient condition". In
73
one dimension stochastic lattice gases satisfying the gradient condi-
tion have been "enumerated" [16]. The bulk diffusion coefficient is now
a somewhat more complicated object but may be computed without solving
for the dynamics. As in the other examples the current-current correla
tion <J (t)J0 (0)> = c 0 o(t). a ,_, p a,_, I emphasize that our method works only if the gradient condition
is satisfied. This holds for interacting Brownian particles, Ginzburg
Landau models and for the zero range process, but only for a particular
class of stochastic lattice gases. For non-gradient lattice gases most
parts of the proof actually go through, but there is one crucial step
missing [ 17]
4. Fluctuation Field, Scaling, Infinite Dimensional OrnsteinUhlenbeck Process
In the tradition of mathematical physics the first inclination
might be to find upper and lower bounds which pin down the behavior of
the two-point function to (3.2). This we have not achieved and it is
not clear ihether it constitutes a feasible program. Instead we formulate
(3.2) as scaling limit which is a concept familiar from many dynamical
problems. Because of the scale invariance of (3.2) we consider then
spatial distances of the order c- 1 and time distances of the order c-2 ,
£~0, and want to show that under this scaling the two-point function
tends to the fundamental solution of the diffusion equation with the
appropriate D(p) and x(p). I emphasize that this hydrodynamic limit
differs from the often studied mean field type and weak coupling li
mits. In the former only time and space are scaled whereas in the latter
also the interaction. In contrast to the Grad limit the density remains
finite as £ ~ o. On a technical, but also conceptual, level it is convenient to in
troduce the density fluctuation field in its scaled form as
i;£(f ,t) d/2 - p) (A) £ :L f(cx)(n _2t X £ ,x
£d/2 {:L f(£ x.(£-2t) j J
- pfdq f(£q)} (B)(4.1)
d/2 - m) (C) £ :L f(cx)(~ -2t X £ ,X
fES'(Rd) is a rapidly decreasing test function. The goal is then to
show that
lim < ~E(f,t)~E(g,O)>p €~0
74
(4.2)
(spatial isotropy assumed and p replaced by m for C). ~E(f,t) is a
generalized random field over RdxR which is stationary in space-time.
At fixed time ~E(f,t) tends weakly to Gaussian white noise with strength
x(p). Since time instants become widely separated as E + 0, it is rea
sonable to expect that also ~E(f,t) jointly has a Gaussian limit. Let
~(f,t) be the Gaussian field with covariance (4.2). Then we would like
to show that weakly
~( f ,t) (4.3)
Since the covariance comes from a semigroup, ~(f,t) is the infinite di
mensional Ornstein-Uhlenbeck process governed by the partial stochastic
differential equation
1 d~(q,t) = 2D(p)~~(q,t)dt + div(dj(q,t)) (4.4)
with initial Gaussian white noise of strength x(p). Here dj(q,t) is
Gaussian white noise in space-time with independent components.
To the models considered one can apply the general theory of Holley
and Stroock [18]. It turns out that, using their results, the same esti
mates which prove (4.2) also give the stronger result (4.3). (We use
here that the equal time, static equilibrium fluctuations have been
studied extensively. Our hypothesis on the interaction will be such that
their convergence to Gaussian white noise is ensured.)
5. Some Results
We quote three theorems only to give the reader an impression of
how much can be done at present.
Theorem 1 (T. Brox and H. Rost [19]). Let ~E(f,t) be the density fluctu
ation field for the zero range process, cf. end of Section 2A) for the
definition. Let the jump rate c:N+R satisfy c(O)=O<c(1) and cis in
creasing but at most linearly. Then with the appropriate X and D (4.2)
holds and (4.3) in the sense of weak convergence of paths measures on
D(R,S' (Rd)).
Theorem 2 (A. DeMasi, E. Presutti, H. Spohn and D. Wick [17]). Let
~E(f,t) be the density fluctuation field for exclusion processes with
75
speed change. Let the jump rates be such that
M
L L pm(x)Txhm(n) m=1 x
( 5.1)
where T is the shift by x, p (•) are functions on Zd with LP (x) = 0, X m zd X m
Lxp (x) = 0, and h are local functions on {0,1} • Let be either d=1
~r ~he potential <l>~e sw:hthate'\II<Pii(exp(e~P//_1)-1) ~ 0.1 with
A= <1>({0}) and 11<1>11 = L /<!>(A)/. Then with the appropriate X and D
(4.2) holds and (4.3) A30,/AJ;;;2 l·n t.he sense of weak convergence of paths
measures on D(R,S'(Rd)).
Theorem 3 (H. Spohn [20]). Let ~E(f,t) be the density fluctuation field
for interacting Brownian particles. Let V be finite range, V(q)=V(-q),
VEC 3 , v;;;o and V(O)>O (or V superstable). Let the initial measure be the
Gibbs measure for the potential V with fugacity O<z<0.28e/dq( 1-e-V(q)).
Then with the appropriate choice of X and D (4.2) holds and (4.3) in
the sense of weak convergence on C(R,S'(Rd)).
6. The Technique
For the zero range process the proof simplifies because the Gibbs
measures are product measures. On a technical level stochastic lattice
gases are most easily handled but the gradient condition is somewhat un
intuitive. For interacting Brownian particles the line of reasoning is
clouded by many technicalities. I therefore decided to illustrate the
method for the case of the Ginzburg-Landau model having the extra bonus
of convincing the reader that this class of models should be added to
the list of Section 5. My remarks should be taken with a grain of salt
because not every step has been worked out in detail. To avoid technica
lities let me assume a compact single site space, say V(q) ~ oo as
JqJ ~ 1, andv;;; a, V E c3(]-1,1[). To save writing I assume V(q) = V(-q),
<¢x> = 0 (i.e. zero magnetic field) and ~ dimension.
¢tis the stochastic process governed by (2.8), (2.9) with initial
measure the Gibbs measure (2.11) (h=O) in the infinite volume limit. Ex
pectations with respect to ¢t are denoted by E and < > refers to expec
tations with respect to the Gibbs measure. t ~ ~E(•,t) (defined in (4.1)
with m=O) is considered as an S'(R)-valued stochastic process. It has
continuous sample paths and the path space i.s C(R,S'(R)). Its path mea
sure is denoted by PE. First one has to establish tightness of the fa
mily {PE/O<E~1} which will not be done here. A limit point of this fa-
76
mily is denoted by P. The idea is to show that any P has to be the so
lution of a martingale problem. Independently one knows that the mar
tingale problem has as unique solution the Ornstein-Uhlenbeck process
with covariance (4.2) with the appropriate coefficients X and D. This
establishes then the desired result.
The martingales to be considered are constructed by the procedure
standard for Markov processes. We have the two PE-martingales
M~(f,t) (6.1)
where (6 1f)(x) = f(x+1)- 2f(x) + f(x-1) is the lattice Laplacian and
3H = -¢ + 2¢ - ¢ + V'(¢x) • Note that ~~ is a local function on 3¢ X+1 X X-1 O'j'
x[-1,1]Z and may be written as T (3H/3¢ ) witS T the shift by x. We X 0 X
want to establish that with an appropriate constant D
t ~(f,t) - /ds~(2Dfn,s)
0
2 Ma.(f ,t) = M1 (f ,t) - 2t/dqf(q)f"(q)
(6.2)
are P-martingales. Since the Gibbs measure < • > is a stationary Markov
chain with a compact state space and a smooth transition probability,
~E(f,t=O) tends to Gaussian white noise with strength X=~<¢¢> as X X 0
E + 0. With this initial condition the martingale problem (6.2) has a
unique solution which is the infinite dimensional Ornstein-Uhlenbeck
process of Section 4 [18].
Clearly for M~(f,t) there is nothing to do. So we are left with
the problem
t lim E(ifds E1/ 2 ~ fi'(Ex){~~ (cfJE-2s)- 2DcfJ -2 }i) = 0 • E + O 0 X o'l'X E S ,x
(6.3)
Life would be easy if the bound, having used stationary, t<!E 112~f"'(Ex) {(3H/3¢)- 2D¢ }i> would tend to zero as E + 0. However E 112~f"l'Ex)
X 112 X (<lH/3¢ ) and E 1 ~f":(Ex)¢ tend jointly to a non-degenerate Gaussian.
X X X Therefore some time-averaging has to be kept. To do this we break [O,t]
into intervals of length E2T with T arbitrarily large. Then after some
rearrangements one finds that (6.3) is implied by
77
0 . (6.4)
(6.4) is the hard problem.
Let Tt be the Markov semigroup of the stochastic evolution, i.e.
(Tt~)(~) = E(~(~tll~t=o = ~). Tt is a continuous, self-adjoint contra
tion semigroup on L2([-1 ,1] 2 ,<•>) = L2. Let D denote the set of local 0
finitely many ~x's , which are c2 functions, i.e. depending only on
with bounded derivatives. For ~ 1 •
fine the scalar product
~2 E D0 with <~ 1 > = 0 = <~2> we de-
(6.5)
Let H be the closure of D with <•I•>. Note that H has a large null 0
space. On an abstract level we may decompose L2 = 1 ~ j(dkH(k) as a di-
rect integral according to the spectral representation of the unitary
induced by the shift T 1 • Then H = H(O). Since Tt commutes with transla
tions, it respects this decomposition. In particular, Tt is a continu
ous, self-adjoint contraction semigroup on H • With this definition we
include (6.4) by
!i~ 00< ~1 [Tt~2 > = < ~1 lx-1/2~o > < x-1/2~ol~2 > (6.6)
-112 I -112 where X ensures that X ~0 > has norm one in H Given (6.6), in
(6.4) we have to set 2D = <aH/a~0>/x = 1/X as claimed.
By the spectral theorem (6.6) we have to establish then that the
Tt-invariant subspace in His one-dimensional. Note that because we work
in H and not in L2 (6.6) differs from the well known mixing.
How do we solve this dynamical problem ? By a miracle it can be
reduced to a purely static, equilibrium problem.
We define the partial dynamics in the interval [m,n] by fixing the
spins outside [m,n] and by setting dj(m-1 ,m,t) = 0 = dj(n,n+1,t) in (~9)
The dynamics of the spins in [m,n] depends on ~m- 1 and ~n+ 1 • We denote b T [m,n] h d. · L · y t t e correspon 1ng sem1group and by [m,n] 1ts generator. Note
h . . . . . [m,n] t at < • > lS 1nvar1ant under every part1al dynam1cs and that Tt
is a self-adjoint contraction semigroup in 12 • Now for
~ 1 , ~2 E D0 with < ~ 1 > = 0 = < ~2 >
<''' IL ,,, >2 "'1 [-n,n]"'2
n-1 L
y=-n
78
n- 1 1/2 L1/2 >2 :S2n L <(L [y,y+ 1] LTxl/1 1) ( [y,y+ 1]l/12 )
y=-n x
n-1 :S2n L
y=-n L <(T l/1 1) (L ~ l/1 )><ljl L l/1 > x 1 [y,y+1]x2 1 2[y,y+1]2
x1 ,x2
(6.7)
Because of the exponential decay of the Markov chain <•> the sums are
well defined.
We extend the inequality (6. 7) to l/1 1 's of the form Ttl/1 1 (to be able
to do this D has to be a domain of essential self-adjointness of L ln 0
H. Under our hypothesis this can be proved) and let t -+ oo Then by the
spectral theorem the right hand side vanishes and on the left hand side
lim Ttl/1 1 = l/1 0 E PH, the Tt-invariant subspace of H. Therefore for all
~0..,.t00PH, l/12 E D0 and n
0 (6.8)
I [ -n n] By continuity then also <ljl0 Tt ' l/12>
t ..,. 00
(6.9)
P · T'f -n,n] · · L2 and [-n,n] proJects onto the t lnvar1ant subspace 1n therefore
n
p[-n,n]l/1 = <l/JIIx==n cpx 'cp[-n,n]c > (6. 10)
where <l/JII· > denotes conditional expectation and cp[ •1 c the configu--n,n
ration in the complement of the interval [-n,n]. Therefore (6.6) holds
provided we can show that for all l/1 E D0
lim P[ 1 l/1 = (1/xl I¢><¢ ll/J> n -+ oo -n ,n o o
(6.11)
ln H. (6.11) constitutes the reduction to a static problem which we
aimed for.
Now we are on safe grounds. Since <•> is a Markov chain particular
methods are applicable. The reader should keep in mind however that this
is so only for the sake of presentation. In general <•> will be some
79
Gibbs measure with a finite range and sufficiently small potential (in
particular d~1). The method to be described refers to the general case.
In fact one of the main tools will be the local central limit theorem
for the magnetization which was one of the topics of the previous Hunga
rian meeting.
Let me reintroduce the magnetic field has in (2.11). The corre
sponding Gibbs measure is denoted by <•>h. Then <w0 lw> = ~h <w>hlh=O I
<w> • The magnetization per spin in ~-n,n] is abbreviated as mn = o n
(1/2n+1) L ¢x' mn = ¢0 in H. Because of the exponential decay of cor-
relation~=tRe convergence of local functions in H may be reexpressed,
at the expense of a volume factor, as convergence in L2 which is easier
to handle. Then (6.11) is 'implied by
lim (2n+1 )<[<wllm ,¢ 1 ,¢ 1> n -n- n+ n->-oo ( 1 /xl<w> 'm lJ 2> o n
0 • (6.12)
Since W E D0 for n large enough the dependence on the configuration out
side [ -n,n] is only through ¢ 1 and ¢ 1• <wllm ,¢ 1,¢ 1> -+ <1/J> = 0 -n- n+ n -n- n+
as n->- oo. Its dependence on mn should be then the second term of (6.12)
with an error o(1/yn) in L2•
Clearly, one needs a good control on the dependence of <wl lm , n
¢_n_ 1, ¢n+ 1> on mn. The idea is to smoothen this dependence by transfe-
ring it to the magnetic field,
(6.13)
where h(m,¢_n_ 1 ,¢n+1) is implicitely defined by
<mnl l¢_n-1'¢n+1>h(m ¢ ¢ ) = m • '-n-1' n+1
For untypical values of m , lm I > a > 0 , by the theory of large n n
deviations (6.13) is exponentially small inn The second equality in
(6.13) reflects that grand canonical.expectations depend exponentially
little on the boundary conditions. The first equality, for typical va
lues of mn' is more delicate and requires a control over the error term
in the local central limit theorem for the magnetization mn • Given the
right hand side of (6.13) we expand at m = 0. The first term in the n
expansion reproduces (1/x)<1j!0 >'mn and the remainder has an L2-norm of
-1 the order n
[. Open Problems
80
In this field it is very easy to state well-defined and physically
interesting problems which are beyond any mathematical reach. As a real
istic and hopefully reachable goal I propose to prove the large space
time behavior of equilibrium time correlations for stochastic evolution
models on pars with the well established expansions in equilibrium sta
tistical mechanics. We have made a first step in this direction but
much remains to be done.
( i) How does one treat models where the gradient condition is violated
(ii) In equilibrium statistical mechanics at low temperatures one may
expand around one of the ground states. How can this analysis be exten
ded to time correlations
(iii) The models discussed here are purely dissipative, in probabilistic
terms the dynamics is reversible as expressed by the symmetry of the
transition probability. In many physical models the dd ft contains in
addition so-called Hamiltonian or reversible (sorry for the terminology)
parts. Examples come from fluids but also from Heisenberg spins which
precess in addition to the dissipative drift - oH/6¢ . A related example
are the interacting Ornstein-Uhlenbeck processes. Common feature of
these models is that the transition probability is no longer symmetric.
How does one prove the asymptotic behavior of time correlat j ons in
these models
Refere:IJces
[1] R.A. Holley and D.W. Stroock, Comm.Math.Phys. 48, 249 (19[6)
[2] R.A. Holley and D.W. Stroock, Z. Wahrscheinlichkeitstheorie verw. Gebiete ]2_, 8[ ( 19[6)
[3] D. Wick, Comm.Math.Phys. §2, 361 (1981)
[4] H. Spohn, Large Scale Behavior of Equilibrium Time Correlation Functions for Some Stochastic Ising Models. In: Stochastic Processes in Quantum Theory and Statistical Physics, ed. S. Albeverio, Ph. Combe and M. Sirugue-Collin. Lecture Notes in Physics 113, p. 304. Springer, Berlin 1982
[5] A. DeMasi, N. Ianiro, A. Pellegrinotti and A. Presutti, A Survey of the Hydrodynamical Behavior of Many-Particle Systems. In: Nonequilibrium Phenomena II, eds. J.L. Lebowitz and E.W. Montroll. North-Holland, Amsterdam 1984
81
[6] K. Kawasaki, Kinetics of Ising Models. In: Phase Transitions and Critical Phenomena, eds. C. Domb and M. Green, vol. 2. Academic Press, New York 1972
[7] T.M. Liggett, The Stochastic Evolution of Infinite Systems of Interacting Particles. Lecture Notes in Mathematics 598. Springer, Berlin 1978
[8] T.M. Liggett, Ann.Prob. 1, 240 (1973)
[9] P.N. Pusey and R.J.A. Tough, J.Phys. A15, 1291 (1982)
[10] P.N. Pusey and R.J.A. Tough in: Dynamic Light Scattering and Velocimetry : Applications of Photon Correlation Spectroscopy, eds. R. Pecora. Plenum Press, New York 1981
[11] J.W. Cahn and J.E. Hilliard, J.Chem.Phys. 28, 258 (1958)
[12] P.C. Hohenberg and B.I. Halperin, Rev.Mod.Phys. 49, 435 (1977)
[13] R. Lang, Z. Wahrscheinlichkeitstheorie verw. Gebiete ~, 55 (1977)
[14] T. Shiga, Z. Wahrscienlichkeitstheorie verw. Gebiete 47, 299 (1979)
[15] W.H. Faris, J.Funct.Anal. 32, 342 (1979)
[16] S. Katz, J.L. Lebowitz, H. Spohn, J.Stat.Phys. ~, 497 (1984)
[17] A. DeMasi, E. Presutti, H. Spohn, D. Wick, Asymptotic Equivalence of Fluctuation Fields for Reversible Exclusion Processes with Speed Change, preprint
[18] R.A. Holley and D.W. Stroock, RIMS Kyoto Publications~, 741 (1978)
[19] T. Brox and H. Rost, Ann.Prob. to appear
[20] H. Spohn, Equilibrium Fluctuations for Interacting Brownian Particles, preprint
* Theoretische Physik, Universitat Munchen, Theresienstrasse 37,
8 Munchen 2, FRG
83
LINEAR AND RELATED MODELS OF TIME EVOLUTION IN
QUANTUM STATISTICAL MECHANICS
A. G. Shuhov and Yu. M. Suhov
1. Introduction
The aim of this paper is to give a review of some re
cent results concerning the problem of constructing and
studying time evolution for quantum systems with infinitely
many degrees of freedom. The first rigorous results in this
direction are due to 0. E. Lanford and D. W. Robinson [17-
-20], see also [13], Ch. 5.3. An approach to this problem
has been proposed by 0. Bratteli and D. W. Robinson [12],
see also [13], Ch. 6.3. Among recent papers we refer to
[1-3], [10-11], [16], [24].
Now the problem of construction of the time evolution
of a state of a quantum system is solved in a satisfactory
way for the classes of lattice spin and lattice fermion
systems where the evolution comes from a group of *-auto
morphisms of an appropriate c* -algebra (time dynamics) .
The main feature of these systems is that they are "locally
finite-dimensional". Other classes of realistic quantum
systems such as quantum particle systems in euclidean space
R v ( both statistics ) or lattice boson system are "lo
cally infinite-dimensional"; for these classes the problem
of constructing the time dynamics is solved only for a de
generated class of models ("linear" fermion models in Rv )
and perhaps has no positive solution in general.
In our context, the problem of studying the time evo
lution is reduced to the question: what is the limit behav
ior of the time-evolved state Qt , t E R 1 , when t->- ±oo ?
This question is of very great interest for understanding
84
the approach to equilibrium and thereby for the mathemati
cal foundation of the Boltzmann-Gibbs postulate in Statis
tical Mechanics. Now the problem of convergence to equilib
rium seems to be very hard and is solved only for a number
of simple models: linear models (both statistics), one di
mensional X-Y model and one dimensional hard rod model.
This list is probably not a final one: in Section 5 we shall
briefly discuss some conjectures which seem to be realistic.
2. Preliminaries
Let V be a complex separable Hilbert space,
be the Fock space constructed over V
+ 00 + W = (l)n=O H~ , ( 1 )
(± indicate the statistics and are omitted whenever there
will be no confusion). As usually, a+(f), a(g), f,g E V ,
denote the creation and annihilation operators in H which
obey the CCR (+) and CAR (-) We shall have in mind two
concrete realisations: V=l 2 (Zv) (lattice systems) and
V = L2 (R v) (continuous systems) . By V0 we denote l~ ( Z v)
and L~(Rv) , respectively (sequences and functions with
compact support ) . In the usual way, one introduces CCR
and CAR C* -algebras A± over V0 (in the fermion case
V0 may be replaced by V ) • The action of the *-automor
phisms group {Sx} of space translations on A is defined
by
(2)
where sxh(·) = h(·-x)
As usually, by a state of a c* -algebra we mean a
linear non-negative normalized functional. In a standard
way one introduces translation invariant (t.i.), ergodic,
even, gauge invariant (g.i.), locally normal (l.n.) and
quasi-free states (see, e.g. [13)). In the boson case one
uses the notions of a C00 -state and analytic state (see 00
[13)). For instance, for a C -state Q the values
85
are defined and for an analytic state Q these values
uniquely determine the state Q itself (in the fermion
case this is given for all states by definition) . The val
ues (3) define the so-called (m 1n) -moment functionals on
( V0 )0 (m+n) (moment form) K (m 1n) m n E Z 1 of a state Q I I + I
Q 1 which have the obvious properties of sesquilinearity and
positive definiteness. The functionals K6m 1n) with
m+n s 2 are called the lower moment functionals (in the
case of a quasi-free state they determine uniquely the
whole state Q ) . In the fermion case the functional K(m 1n) ®(m+n) Q
may be extended to V and hence is bounded
II (m 1 n) ., 1 ( KQ II s 1 I m1n E z>t) .
A convenient topology on a set of states is defined
by means of convergence of the (m 1n) -moment functionals
K(m 1n) In the fermion case this is merely the w* -Q
topology.
In the sequel we shall intensively use "weak depen
dence" properties of a given state Q . For m1n E Z~ and
two orthogonal subspaces V ( 1) 1 V ( 2 ) c V we set
a(mln)(V(1llv(2)) = max(m11n1) sup<V(1)1V(2)) Q
I (m1n) (1) (1) (2) (2) (2) (2) (1) (1) KQ (f1 I ••• If lf1 I ••• If - ; g1 I ••• lg - lg1 I ••• lg ) -m1 m m1 n n1 n1
(m1 1n1) (1) (1) (1) (1) (m-m1 1n-n1) -KQ (f1 1 ••• 1fm1 ;g1 1 ••• lgn1 )KQ x
(2) (2) (2) (2) 'I X (f1 1 ••• 1 f ;g1 1 ••• 1g ) m-;n1 n-n1
The maximum in RHS of (4) is taken over all m1=0 1 •.. 1m
and n 1 =o~··· 1 n with m1+n 1 > 0 1 the supremum is taken
11 f 1( 1) I • • • If ( 1 ) ( 1 ) ( 1) V ( 1 ) over a vectors 1 g 1 1•••1g E m1 n1
f ( 2) f ( 2 ) ( 2 ) ( 2 ) E V ( 2 ) . th 1 1 1•••1 m-m 1g1 1•••1gn-n w1 norm s .
1 1 The following case is particularly interesting:
( 1) ( 2)
(4)
V =.t2 (I(xlr)) or L2 (I(x 1r)) 1 V =.t2 (CI(x 1s+r))
or L2 (CI(xls+r)) where xEZv or Rv 1 S 1r>O and I(x 1u)
is the cube on zv or in Rv centered at x with the
edges parallel to coordinate axes and the edge length u
86
In this case we set
where sup is taken over all subspaces of the form men
tioned above where s, r are fixed and xEZv or Rv is
arbitrary. The quantity a6m,n) (r,s) characterizes the de
crease of the "space correlations" in the state Q .
3. Linear models (groups of Bogoliubov transformations):
the fermion systems
Let T( 1 ) be a bounded linear operator and T( 2 ) a
bounded antilinear operator in V satisfying the relations
(T( 2))*T( 1)+(T( 1))* T( 2) = T( 1) (T( 2))*+T(2) (T( 1))* = 0 (5)
(T( 2))*T( 2)+(T(1))* T( 1) = T{ 1) (T( 1))*+T( 2) (T{ 2))* = 11 (6)
Then the formula
(7)
defines a * -automorphism T of the c* -algebra A
This automorphism is called the Bogoliubov transformation,
or linear canonical transformation (LCT) .
Assume that 'll"t = (T!'j), i,j ~ 1,2, tER 1 , is a
group of bounded operator (2x2) -matrices with T~' 1 = = T2,2 T(1) T1,2 = 2,1 (2) . f . (5 ) ( 6 ) t t , t Tt = Tt , sat1s y1ng - .
Such a group generates the corresponding group of Bogoliu-1 bov transformations (LCT) {Tt, tER } . This is the time
dynamics under consideration. We shall suppose that the
group {Tt} commutes with space translations Sx on A xEZv or Rv Given a state Q , we define
Tt*Q(A)
Our aim is to examine time-evolved states T *Q as t
t+±oo
Consider the following condition (A) on the initial
state Q and the group of operator matrices {'ll"t} generat
ing Bogoliubov transformations.
(A) For every tER 1 there exists a finite or count-
87
able orthogonal decomposition
that
v = v!tl (j) v!tl (j) ••• 0 1
A1) for all 1 m,n E z+ with m+n;:: 2
where
where
(m,n) (") = (m,n) (V(t) "' V(t)l ct .._ aQ .e. , "'.t•:.e.•,:o,.e. .e.•
A2) for every vector hEV0
lim sup dt(h,.t) = 0 , t+±oo .t;::O
max II'~~ (t) T2~lhll , a>.=1,2 V.e_
is the orthogonal projection
such
1 A3) for all m,n E z+ with m+n;:: 3 and all f 1 , ... ,
fm' g1, ... ,gn E Vo
where
Theorem ([25]). Let an initial state Q and a
group {Tt} satisfy condition (A). Then the states
tER 1 , converge to a quasifree state P as t+±oo
Tt*Q ,
iff the (a?-1, a?. 2)
lower moment functionals KT *Q , a.>. 1 +~ 1,2 , converge t
to the corresponding functionals 0
This theorem reduces the problem of convergence of
states Tt*Q to the more simple question about convergence
88
of the lower moment functionals. The question of conver
gence for lower moment functionals may be solved separately
(see Subsection 3. 4) . However the conditions of Theorem 1 are
imposed on the pair ( Q 1 {'l' t}) 1 while from the physical
point of view it is convenient to deal with separate con
ditions on Q and the group {~t} .
The group {Tt} is uniquely determined by its infi
nitesimal generators
c (8)
The Fourier transforms of these operators are given by the
equalities
Bf !6l b(6)f(6) 1 Cf(6) ~ c(6)f(-6) (9)
where b is a real and c an odd function on [-n 1 n)v
or Rv Let
~(b(6)±b(-8)) 1
w(8) = (b:(8)+[c(8) [2 ) 2 ( 10)
w±(8) = b_(8) ± w(8)
The Fourier transform of the operator
plication operator
( 11 )
is the multi-
itb ( 8 ) [ ib ( 8 ) ] T~ 1 lf(8) = e - cos(tw(8)) + w~ 8 ) sin(tw(8)) f(8) I
and that of is defined by the equality
itb (8) T( 2 lf(8) = ie - c(G) sin(tw(8))f(-8)
t w( 8)
3.1. One-dimensional case
Let v=1 • We shall suppose now that
for some given vE=2 1 3 1 ••• the function
intersection
where
b 1 c E C 1 and + v +1
w EC E and the E
( 12)
89
6J.(w) = {8: dj w(8) = o}. d8j
It is convenient to choose the minimal
(13)
with these
properties. For ~s=2 the above condition was formulated
in [9]. As to the initial state, we shall suppose that for
all m,n;::: 0 there exists d = d (m,n) > 0 such that
S d (m,n) lim sup aQ (r,s) s->-oo r>O
0 ( 14)
Theorem 2 ([21]). Let V = t 2 (z 1) or L2 (R1) . If a
group {Tt} and an initial state Q satisfy the above con
ditions, then the assertion of Theorem 1 holds. o
3.2. Multi-dimensional case: coranks of singularities
In this subsection we consider the general case v;::1
Suppose now that the functions b, c E C"" Set
det d 2 w (8) = 0} E
c=+ , ~ - ( 15)
Suppose that for all yERv the set {8:gradw±(8)=y} con
sists of a finite number of points. Set
p = max max s=± 8E6 2 (ws)
corank d 2 w (8) E
( 1 6)
Theorem 3 ([21]). Suppose that v>3p and the initial
state Q satisfies the following condition: for all
m,n;::O and j>O
3pv 1 vp ll. m v-3p (m,n) r . -v-3pJ - 0
s aQ l' s, J s -s->-oo
( 17)
Then the assertion of Theorem 1 holds. o
Remark. In the lattice case ( V = t 2 ( Z v)) the set 62 (w±)
is always non-empty. Hence, Theorem 3 is applicable only
for v~4 . In the continuous case (V=L 2 (Rv)) the asser-
tion of the theorem remains true when the sets 62 (w±) =0
in this case one sets p~O If v=1 and 62 (w±)=0 , we
can suppose that Q satisfies ( 14) with d=O which is
90
weaker than (17) with p=O .
3.3. Multi-dimensional case: local singularity types
In this subsection we shall use a more detailed in-
formation about the functions in a neighbourhood of the
sets 13 2 (w±) Suppose that for every e0 E 13 2 (w±) the
function F± e~ -e.grad w±(e 0 )+w±(e) has a simple or
parabolic singularity at the point e=e 0 (the singularity
of one of the types Ak (k~2) , ok (k~4) , E 6 , E 7 , E8 , P 8 ,
x9 , J 10 (see, e.g., [5], [6] and the books [7-8])). It is
convenient to introduce
sup E(o(e )) e r::6 2 (w+l 0
0 -
( 18)
Here o(e 0 ) ~ o±(e0 )
e0 , and E(o) is the
type o (see [4], [6]
is the singularity type of F± at
so-called index of singularity of
and the books [7-8]). For simple and
parabolic singularities the index E is given by the fol
lowing table (see [6], [8], [14-15]):
1 1 1 2 2 2
Theorem 4 ( [21]). Suppose that v>6E where E =
max E(w±) and an initial state Q satisfies (17) with
p replaced by 2E . Then the assertion of Theorem 1 holds.
0
Remarks. 1 . For v;::6 the above conditions on the func
tions w± describe a "generic" situation (in the class
C"" ) , see [ 6] , [ 7] .
2. If v=2 then the theorem is applicable only if the
functions have a singularity of a type Ak , 2s;ks;4 , at
every e0 E 13 2 (w±)
3.4. Convergence of the lower moment functionals
In this subsection we discuss briefly the question of
convergence for lower moment functionals K (m,n) , m+n::;; 2 , T *Q
t as t~±oo . The reader can find the details in [21]. For the
91
sake of simplicity we consider the one-dimensional case al
though there is no restriction on the space dimension.
First, suppose that the initial lower functionals
K(m,n), m+n$2, are S -invariant, xEz 1 or R1 . It is a o o X (1 0) (0 1)
easy to see that ln thlS case K a' =K a' =0 . By the
Riesz theorem, one can write
K( 2 ,0) (f g) = <f M(al g> a , , 1,1 K ( O ' 2 ) ( f g) = <M (a) g f > , a , 2,2 ,
K(1,1) (f ) a ,g <f M(a)g> , 1, 2 f ,g E V
M<al and 1 , 1
( 19)
where M<al is a bounded linear operator,
M<al ar~'~ounded antilinear operators in V commuting 2, 2 1 1
with Sx , xEZ or R . In addition, it is convenient to
use the linear operator M~~~ = 1-(M~~~l* . Denote by ~(a) the operator (2x2) -matrix with entries M1~~ , 1si,js2
Let us introduce also the operator (2x2) -matrix
ID =(Di,j) , 1si,j$2 , with o 1 , 1 =o 2 , 2 =B , o 2 , 1=o 1 , 2 =C
(this is the generator for the group {Tt} , see (8)).
Passing to the Fourier transform we write the matrix fu in
the form
A
where ID0 is the diagonal matrix which consi·sts of the
multiplication operators by the function b+
Suppose that for some given ~=1,2, ..• the function
wE C~+ 1 and the intersection
Then
matrix
n~+1 j = 1
6 o (w) = 0 . J
(Tt*Q) :M converges weakly as t+±oo
with the Fourier transform ~
to the limiting
Suppose now that initial lower functionals K(~,n) m+n ::s 2 , have the following "periodicity" property: for
some fixed zEz 1 or R1 + +
92
K (m,n) (h) = K (m,n) (S h) , m+n=1 , hE:V , kEZ 1 and the Q Q zk
t (Q) 1 .. 2 t 'th th t opera ors M .. , ~1,JS , commu e w1 e space rans-
lations Szk1 '1this also implies K(6,0)=K(g, 1 )=0 ).
Under some natural conditions of the non-degeneracy of the
functions w± which are omitted from this text one can
check that the matrix (T t *Q)
~ converges as t+±oo to a
:M • The limiting matrix ~ may be obtain-limiting matrix ed from N (Q) by some space-averaging procedures (which
related to periodicity properties of the func
see (11)). Compare with [9], theorem 4.2.
are closely
tions
Finally, the periodicity of the initial lower func
tionals K (~, n) , m+n::; 2 , may be replaced by more general
"alm::>st periodicity" conditions. The same statements hold for
the multi-dimensional case. In this case the functions w±
should satisfy conditions which correspond to the condi
tions of Subsection 3.2.
In the one-dirrensional case it is possible to construct
examples of initial lower moment functionals K(~,n) m+n ::; 2 , for which
lim K+m~~) (f,g) f lim t++oo t t+-oo
K (m,n) (f ) T *Q ,g
t
(compare with [9], Theorem 4.1).
4. Linear models: the boson systems
, f,gEV,
The boson case is more difficult. First, a Bogoliubov
transformation (7) created by a pair of operators T( 1),
T( 2 ) , in general, does not preserve the CCR C* -algebra
A+ (this is the case,whenever either T( 1 lvo ¢ V0 or
T( 2 lvo ¢ V0 which is typical for physically interesting
examples). However, a simple idea will allow us to intro
duce a linear time evolution {Tt*Q, tER1 } of an initial
state Q without using the time dynamics on A+ .
Let us give some definitions. A coo -state Q on A+
is called bounded if the moment functionam K(m,n) may be
extended to v181 (m+n) and hence are bounded, m~ n E Z! . Let
93
Tt=(T~'j) , i,j=1,2 , tER1 , a group of bounded operator 1 1 2 2 (2x2) -matrices be given where the operators Tt' =Tt'
=T~ 1 ) :V+V are antilinear and T~ 1 ) ,T~ 2 ) satisfy (5)-(6)
A bounded quasifree state P is called {Tt} -invariant if
the lower moment functionals K (m' n) , m+n ,S 2 , satisfy the p relations
K(0,1) ( )=K(1,0) (T(2) ) K(0,1) (T(1) ) p g p t g + p t g
(20)
etc. These equalities express only the fact that the lower
functionals do not change under the "transformations" (7). Such
a definition seems to be reasonable since these relations
imply that all (m,n) -moment functionals K(m,n) do not p
change under the "transformations" ( 7) •
We say that a C00-state Q1 is majorized by another
c""-state a2 if for any nEz! and f 1 , ... ,fnEV0
(21)
A bounded state Q is called stable if it is majorized by
a {Tt} -invariant state (which is called a {Tt} -inva
riant majorant for Q ) .
Let Q be a bounded state. We may define the family
{(K(~,n))t} , m,nEz! , tER1 , of "time-evolved" moment func
tionals by using the formulas analogous to (20). In other
words, the functionals (K(~,n))t describe the "time evolu-
tion" of the initial functionals K (~' n) under the "transforma
tions" (7). The precise definition is rather complicated and
[25]. The is given in 1 Qt , tER , for which
problem is to reconstruct the states (Km,n) ~K(m,n) . Such a family
Q t Qt 1 {Qt,tER }
state Q .
defines the linear time evolution of the initial
94
For the sake of simplicity we consider the particular
case where T~ 2 )=0 (i.e., the matrix Tt is diagonal). We
assume also that the operators T~ 1 ) commute with the
space translations. Then any bounded t.i., g.i., quasi
free state P is {Tt} -invariant. Such a state is called
regular. The only non-zero lower moment functional of the
state P is K(;, 1) . The corresponding bounded linear
operator in V is denoted, as in the fermion case, by
M~~~ (see (19)). Given a bounded Aczv or Rv , we de
note by V(A) the subspace of V consisting of elements
with support in A and by nA the orthogonal projection
V-+V(A)
Theorem 5 ( [25)). Let Q be a l.n., q.i., stable
state with a {Tt} -invariant regular majorant P satis
fying the following conditions: (a) ilK (;' 1 ) II < ~ , (b) for
any bounded A the operator
is of trace class. Then for any
unique l.n., g.i., stable state
ooK (m,n) m,n E: Z 1 . The mappings
tER1 there exists a Qt such that (K(m,n))
Q t Tt*: QI~Qt , tER1 , have
T* = T* T* t t E R 1 t1+t2 t1t2' 1'2·
D
Qt +
the group property:
on
By construction, the states
tER 1 .
Tt*Q depend continuously
Conditions (a) and (b) are sufficient, but not neces-
sary. Roughly speaking, it means that the particle density
in the regular state P is relatively small (for instance, in
the case of the usual free time evolution we work outside
the region of Bose condensation) . An interesting question
is to find weaker sufficient conditions.
Having the reconstruction theorem, one can repeat the
analysis of convergence of the states T t *Q as t-+±oo given in
thefermion case. We omit the details (see [25)).
5 Degenerated one dimensional models
In this section we discuss some quantum models with
interaction. A natural conjecture is that in a "generic"
95
situation the time-evolved state Qt of a large particle
system converges as t+±oo to an equilibrium Gibbs state
corresponding to the interaction potential (the Boltzmann
Gibbs postulate). For linear models, as we have seen in the
preceding sections, such a conjecture is false: the set of
limiting states consists not only of equilibrium Gibbs
states but includes many other time-invariant states. This
is related to the fact that a linear motion which corres
ponds to a quadratic Hamiltonian has many invariants (this
is, in a sense, a simple example of quantum completely in
tegrable systems) .
Among the i~teracting systems there are models which
may be reduced in some way to linear ones. We shall con
sider two classes of such models which are probably the
simplest ones. The first class consists of the X-Y rrodel and
its generalizations, the second class is formed by hard
rod models in space and on lattice. Both classes are con
sidered in one dimension. We believe that next examples of
such type will be the one dimensional X-X-X model (iso-1 tropic spin 2 Heisenberg model) and the one-dimensional
continuous boson model with 8 -interaction. All the models
we have mentioned are examples of what is called now quan
tum completely integrable systems (moreover, ones are speak
ing about systems which are integrable by the (quantum)
Lax method (the method of (L,A) -pair)). This list may
probably be continued further on, but the corresponding
models seem yet to be too hard for rigorous treatment.
5.1. The X-Y rrodel andits generalizations
The c* -algebra of a quantum spin system (spin ~) on
the lattice z1 is defined as the infinite tensor product
m = M 3Z 1
where M is the complex (2x2) -matrix algebra. The local
* -subalgebra of
o.Y, o.z denote
m is denoted by m0 • As usually, ojx'
the Pauli matrices associated with the J J 1
point jEZ . Consider the derivation of m which
is defined by
96
(22)
where HX-Y is the formal Hamiltonian of the X-Y model
(23)
Although (23) is a divergent series, formula (22) is cor
rect due to the locality of A . The group of * -automor
phisms {W~-Y, tER 1 } of ffi generated by oX-Y describes
the time dynamics in the X-Y model. If Q is an initial
state of m , then its time evolution is given by
The problem under consideration is: what happens with X-Y
Qt as t-+±oo ?
Let ¢ be the automorphism of ffi defined by the
formula
¢(A) rr':" J=-oo
a~ A II~ a~ J J =-oo J
(24)
(25)
(this definition is correct due to the locality of A ) .
Let ffi(+) be the * -subalgebra of m consisting of ¢-in
variant elements.
In the analysis of the one-dimensional X-Y model the
key role is played by the so-called Jordan-Wigner transfor
mation. This transformation is formally written in the form
+ Z X a.+a. +-+II . aSaJ.,
J J S<]
and generates the * -automorphism of c* -algebra m ( +)
and the even c* -subalgebra Aev of CAR c* -algebra A
This * -automorphism commutes with the space translations
and establishes the one-to-one correspondence between the
groups of * -automorphisms of both c* -algebras as well as
between their states. It is convenient to consider the in
duced correspondence between ¢ -invariant states of spin
~ c* -algebra m and even states of CAR c* -algebra A X-Y
It is easy to check that the tirre-evolved state Qt (see
( 24) ) is ¢ -invariant iff the initial state Q is ¢ -inva
riant. The transformation of Jordan and Wigner maps the X-Y
97
dynamics into the LCT -dynamics which is defined by the
infinitesimal generators B and C (see (7)-(8)) such
that
b(8) =2 [h0 -(a+B)cos8], c(8) =2i(-a+B) sine, 8E [-71,71)
We shall consider the more general situation. Let
{Ut, tER 1 } be any group of * -automorphisms of the C* -al-1 gebra ffi(+) which induces a group of LCT {Tt, tER } of
the c* -algebra A under the Jordan-Wigner transforma
tion. Given a ¢-invariant state Q of the c* -algebra
m , we introduce the family of time-evolved states
{U~Q, tf:R1 } . 1
Theorem 6. Suppose that a group { Ut, tER } of * -automorphisms of ffi(+) satisfies the above condition and
the corresponding group of LCT {Tt, tER 1 } of A satis
fies conditions of Theorem 2. Let Q be a ¢ -invariant
state of m which corresponds to an even state Q0 of A
satisfying conditions of the same theorem. Then the states
U~Q converge as t+±oo to a ¢ -invariant state P of m which corresponds to an even quasifree state P0 of A
if for all j 1 , j 2 E Z 1 with j 1 <j 2
lim t+±oo
• =x, y . D
The proof of this theorem is evident. Using the Jordan
Wigner transformation we reduce the problem of convergence
for the state U~Q to the problem of convergence for the state
T~Q0 • The latter is solved by Theorem 2.
Remark. For the proper X-Y model this theorem is
applicable whenever (Ba) 2 +h 2 f 0 . 0
In the same way we reduce the question of convergence
for
to the question of convergence for the lower rroment functionals
of the state T~Q0 • See Subsection 3.4.
98
5.2. Hard rod model
In this subsection we discuss the one-dimensional con
tinuous hard rod model. The lattice version of this model
is simpler.and is to be discussed later. The details and
proofs of the theorems which follow may be found in [23].
Given d>O , we introduce the n -particle Hamiltonian
of the model by the formula
H (n) (d)
where 1 xjER , j=1, ... ,n, and
{+oo , O::;r:s;d
<Pd (r) = 0 , r>d
(26)
(27)
is the pair interaction potential of hard rods of the length
d . The operator ( 26) is considered in the subspace Hn [d] c
c Hn which consists of functions vanishing whenever
min [x. -x. I < d . We shall use the Dirichlet boundary conJ1 ]2
ditions on the set
min [ x . -x . I =d} ]1 ]2
(28)
H(d) denotes the Hamiltonian of the system with ar
bitrary number of particles
H(d) = €& 00 H(n) n=O (d) ( 29)
The first problem is to define the time evolution of a
given initial state Q . For the model under consideration
we are not able to construct the one-parameter group of
* -automorphisms of c* -algebra A nor its appropriate
subalgebra (the time dynamics). As in Section 4, we proceed
in a round-about way. For a given initial state Q which
satisfies certain {W h.r.*Q tER1} t _,
conditions we define a family of states
and declare this family to be the time
evolution of Q . A reason for this is that the states
Wth.r.*Q give a (unique) solution of the Liouville equa
tion. For precise statement we need some definitions and
constructions.
99
1 Given 1\~R , we introduce the Fock space HV(/\) =
rJ/'" H in the volume 1\ (cf. with (1)). HV(') [d] n=O n,V(l\) " denotes the subspace of HV(/\) consisting of vectors
f = (fn' n?:O) such that
whenever
If /\=R 1 , we omit the index V(/\) in these notations. +
It is convenient to identify both spaces HV(/\) with
the Hilbert space L2 (C(/\) ,A/\) where C(/\) is the collec
tion of finite subsets Xc/\ (including the empty set) en
dowed with the natural topology and the corresponding Borel
CJ- algebra /./\ , A/\ is the Lebesgue-Poisson measure on
C(/\) . The subspace HV(/\) [d] thereby coincide with the
subspace of L2 (C(/\) ,A) which consist of functions with
support on the subset
C(/\,d) = {XEC(l\): minx,x'EX:x;tx' :x-x' i ?:d} .
Given Ac/.1\ , we denote by rr(A) the orthogonal projector
in L2 (C(/\) ,A/\) onto the subspace of functions which van-
ish outside A
The next
cally finite"
the property:
object
subsets
xnc is
we need
of R1
finite
is the collection C of "lo-
, i.e., subsets XcR1 having
provided CcR1 is a bounded
set (it is clear that such X should be either finite or
countable) . The set C is endowed with the natural topo
logy; the corresponding Borel CJ -algebra is denoted by /.
The CJ -algebra L is generated by the family { L ( /\) } where 1 \ ( 1\) /\cR is a bounded (Borel) set and L is the CJ -subal-
gebra of L generated by the map
\ ( /\) Clearly, L is isomorphic to Ll\ and we do not dis-
tinguish these CJ -algebras further on.
Given XEC it is convenient to label the points
xEX by integers j so that (a) index 0 is attached to the
point of X with the smallest non-negative coordinate,
(b) the indices of next neighbour points differ by 1,
100
(c) the indices of points increase with the coordinates.
Correspondingly, we shall use the notation xj(X) .
In the usual way one introduces the action of the
space translation group {Sx' xER1 } on C and the defi
nition of translation invariant (t.i.) probability measure
on (C,L) . Every such measure is uniquely determined by
its restrictions on the o -algebras LA Now let Q be a l.n. state. Given a bounded AcR 1 ,
we set
This defines the probability measure on
probability measure ~Q on (C,L) whose restrictions on
LA coincide with~(~) is called the diagonal measure of
the state Q . It is clear that if a state Q is t.i.,
then the measure ~Q is t.i., and if Q is ergodic, then
~Q is ergodic. If Q is t.i., then the formula
k( 1 ) (A) = J ~Q(dX)Card XnA, AcR1 , Q c
defines the t.i. (Borel) measure
has the form
on which
k61 ) (dx) = aQ•dx ,
-1 where aQ E R + = [ 0, +00 ] • The number aQ is called the
particle density of the state Q .
A l.n. state Q is called d -admissible if the
diagonal measure ~Q is concentrated on the set C[d]
where
C[d] = {xEC Card xnR! = card xnR1 = oo ,
x. (X) -x. 1 (x) > d for all jEZ 1 and J J-
Lk<O(~(X)-~-1(X)-d) = Lk>O(~(X)-~-1(X)-d)=oo}
It is clear that if a t.i. state Q is d -admissible, then -1
aQ<d . Given a t. i. d -admissible ergodic state Q , one can
-1 define a new t.i. ergodic state CQ with acQ=daQ(1-daQ)
101
which is called the contraction of the state Q . Converse
ly, if Q is a t.i. ergodic state with a finite particle
density aQ , then one can
godic state ~Q with ~Q
the dilatation of the state
mutual inverses:
define a t.i. -1
daQ ( 1 +daQ)
Q . The maps
CDQ = Q , IDCQ = Q .
d -admissible er
which is called
C and ~ are
The precise definition of the maps C and ~ may be found
in [23]. We want to emphasize that these maps are not in
duced by any transformation of the c* -algebra A
In addition, one can introduce a natural map 0 which
transforms states of
CCR c* -algebra A+
CAR c* -algebra A into states of
(see [23]). The definition of all the
maps C, ~ and 0 essentially uses the one-dimensional
structure of the system.
(2X2)
where
In what follows we denote by T~ree the operator
-matrix with T( 2)=0, T(1)=exp(it-21 ll), tER1 , t t
is the Laplacian in V = L2 (R1 ) . Corres-
pondingly, yfree tER1 t , , denotes the corresponding Bogo
CAR c* -algebra A . Let Q liubov transformations of
be a t.i., d -admissible ergodic state of A We set
and call the family of states the hard rod time
evolution of Q • If Q is t.i., d -admissible ergodic
state of CCR c* -algebra A+ , the corresponding defini
tion is as follows:
Theorem 7 ([23]). The mappings w~·r·*, tER1 , trans
form the set of t.i., d -admissible, ergodic states into it
self. They are continuous and have the group property:
wh. r . * = wh. r. * wh . r. * 1 t1+t2 t1 t2 't1,t2 E R
The states W~ • r · *Q depend continuously on t E R 1 . o
102
Our definition is justified by the following fact. + +
One can choose subsets D- c A- having the property that
any t. i. , d -admissible, ergodic state is uniquely deter
mined by its restriction onto D± and are such that for
any and an above-mentioned state Q the function
is of class c1 and satisfies the Liouville equation
~ Wh.r.* Q(A) Wh.r.*([H A]) dt t = t (d) '
Moreover, this is a unique (in some sense) solution (for
the details see [23]).
Now we turn to the problem of convergence when t+±oo
A natural candidate for the role of invariant state is a
state P of the form DP0 in fermion case and e-1DP0 in
boson case where P0 is a t.i., g.i. quasifree state
(which is invariant with respect to the free time dynamics free { T t } ) . Such a state P will be called d -equilibrium.
It is completely determined by its (1,1) -moment func
tional K( 1 • 1 > p Theorem 8 ([ 23]) . Let Q be a t. i., d -admissible,
ergodic state. Suppose that the contraction CQ in fermion
case and the state C8Q in boson case satisfies condition
(14). Then the states w~·r·*Q , tER1 , converge as t+±oo
to the d -equilibrium state P defined by the relation
K(1,1) = K(1,1) p Q
Proceeding in the same way one can treat the lattice
version of the hard rod model. The Hamiltonian (26) is re
placed by its discrete analog here. The main feature of the
lattice hard rod model is that the time evolution may be
constructed in a "direct" way for any d -admissible state
(although we are not able to construct any reasonable time
dynamics). The details will appear in [22].
103
6. Concluding remark: the validity of conditions of
Theorems 1-8
The problem to give reasonable examples of states
which satisfy conditions of theorems formulated above is
not only of academic interest. It is connected with the
physical background of the approach we adopted in this
paper. We shall not dwell on this problem here with all
the details. The main class of such examples is constituted
by the so-called Gibbs states. Up to now the theory of
Gibbs states has been developed mainly for spin systems
(see, e.g. [17-20]) using the KMS boundary condition. For
the "locally infinite-dimensional" systems there are only
particular results. However, even for particular Gibbs
states constructed so far the problem to verify conditions
of theorems under consideration is not trivial. This is
especially true for theorems from Subsection 5.2. For the
details see the references cited above.
References
[1] Anshelevich V. V. First integrals and stationary states of quantum spin dynamics of Heisenberg. (Russian) Teoret. Mat. Fiz. 43, no. 1, 107-110 (1980).
[2] Anshelevich V. V., Gusev E. V. First integrals of one dimensional quantum Ising model with diametrical magnetic field. (Russian) Teoret. Mat. Fiz. 47, no. 2, 230-242 (1981).
[3] Araki H., BarouchE. On the dynamics and ergodic properties of the X-Y model. J. Stat. Phys. 31, no. 2, 327-346 ( 1983).
[4] Arnold V. I. Integrals of quickly oscillating functions and singularities of projections of Lagrange manifolds. (Russian) Funktsional. Anal. i Prilozhen. 6, no. 3 , 61 -6 2 ( 1 9 7 2) .
[5] Arnold V. I. Normal forms of functions near degenerated critical points, Weyl groups Ak, Dk, Ek and
Lagrange singularities. (Russian) Funktsional. Anal. i Prilozhen. 6, no. 4, 3-25 (1972).
[6] Arnold V. I. Remarks on stationary phase method and Coxeter numbers. (Russian) Uspekhi Mat. Nauk 28, no. 5, 17-44 (1973).
[7] Arnold V. I., Varchenko A. N., Gussein-Zadeh S.M. Singularities of differentiable mappings, vol. I. (Russian) "Nauka", Moscow, 1982.
104
[8] Arnold v. I., Varchenko A. N., Gussein-Zadeh S.M. Singularities of differentiable mappings, vol. II. (Russian) "Nauka", Moscow, 1984.
[9] Boldrighini C., Pellegrinotti A., Triolo L. Convergence to stationary states for infinite harmonic systems. J. Stat. Phys. 30, no. 1, 123-155 (1983).
[10] Botvich D. D. Spectral properties of fermion dynamical systems. (Russian). Diss. Thesis. Moscow State University (M. V. Lomonosov), 1983.
[ 11] Botvich D. D. , Malyshev V. A. Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas. Commun. Math. Phys. 91, 301-312 (1983).
[12] Bratelli 0., Robinson D. W., Green's functions, Hamiltonians and modular automorphisms. Commun. Math. Phys. 50, 133-156 (1976).
[13] Bratelli 0., Robinson D. W. Operator algebras and quantum statistical mechanics, vol. II. SpringerVerlag, New York -Heidelberg - Berlin, 1981.
[14] Colin de Verdiere Y. Nombre de points entiers dans une famille homothetique de domaines de Rh. Ann. Scient. Ecole Norm. Sup. 4eme ser. 10 no. 4, 559-576 (1977).
[15] Duistermaat J. T. Oscillatory integrals, Lagrange immersions and unfoldings of singularities. Commun. Pure Appl. Math. 27, no. 2, 207-281 (1974).
[16] Gusev E. v. First integrals of some stochastic operators of statistical physics. (Russian) Diss. Thesis, Moscow State University (M. V. Lomonosov), 1982.
[17] Lanford 0. E., Robinson D. W. Statistical mechanics of quantum spin systems III. Commun. Math. Phys. 327-338 ( 1968) .
[18] Lanford 0. E., Robinson D. w. Approach to equilibrium of free quantum systems. Commun. Math. Phys. 24, 193-210 ( 1972).
[19] Robinson D. w. Statistical mechanics of quantum spin systems. Commun. Math. Phys. 6, 151-160 (1967).
[20] Robinson D. W. Statistical mechanics of quantum spin systems II. Commun. Math. Phys. 7, 337-348 (1968).
[21] Shuhov A. G., Suhov Yu. M. Ergodic properties of groups of Bogoliubov transformations of CAR C*-algebras. (Russian) Izv. Akad. Nauk SSSR, ser. Mat. (to appear) .
[22] Shuhov A. G., Suhov Yu. M. in preparation.
[23] Suhov Yu. M. Convergence to equilibrium state for one dimensional quantum system of hard rods. (Russian) Izv. Akad. Nauk SSSR, ser. Mat. 46, no. 6, 1274-1315 (1982).
[24] Suhov Yu. M. Convergence to equilibrium for free Fermi--gas. (Russian) Teoret. Mat. Fiz. 55, no. 2, 282-290 ( 1983) .
105
SYSTEMS WITH RANDOM COUPLINGS ON DIAMOND LATTICES
P. Collet
I. Introduction
It is by now well established that renormalization
group techniques provide a very powerful tool for the stu
dy of models in statistical mechanics. These ideas were
first introduced to study the problem of phase transitions,
but they have proven to be useful in many other contexts
like the high temperature phase in constructive quantum
field theory, dynamical systems, turbulence, etc ... The
main difficulty in a renormalization group analysis is to
derive the renormalization transformation (sometimes called
the renormalization group for historical reasons). This is
sometimes so difficult that one has to abandon the idea of
u::oing the simple general scheme which needs a lot of infor
mations about this transformation. Migdal and Kadanoff tried
to develop a general technique of approximation, the main
idea being to obtain a tool to study gauge invariant pro
blems. One of the advantage of their approximation is that
the renormalization transformation can be written explicitly.
This is very convenient for numerical and theoretical consi
derations (see for example [4] ) . Unfortunately, it is very
difficult to appreciate the quality of the approximation.
Nevertheless, it was soon realised that one can cons
truct models of statistical mechanics on so-called diamond
106
lattices for which the r.Ugdal Kadanoff renormalization
transformation is exact. This idea has been applied to
obtain rigorous results in various areas of physics where
they are otherwise very hard to get. It is believed that
regular lattices have properties which are not too different
from those of diamond lattices. Without trying to be exhaus
tive, we can list a few applications. Percolation problems
were considered in [3] , while problems associated with the
Schrodinger equation were investigated in [19] , [23] , and
[11]. The interested reader may also refer to the litteratu
re about fractal lattices for very similar technisues. Ano
ther important area is the theory of spin glasses. Models
with complicated (chaotic) phase diagrams were first cons
tructed in [18] (see also [14] and [10] ) . It was then
realised by different groups that one can also treat models
with random couplings. Numerical computations were performed
by Benyoussef and Boccara [2]. Their result suggests a lower
critical dimension equal to 4. Pott's like models were con
sidered in [1] , and Ising like models in [15] , [5] , [7]
and [8] . In the remainder of the present paper we shall
concentrate on the study of these Ising like models.
In chapter II we shall explain the construction of the dia
mond lattices and of the two models we have considered (mo
del A and model B) . Chapter III will be devoted to the
renormalization analysis of model A, and chapter IV to the
study of some spin observables. Chapter V contains some
informations about model B.
107
II. Construction of the models
We shall first construct the diamond lattice (see [16]).
we shall choose once for all an integer n ~2. The construc
tion of the lattice is recursive (seethe figure). To explain
the basic induction step, westart with a lattice composed of
one bond and two sites A and B. We transform this lattice
into alattice with n+2 sites A,B,B1 , .•. Bn' and 2n bonds which
connect the sites (Bi) 1 .;;;i .;;;n to the sites A and B. We now
apply this basic process to each bond previously constructed.
We shall denote by LN the lattice obtained after N successi
ve applications of the basic construction.
If n=2d, dEN, it is possible to consider the above
lattice as a regular ~d lattice where some of the sites have
been identified. This is done as follows. Let ~ be a direc
tion of bond in the ~d lattice. We only keep those bonds
which are parallel to e. We consider a parallelipipedic box
BN of size 2N in the ~ direction and 2N-l inthe other
~d directions. Let i 1 be the ~ coordinate of a lattice
point i inside the box. We shall assume that 0 .;;;i 1.;;; 2N-1.
Let r be the largest integer such that 2r divides i 1 . Then
the set of points in the box BN with ~ coordinate i 1 is
divided into 2(N-r) (d- 1 ) hypercubes of dimension d-1 and
size 2r, and the sites inside each hypercube are identified.
In this way we obtain the lattice LN_ 1 . We now describe two
models with random couplings which we shall call model A and
model B
Model A
To each site i of the lattice LN we associate an
Ising variable Si= ±1, and to each bond <l,jl we associate
a random variable ~ .. which is an independant copy of a ]:,.J.
108
A B ===)~ A B
Recursive construction of the lattice
109
given random variable ~ . We shall consider the hamiltonian
HN( f._, £) given by
nearest neighbours
Model B
This model is very similar to model A, except that we
have imposed some
At each site ~ ,
the coupling~· . ~·1.
relations between the random couplings.
we have again an Ising variable S., but ~ d
is defined as follows assuming n=2 .
We have ~ = (i 1 , ... , i 2) where i 1 is the coordinate of i
in the ~ direction. For each value of i 1 we shall take
the couplings ~. . with j 1 =i 1 +1 identical and equal ~·.1
to an independant copy of the given random variable ~ . It
is easy to extend this construction to the case of a general
n. The hamiltonian is the same as for model A. From now on
we shall assume that the random variable ~ satisfies
E(~)= 0 where E denotes the expectation (see [12] for
the case E(~) t 0~ For both models we first consider sepa
rately each possible configuration of random couplings. We
then construct the Gibbs state with probability z- 1
e-SHN(~,~) for the configuration S of spins. Next we con
sider the thermodynamic limit N -+ + oo, and should ask for
informations about physical observables which are true
almost surely, with respect to the distribution of couplings
(see [13] ) . We shall only deal with average properties
which are simpler to analyse.
The renormalization group program is now easy to imple
ment. We described above the transition from a lattice LN
to a larger lattice LN+ 1 . The renormalization operation
can be thought of as the inverse transition, i.e. replacing
a model on lattice LN+1 by a similar model on lattice LN.
In this operation, the couplings will be modified, and their
transformation is called the renormalization mapping. In
110
the present case, the transition from lattice ~+1 to lat
tice LN is performed by summing over the spins at the
last level of LN+1 ' i.e. the spins which are on the sites
newly created when going from lattice LN to lattice LN+1.
We shall consider for the moment that the inverse tempera
ture S has been included in the coupling. We obtain an
equivalent model on a lattice LN, where the random coup
lings are copies of a new random variable~·. The transfor
mation ~ +~' is the renormalization transformation.
Obtaining the expression of ~· as a function of~ is an
easy exercise in Ising spins algebra and we refer to [5]
for the details. It is,important to notice that the reduc
tion of the renormalization operation (summing over the
spins) to a simple transformation ~ +~' is a special
feature of the particular lattices and models \ve are looking
at. For ~d lattices this operation can be very complicated
(see [17] for a similar problem). We now give the rules for
constructing ~·.
Case of model A
Given ~ , one constructs 2n independent copies of s
~1' ~2•···•~n'~ 1 .~ 2 , ... , Sn· ~· is now given by
~I R(~)
n
2: th- 1 (thsi thsi> j=1
Case of model B
( 1) •
Given s one constructs two independant copies ~l and
~2 . The formula for s' is
s' R(O (2)
These formulas give the ne~1 couplings in term of the old
ones. However, in our case, these couplings are random varia
bles and it is sometimes useful to think of the renormaliza
tion transformation as an action onthe probability law of
the random variable S· The renormalization analysis can now
proceed along the usual steps. We shall be interested in the
111
behaviour of the sequence of random variable (~p) given by
~p+l = R(~p) and ~ 0 = S~ . To do so, we shall determine the
fixed points and local behaviour of the renormalization
transformation. We shall then look at some adapted physical
observables and determine their critical behaviour. Notice
that if E(~0 ) = 0, formulas {1) and (2) imply that E(~p) =0
for every p. We shall first consider model A which is
better understood and then give some indications about model
B.
lt2
III. Re~ormalization of model A, large d expansion.
As mentioned in the previous paragraph, one may inter
pret the quantity log2n as a dimension of the lattice. Loo
king at formula (1) it is tempting to consider the case of
large n (i.e. large d) and to apply the central limit
theorem. As we shall see, this simple idea turns out to be
a very powerful method of investigation. We shall first give
a heuristic argument in the case of large n. Assume first
E, is a1most surely small. Then \ve can replace ( 1) by the
approximate recursion relation
n ~
E, I =.L E,. r. 1 .L "1.
If o = E (E, 2 ) 112 and o 1 = E (E, ' 2 ) 1 / 2 (we have assumed
E (E, ) = E (E, 1 ) = 0) , we obtain
0 1 = n1/2 0 2
Therefore, if
wing behaviour
is the variance of E, we have the follop
i) If 0 < n- 112 (high temperature), o ~ 0 (E, ~ 0 a.s). 0 p p
ii) If 00 > n - 1 / 2 , op ~ +oo and we cannot draw any con-
elusion since our approximation (E,p small) breaks
down.
iii) If - 112 = n- 1/ 2 o 0 n , op v p , i.e. we have a
fixed point. Horeover
do 1 I do -1/2 = 2 • o= n
If we reintroduce the temperature dependence we obtain
113
cr = SE(s 2) 112 , and there is a phase transition at inverse
t~mperature S= n-l/2 E(s 2)-l/2 with critical index toequal 2.
These heuristic results are confirmed (up to small correc
tions) by the rigorous analysis. Our first result concerns
the behaviour at large temperature.
Theorem 1 Given s~etric and of variance 1, for
every S <(SOn)-l/ 2 we have
Proof We observe that
where s'P is an independant copy of sp·
From I th -l (thx thy) 1 2 .;; 25x 2 y 2 v x,y in m, we
deduce
and the result follows easily.
Theorem 1 tells us that if the temperature is large enough,
the renormalization drives the coupling almost surely to
zero, i.e. to the infinite temperature fixed point.
We now investigate the low temperature behaviour.
Theorem 2 : Let s be a symmetric random variable of varian
ce 1. Let Q -=-E(Inf(s 2, s' 2))/8 where s' is an inde_pendent
copy of s (note that Q I 0 since s has a non zero
variance). There is a constant 110 = H0 (Q) > (320) 2 such that
for S large enough, and for n >H0 , the sequence
s = Rp(Ss) has the following properties p
E(s~)l/2 > (nl/2/320)p SQl/2 (i.e. crp _. + oo).
2) If the probability density of sis in L co, the
density f of s satisfies p - p
114
where K is a universal constant.
Proof Let n and n' be two independant and identical
random variables. We define a new random variable n by
and we shall estimate the variance a of n . It will turn
out that we shall also need the skewness a of n which is
given by
~ 1~13 ~3 a = E ( n ) fa •
From the inequality
We now use the fact that n will be identified with ~p· This
is a random variable which is the sum of n independant n
random variables. Let be the random variable [ 1
where n i , 2 .;; i.;; n are independent copies of n i. We shall
denote by F the distribution function of n. We have the
Berry - Essen inequality [22]
IF(xn1/2 &l - tl>(x) I .;; & n-1/2 Vx EJR,
where <!> is the normal distribution, 6 and & are the va
riance and skewness of n1 . Assume now the variance a of n
satisfies a > 9 (notice that a= n 1 / 2 0) I and also
I F(x a) - <l>(x) I .;; 1/40 vx Em.
Then standard probability estimates imply that cr< a/320 and
a< (320) 3. Therefore, if n is so large that 40n- 1 (320) 2<1,and
115
n 112 >320, we can use recursively the above estimates.
This proves 1) provided the initial random variable s 0 = Ss
satisfies the recursive hypothesis. This is insured by ta
king S large enough. Notice that op + +oo does not
imply that the random variables sp tend to infinity almost
surely. This result follows however from an estimate on the
probability laws like 2) which is proven using a more pre
cise version of the Berry-Esseen inequality due to Sahaida
rova (see [22]). We refer the reader to [5] for a complete
proof.
We now come to the more delicate case of the critical
behaviour. We shall only deal with random variables s which -1/2 2 have a density of the form (2ns) exp(-x /2s) +q>(x) where
s belongs to IR+ and 4> belongs to a function space H'
defined by
H' {q> E Lool f x 8 I4>Cx)! 2 dx< + 00 I and fq>(x) dx
fx 2 q>(x) dx = 0} .
We shall denote by H the set of couples (s,q>) as above.
define II Xk4>11 p to be the L p norm of the function
k x -+x q>(x), 114>11H I and ll(s ,4>)11 H are defined by
4 + II X 4>11 2 I
We
We shall think of the renormalization transformation as a
map from H into itself, and we shall look for a fixed point -1/2
near (n , 0). We now need the expression of the renorma-
lization tranformation in the space H. We define first a map
s1 from H to the set of probability densities by
Notice that the above expression defines a function in L1 .
However this function is a probability density only if 4>
satisfies some special conditions. Let f = s 1 «s,q>)), we
116
shall denote by n the random variable with probability density f.
s (f) will denote the probability density of the randan variable
th -l ( thn thn 1 ) where n 1 is an independent copy of n . The fonnula
for f = S(f) is easily seen to be
S (f) (x) (l-th2x) J dy [y[-1 (1-l)-1 (1-y-2 (thx) 2)-1 x
[thx["'IYI q
X f(th-1y) f(th-1 (thx/y) ). (3)
We now define a map T among probability densities by
f + f Tf given by
Tf f * f* ... * f (n fold convolution) .
Finally there is a map T1 from probability densities back
to H : (s,~) = T1 f which is given by
s = n J x2 f(x) dx, and ~(x) = (s/n) 112 [f (xn- 1/ 2)
- (2JT) - 1/ 2 exp (- x 2 I 2ns) ] .
Notice that T1 is roughly the inverse of s1 . It is easy
to verify that as a mapping from H to H the renormalization
transformation R is given by the product M =T1 oToSoS 1 . We
are now able to state the result for the critical behaviour.
Theorem 3 Let B = { (s,~) EH 10 II (s-1,~)11H.;; (log n) /n}.
The map M from B to H has a unique fixed point (s0 ,~0 )
in B for n large enough. Horeover
1 +O((Log n) 10;n1/ 4)
0 ((Log n) 10 /n1/ 4)
and the function s1 (s0 ,~0 ) is a positive function.
The linear operator has in H a simple
117
eigenvalue equal to 2 + 0 (n- 1/ 2). The rest of the spectrum
is contained in the open unit disk.
Using this result, one can repeat the usual renormalization
group analysis in the space H i.e. one first constructs
the local stable and unstable manifolds at the fixed point
and proves the universal scaling behaviour (see [6) or [9]).
The 1/n expansion w.as investigated in [ 15).
We now give the main ideas for the proof of theorem 3
(see [5) for more details). As mentioned before, if n
is large, we expect the fixed point to be nearly gaussian,
i.e. near to (1,0). This suggests to use the inverse func
tion theorem. We implement this ideas as follows. We have
DM -]__ (s,(jJ) Ii
M* + M' ( S 1 (jJ) where M* = {: -~)
and M' (s ,(jJ) for (s,(jJ) in
is a linear operator of norm less than (log n) 2
B. We define a map ¢ from B to H by
It turns out that ¢ is a contraction from B to B (D¢ has
norm smaller than 1/2 in B) , and the fixed point of ¢ in B
is a fixed point of M.
There are two technical steps in the above argument. The
first one is to prove that ¢((1,0)) is near to ( 1 1 0) for
large n. The second one is to show that D¢ has norm less
than 1/2 in B. The proofs of these two assertions are
very similar and we shall only give some indications about
the first one. We first observe that only T and S give
non-trivial contributions to the operator M(T1 and s1 )
are trivial linear operators). The good part is T, although
this is a very nonlinear operator (n convolution) . T is al
most a projection onto gaussian functions, the correction
being of order n- 1/ 2 . We have the following lemma
Lemma 4 Let f = n-1/2 f*n (n- 112 .) , where f satisfies
118
llfll 1 + II fll 2 4~
+ II X fill <oo and +co
J x f(x) dx 0 •
Then, for n large enough
f(x) = (2rr s-l)-l/2 exp(- x 2 s/2) + ~(x)
and II~ II H , .;;; 0 ( n -l I 2) .
This lemma is very much in the flavor of the results about
corrections to the central limit theorem. Unfortunately it
does not seem to have already appeared in the litterature
under this functional form. The proof consists in estimating
the Fourier transform of f (and of its first few deriva-1/2 tives). For ltl .;;; O(n ) one performs the usual expansion
and cancella.tions of the usual proof of the central limit
theorem. For ltl ~ 0(n1/ 2 ) one uses a general result of
Statulyavichus (see [5] for details). In order to be able
to apply this lemma, we have to show that if (s,~) is in H,
ss1 (s,~) satisfies the hypothesis. This is the main reason
for the choice of the space H. The proof relies on various
L estimates about formulas (3) (see [5] ) . It is worth p
mentioning that the function ss1 (1,0) is not at all like a
gaussian. It is infinite at x = 0. However, the operator T
is so strongly contracting toward gaussians that for n
large enough this bad behaviour desappears.
119
IV 5oin observables
We now come to the more concrete part of the renormali
zation analysis, i.e. the analysis of those physical obser
vables which behave in a particular way under the renorma
lization. Notice that various boundary conditions can be
obtained by fixing the two extreme spins. We shall give here
some details about the averaged spin, the same ideas can be
used to analyse some correlations. The results are summari
zed in the following theorem.
Theorem 4 Under the conditions of Theorem 2, we have
in the thermodynamic limit and for any boundary condition
i) E( < s > ) = 0 at every temperature
2 ~: if 6 is small enough
ii) E( <S> ) = if 6 is large enough.
This theorem proves that the Edwards-Anderson order parame
ter has a phase transition. Using the results of chapter III,
it is also possible to compute the critical exponent. Recent
works based on the replica symmetry breaking ( [21] , [20])
have shown that in the 5herrington-Kirkpatrick model, the random variable v- 1 l: 5~ 1 ) 5~ 2 ) (where 5~l) and 5~ 2 )
l l l l i
are the spins of two independant replicas and V the num
ber of spins) has a non trivial distribution in terms of
the couplings. In the present case, it was shown in [15]
that this is not the case. It was also shown in [15] that
thereplica technique cannot be applied to this model (at
least in it's initial form). We now give some ideas about
the proof of theorem 4. We shall obtain for tije thermal
average of a spin a recursion relation associated to the
renormalization of the model. Let s 0 be a spin which is on
120
a lattice site of LN+l which does not belong to LN. S0 has
two neighbouring spins which belong to LN. We shall denote
by s1 that neighbour which is not on a site of LN_ 1 and by
Si the other one. LetS~ be equal to s1 or to Si· If A and
B are some constants, it is easy to verify that the renor
malization acts as follows
< AS0 +B S~ > ~+l
where
A'= B+AX(l-X' 2) (1-X 2x• 2 ) -l, B' = AX'(1-X2) (1-X2x• 2 ) - 1 if
s~ =:::: 1 .
A'=AX(1-X' 2)(1-X2x·~- 1 , B'= B+AX'(l-X2)(1-X2x• 2)- 1 if
s~ = si,
X =th~; , X'=th~;', where 1; and ~;:' are the couplings between
S0 and s 1 and S0 and Si respectively.
It should be obvious that iterating infinitely many ti
mes the above relations, one should be able in principle to
compute the thermal average of a spin in the thermodynamic
limit. Starting with a lattice LN, the recursion ends with
a lattice L0 of two sites, where the thermal average can
be computed explicitly. We summarize the argument by the
following formula
AN and BN are random variables which have (complicated) ex
pressions in terms of the couplings. The behaviour of these
random variables can be controlled as follows. Given a star
ting site s0 on lattice LN, we shall denote by (Ap)O <p< N
(BP) 0 <p< N the sequences of random variables produced by
the above recursion and with initial conditions A0 = 1,B0 =0.
The follm-ling lemma summarizes some useful properties of
and B . p
Lemma 5 : Under the hypothesis of Theorem 2, and for 0 < p < N
121
we have
0
is large enough, we have
i) and ii) are easy consequences of the gauge symmetry
and of the recursion formulas. iii) is more technical and
\·7e refer to [ 5] for the proof. Theorem 4 is now an easy
consequence of Lemma 5. In the case of high temperature, we
have X ~o exponentially fast by Theorem 1. We get smaller p
and smaller coefficient in the recursion relations and it is
easy to show that AN and BN tend to zero if N tends to in
finity. For the low temperature, we have to use Theorem 2
instead which implies that o ~+co exponentially fast. The-p
refore, if o 0 is large enough, using iii) of Lemma 5, we
get a uniform lower bound y >0 for E(A2 + B2). This implies p p
which is ii) of Theorem 4. At this writing no information
has been derived in the presence of a magnetic field. This
is due to the fact that no simple recursion relation with
simple estimates (like in Lemma 5) has been derived up to
now.
122
v. Some results about model B
As we shall see, this model is less understood than mo
del A. We shall use the hyperbolic tangents of couplings to
describe the system. The recursion relation takes the follo
wing form. Let X0 = th(B~), and define recursively a sequen
ce of random variables (with values in [-1,1])Xp = RP(X0 ),
p Em, by
X = q(X X' ) p+1 p p
-1 where x' is an independent copy of X and q (s) = th (nth (s)) • p p We shall assume that n ~2, and show that for certain dis-
tributions of ~the model has a phase transition.
Theorem 5 : Assume that for some number c >0, we have
c <I~ I -1 < c almost surely.
i)
ii)
for B small enough
for B large enough
X -+ 0 a.s. p
J XP/ -+ 1 a • s .
Proof It is easy to verify that if /X J c [a,b] a.s., then 2 2 p 2 Xp+1 c [q(a), q(b )] almost surely. The map t -+q(t) has
two attractive fixed points on [ 0,1] : t = 0 and t = 1, and
there is also a repulsive fixed point t 0 • Moreover the map
is monotone. Therefore, if Jx0 Jc[O,t0 [ a.s., then Xp-+0 a.s.
and if JX0 J c ]t ,1] a.s., we have JX I -+1 a.s. 0 p
we also observe that these convergences to 0 or 1 have
an exponential rate. Therefore one can repeat the argument
of chapter IV to analyze the behaviour of the spin observables (see [7]). The following result implies that some ini
tial random couplings do not give rise to models with phase
123
transition.
Theorem 6 : If for some o E [ 0, 1/128 [
I X I < o has probability larger than 4 o , 0
This result also indicates that numerical
, the event
then X -> 0 p
computations
a.s.
must
be done with extreme care since a small error near 0 can
destroy a phase transition.
The critical behaviour seems to be much harder to ana
lyze, and we have not yet obtained rigorous results. Instead
we have studied an approximate version of the renormaliza
tion transformation which is built as follows. We replace
the probability density by a ne\'l one which is constant :on
the sets [-2-q, -2-q- 1 [U] 2-q- 1 , 2-q]. We shall denote by
f(z) the generating function of the sequence of numbers ob
tained this way. f is a function which is holomorphic in
the unit disk, has positive (or zero) Taylor coefficients in
z =0, and satisfies f(1) =1. The approximate renormaliza
tion transformation R is given by
(Rf) (z)
It is easy to verify that Rf has the same properties as f.
R has only three fixed points : f(z) =1 (low temperature),
f(z) =z and f(z) =zoo (=0 if I zl < 1, oo if I zl > 1) which is
the high temperature fixed point. The global behaviour of R
in the set of admissible f's is completely given by the fol
lowing theorem[8].
Theorem 7 : Assume f is holomorphic in the unit disk, with
positive (or zero) coefficients at z=O and satisfies f(1)=1.
Then
i) If f does not extend to a function holomorphic up
to z=2, or if f(2) -2f' (2) <0, then Rpf(z) ->Z00 if p ->+ oo
ii) If f(2) -2f'(2) >0, or f(2) -2f'(2) =0, but f is
not the function z -> z, then Rpf -> 1 if p -> + oo .
It is easy to verify that f(2)-2f' (2) =0 is an invariant sur
face which plays the role of a critical surface in the usual
renormalization analysis. However, except the fixed point
124
f(z) =z, all the models on this surface are in the law tem
perature phase. In particular it does not seem possible to
define critical indices. We conjecture that model B has a
similar behaviour.
125
References
[1] D. Andelman, A. N. Berker, Scale-invariant quenched disorder and its stability criterion at random critical points. Preprint MIT (1983).
[2] A. Benyoussef, N. Boccara, Real space renormalization group investigation of three-dimensional Ising spin glasses. Phys. Lett. 93A, 351-353 (1983). Existence of spin-glass phases for three and four dimensional Ising and Heisenberg model. Preprint CEA-Saclay (1983).
[3] V.P. Bovin, V.V. Vas'kin, I. Ya. Shneiberg. Recursive models in percolation theory. Journ. Theor. and Math. Phys. 2i• 175-181 (1983).
[4] P.M. Bleher, E. Zylas, Existence of long range order in Migdal's recursion relation. Commun. Math. Phys. 67, 17-42 (1979). -
[5] P. Collet, J-P. Eckmann, A spin glass model with random couplings. Commun. Math. Phys. 22, 379-407 (1984).
[6] P. Collet, J-P. Eckmann, A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics. Lecture Notes in Physics 74, Springer, Berlin, Heidelberg, New York (1978).
[7] P. Collet, J-P. Eckmann, Y. Glaser, A. Martin, A spin glass with random couplings. Preprint IHES (1983).
[8] P. Collet, J-P. Eckmann, Y. Glaser, A. Martin, Study of the iterations of a mapping associated to a spin glass model. Commun. Math. Phys. 2i• 353-370(1984).
[9] P. Collet, J-P. Eckmann, O.E. Lanford III, Universal properties of maps on an interval. Commun. Math. Phys. ]_£, 211-254 (1980).
[10] B. Derrida, L. De Seze, C. Itzykson, Fractal structure of zeroes in hierarchical models. Journ. Stat. Phys. 33 559 (1983).
[11] E. Domany, S. Alexander, D. Bensimon, L.P. Kadanoff, Solutions to the Schrodinger equation on some fractal lattices. Phys. Rev. B28, 3110-3123 (1983).
[12] B. Derrida, E. Gardner. To appear.
[13] S.F. Edwards, P.W. Anderson, Theory of spin glasses. J. Phys. F5, 965 (1975).
[14] B. Derrida, J-P. Eckmann, E. Erzan, Renormalization group with periodic and aperiodic orbits. Journ. Phys. A16, 893 (1983).
126
[15] E. Gardner, A spin glass model on a hierarchical lattice. Preprint spht-84-44 CEA-Saclay (1984).
[16] lL Kaufman, R.B. Griffiths, Spin systems on hierarchical lattices. Introduction and thermodynamic limit. Phys. Rev. B26, 5022-5032 (1982).
[17] A. Kupialnen, These proceedings.
[18] S. Me Kay, A.M. Berker, S. Kirkpatrick, Amorphously packed, frustrated hierarchical models, chaotic rescaling and spin glass behaviour. J. Appl. Phys. 53, 7974-7976 (1982).
[19] J-M. Langlois, A-M. S. Tremblay, B.W. Southern, Chaotic scaling trajectories and hierarchical lattice models of disordered binary harmonic chains. Phys. Rev. B28, 218-231 (1983). -
[20] M. Mezard, G. Parisi, Self averaging correlation functions in the mean field theory of spin glasses. Preprint LPTENS 87-4, Paris (1984).
[21] M. Mezard, G. Parisi, M. Sourlas, G. Toulouse, M. Virasoro, Nature of the spin glass phase. Phys. Rev. Lett. 52, 1156-1159 (1984).
[22] V.V. Petrov, Sums of independent random variables Springer-Verlag Berlin, Heidelberg-New York (1975).
[23] A-.1>1. S. Tremblay, B.W. Southern, Generating function and scaling for the density of states of fractal lattices. Preprint (1983).
P. Collet
Centre de Physique Theorique Ecole Polytechnique
91128 Palaiseau France
127
ON THE DIFFUSION IN DYNAMICAL SYSTEMS
L. A. Bunimovich
In the recent years there has been a considerable
progress in understanding the equilibrium mechanisms of
statistical mechanics. This progress has been achieved with
help of methods of the ergodic theory. For some realistic
systems there have been proved such properties as ergodicity,
mixing, K and Bernoulli property. At the same time the
same progress has not been obtained in the field of non
equilibrium statistical mechanics. It is connected with the
following well-known fact: stochasticity may be strong or
it may be weaker. In other words, the onset of stochasticity
is not sufficient in itself to explain the hydrodynamical
laws from those of statistical mechanics. So dealing with
the problems of transport theory one must consider the most
delicate property in the hierarchy of statistical proper
ties of dynamical systems - the speed of mixing (or the
rate of correlation decay) . Knowing this characteristics
one can calculate the values of transport coefficients for
the considered dynamical system by Green-Kubo formulae [14],
can obtain some information about its energy spectrum etc.
Quite recently this property was formulated as "expo
nential decay of correlations". Really there was a general
belief that, in stochastic systems, correlations bf(n) =
= f M f (Tn x) f (x) d)l must decay exponentially fast. Here M
is a phase space of a dynamical system under consideration,
x - its point, T - a transformation which defines this
system, )1 - an invariant measure on M and n a (discrete)
time. Of course, it was clear that this property in such a
general formulation fails even for the "most stochastic"
128
classical dynamical systems as the hyperbolic automorphisms
of tori [7]. One must consider not all functions f(x) on
the phase space of the considered dynamical system but only
sufficiently "nice" (for instance, continuously differenti
able or Holder) functions f(x) . The physically natural
functions arising in realistic models belong, as a rule, to
these classes.
But the numerical computations carried out in the late
sixties have shown that the integrands in Green-Kubo for
mulae can exhibit long-time algebraic decay [14]. These re
sults have forced to reconsider previous intuitive ideas
that the relaxation processes in dynamical systems of sta
tistical mechanics are controlled by processes localized in
space (on scales of mean free path) and in time (on scales
of relaxation time) .
From the general point of view, the description of
dynamical processes in many-body systems by kinetic equa
tions can be considered as approximating a non-Markovian
stochastic process by a Markovian one. The fast decrease of
correlations indicates that the system is close in some
sense to a Markov chain. The main tool for the rigorous in
vestigation of this approximation is the method of Markov
partitions [16].
In [5] such a partition was constructed for the two
dimensional Lorentz gas with a periodic configuration of
scatterers and a bounded free path. With the help of this
partition it was shown in [6] that, in this system, the
correlations of smooth functions decay quasi-exponentially.
We shall consider below the problem of approximation of
such dynamical system by the process of random motion.
Let us consider the two-dimensional Lorentz gas con
sisting of a single point particle moving in a triangular
array of immobile disk scatterers (Fig. 1). Outside all
scatterers the point particle moves with a constant veloc
ity and at the moments of reflections it changes its veloc
ity according to the usual law of elastic collision. With
out any loss of generality one can assume that the radius
of the scatterers equals while the lattice spacing is
129
Figure 1.
130
2+W . If W=O the moving particle is trapped in a bounded
triangular region formed by three touching disks. If 0 < 4 < W < -- 2 , the free path of the moving particle is /3
bounded by the value 2/3 . In this case, the correspond-
ing dynamical system is called a Lorentz gas with a finite
horizon. For W > _!_- 2 , the particle may move arbitrary /3
far between collisions. J. Machta and R. Zwanzig [11] bas-
ing on results obtained in [6] have considered at physical
level the behaviour of the diffusion coefficient D for
this system [9] in the limit W+O . Their approach was
based on the natural idea that, at high densities, the
exact motion of the particle can be replaced by a random
walk between triangular trapping regions. (A single such
trapping region is shown by cross-hatching in Fig. 1.) This
assumption means that the sequence of traps visited by the
moving particle is a Markov process.
The phase space M of this dynamical system consists 1 2 of points x = (q,v) , where q = (q ,q ) is the position
and v = (v1 ,v2 ) is the velocity of the particle. The flow
corresponding to this system will be denoted by {St} . We
shall consider a natural special representation [10] of the
flow {St} . Namely let M0 be the space of points
x = (q,v) such that q belongs to the boundary of one of
the scatterers and v is directed inside the scatterers.
We denote by T0 the transformation of M0 into itself
which arises when the point xEM0 moves along its trajec
tory till the next reflection from a scatterer and T0x = 2 = (q1 ,v1l, where q 1 E::JR
flection takes place and
is the point where the next re
v1 is the velocity after the
reflection. If q is a point of the boundary of a scat
terer then n(q) is the unit normal vector directed out
wards the scatterer and cos¢= (n(q),v) , ; ~ ¢ ~ 3; ,
where (•,•) denotes scalar product. Thus, for every scat
terer Ki , i=1,2, ... , we can introduce natural coordi
nates r , ¢ on the set Kj_ c M0 of points x = (q,v)
qEKi , where r is a cyclic coordinate along the boundary
Ki and ¢ measures the angle between n(q) and v . Let
131
u0 be the trapping region with center in the point (0,0)
Let us consider the differential d~ 0 of the measure on
the set u0 n (~Ki) such that its restriction to Ki is
proportional to [cos ¢ldrd¢ . i Let fi(x) = q , x = (q,v) , i=1,2 . We denote by
gi(x) , i=1,2 , the function which equals to the i -th
coordinate of center of a trapping region containing the
point x . If x belongs to the boundary of two neighbour
ing trapping regions, then, by definition, the value of
gi(x) , i=1,2 , equals to the i -th coordinate of center
of a trapping region with the less second coordinate.
Let us consider the following expression
(i) n 2 V = JM (f. (T 0 x)-f. (x)) d~ 0 n 0 1 1
( 1 )
Theorem. For any W , 0 v!il
4 < W < -- 2 , there exists the
limit limn+oo _n_ = V (i) (W) n
/3 and as W-+0 the value
V (i) (W)
w tends to a limit v (i) > 0 •
It is easy to see that
large n by
v(i) can be replaced (for n
V-(i) n 2 n = 1M0 (gi(To x)-gi(x)) d~o
m We shall consider the function hi(T 0 x) m+1 m gi(T 0 x)-gi(T0 x) . So we can write
v (i) n --n
(2)
(3)
In what follows we will denote all constants which do
not depend on W simply by const.
The new point arising in this problem comparing with
the common situation in the periodic Lorentz gas with a
finite horizon is the following. The dynamical system con
sidered here is not uniformly hyperbolic [13] in the limit
W-+0 . Indeed, in this limit the free path of the moving
particle is not bounded away from zero and the components
of the boundary of the trapping region touch each other.
132
(The last property makes this system essentially different
from billiard systems considered in [4] .) Therefore in the
phase space of our system there is a "bad" subset sitting
in the neighbourhood of straight segments belonging to the
boundary of a trapping region (Fig. 1). Such segments will
be called boundary segments.
It is easy to see that the first term in the right
hand side of ( 3) equals to const • W . So the main point is
to estimate the second term.
The set of all trajectories which intersect on the
given step one of the boundary segments can be decomposed
into two subsets. The first subset consists of all such
trajectories which intersect the same scatterers as the cor
responding boundary segment. The second subset consists of
all other trajectories. From elementary geometrical consi
derations one can see that the measure of the second sub
set equals to const·w2 . So only the first subset must be
considered.
All trajectories can be decomposed into series of re
flections from the boundaries of scatterers taking place in
one and the same trapping region. According to this decom
position, for any point x E M0 , the sequence h (x),
h(T0x), ..• , h(T;x), .•• is decomposed into such segments
that all elements in any of them besides the last one are
equal to 0 .
Now one must take into account that in each such
series there is at least one reflection with a free path
not shorter than a>O , where a does not depend on W •
Therefore from the general theory of dispersed billiards
[15] a corresponding coefficient of expansion of neighbour
ing trajectories in phase space is bounded from below by
1+y , where y does not depend on W , too. So an applica
tion of methods derived in [6] allows us to obtain for the
second term an estimation from above by const. W . This al
ready gives the assertion of the theorem formulated above.
The last estimation can be improved if we use the fact
that a typical trajectory is sitting in a trapping region
during a sufficiently long time. Really it follows from the
133
ergodicity of the dynamical system under consideration
that the average rate for leaving a trap is const. w- 1
It can be shown that the second term in (3) has an order
o(W) as W+O In order to obtain this estimation one
must use some sufficiently subtle properties of the Markov
partition of this dynamical system. The approach based on
replacing the system under consideration by a random mo
tion between trapping regions [11] allows to obtain estima
tion of the first term in (3), only. Computer simulations
[111 have shown that it is relevant only for very small
values of W • The rigorous approach based on Markov parti
tions allows us to obtain correction to this estimation, in 4
principle, in the whole segment O<w<13-2 •
The new method for the construction of a Markov parti
tion for billiard systems proposed in [3] can be applied
to the Lorentz gas in arbitrary (but finite) dimension.
Besides this Markov partition has the same properties as
the one constructed in [5] for the two-dimensional case.
It allows us to prove the same theorem for the periodic
Lorentz gas with triangular lattice of scatterers in hi~
dimensions, too.
It will be interesting to compare the limiting be
haviour, as W+O , of the Lorentz gas when the sets of
centers of the scatterers form triangular and square lat
tices on the plane E 2 • These two dynamical systems essen
tially differ due to the unboundedness of a free path length
in the second one. The rate of correlation decay for the
Lorentz gas with an infinite horizon and a periodic con
figuration of scatterers was treated in [2]. In particular,
it was proved that, in this system, the decrease in the cor
relations is polynomial but, nevertheless, the diffusion co
efficient exists and is positive.* Besides in [2] it was
shown that the diffusion coefficient D in the low den
sity limit depends on the density of scatterers nonanali
tically. Earlier this result was obtained at a physical
*Quite recently the numerical simulation of this system
which confirms an algebraic decay has been performed [8] .
134
level for the hard spheres gas and the Lorentz gas [14].
It seems that, in the high density limit, the diffusion co
efficient D depends on the density analytically. There
fore it may be the case that, in spite of the general opin
ion, the expansion of the diffusion coefficient into a
power series with respect to the density does not exist
for small densities but does exist for large densities of
particles.
In conclusion, we shall discuss the question of the
integrability of the velocity autocorrelation function
bv(n) . Recently some authors have obtained for some models
by considerations at a physical level or numerical computa
tions that lb (n) I ~ l [12, 17]. Then it has been con-v n eluded that the diffusion coefficient does not exist. But
the symmetry properties of these systems which provide the
existence of D were not taken into account. Let us con
sider these systems.
The first system is the so called "stadium", i.e. the
billiard in the region in m2 bounded by two semi-circles
of one and the same radius and two straight segments tan
gent to them parallel [1]. It was shown [17] analytically
(at a physical level) and numerically that, in this system,
lbf(n) I ~ * . It can be proved rigorously that lbf(n) I < -1+E < const. n , where E>O • Thus, for a general function
in the phase space of this system, bf(n) is nonintegrable.
But for the velocity it can be proved that lb (n) I < -2+E V
< const. n . Really the main contribution to bf(n) is
made by trajectories which spend a long time in neighbour
hoods of the family of periodic orbits which are perpendi
cular to the boundary line segments. But it is easy to see
that, in the case f (x) = v , after two consequtive reflec
tions from the boundary the moving particle has almost op
posite velocities. It gives the desired estimation. The
same arguments can be applied to the billiard system con
sidered in [ 1 2] .
135
References
[ 1] Bunimovich L. A. On the ergodic properties of nowhere dispersing billiards. Comm. Math. Phys. 65 (1979), 295-312.
[ 2] Bunimovich L. A. Statistical properties of the Lorentz gas with infinite horizon. Proc. of 3rd Int. Vilnius Conf. on Probability Theory, 1981, 1, 85-86.
[3] Bunimovich L.A. Some new advancements in the physical applications of ergodic theory. In "Ergodic Theory and Related Topics" (ed. by H. Michel), Akademie-Verl., Berlin, 1982, 27-33.
[4] Bunimovich L.A., Sinai Ya. G. On a fundamental theorem in the theory of dispersed billiards. Math. USSR Sbornik 1 ~ (1973), 407-423.
[ 5] Bunimovich L.A., Sinai Ya. G. Markov partitions for dispersed billiards. Comm. Math. Phys. 78 (1980), 247-280.
[6] Bunimovich L.A., Sinai Ya. G. Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78 (1981), 479-497.
[7] Crawford J., Cary J. Decay of correlations in a chaotic measure-preserving transformation. Physica 6D ( 1 983) , 223-232.
[8] Friedman B., Martin R. F., Jr. Decay velocity autocorrelation function for the periodic Lorentz gas, Preprint, 1984, 13 p.
[ 9] Hauge E. H. What can one learn from Lorentz models? In "Transport Phenomena" (ed. by L. Garrido), Springer, Berlin, 1974, 337-367.
[ 10] Kornfeld I. P., Sinai Ya. G., Fomin s. V. The ergodic theory. Moscow, Fizmatgiz press, 1980.
[ 11] Machta J., Zwanzig R. Diffusion in a periodic Lorentz gas. Phys. Rev. Letters 50 (1983), 1959-1962.
[ 12] Machta J. Power law decay of correlations in a billiard problem. J. Statistical Phys. 32 (1983), 555-564.
[ 13] Pesin Ya. B., Sinai Ya. G. Hyperbolicity and stochasticity of dynamical systems. Soviet Math. Surveys 2 (1981), 53-116, Gordon and Bridge.
[ 14] Resibois P., DeLeener M. Classical kinetic theory of fluids. John Wiley and Sons, 1977.
[ 15] Sinai Ya. G. Dynamical systems with elastic reflections. Russian Math. Surveys 25 (1970), 137-190.
[ 16] Sinai Ya. G. Markov partitions and C-diffeomorphism. Functional Anal. Appl. 2 (1968), 61-82.
[ 17] Vivaldi F., Casati F., Guarneri I. Origin of long-time tails in strongly chaotic systems. Phys. Rev. Letters 51 (1983), 727-730.
137
¢! with negative coupling
A. Kupiainen
1. Introduction
Construction of a Quantum Field Theory model in four space
time dimensions, which has a non-trivial S-matrix, has
for a long time been one of the foremost problems of
mathematical physics. In Constructive Field Theory this
program has been carried out for space-time dimensions two
and three for the superrenormalizable scalar ¢4 model [1],
whereas the four dimensional renormalizable case has so
far defied all attempts (see however [2] and [31 for the
construction of a planar theory). The ¢4 interaction in
four dimensions is perturbatively renormalizable, i.e. the
S-matrix has an (divergent) expansion with finite coeffici
ents in powers of the physical coupling constant: a non
trivial result of perturbative renormalization theory. In
particular this expansion, if asymptotic to some ''true"
theory, predicts non-trivial scattering. However, already
three decades ago, it was speculated by Landau and others
( [4], [5], for a review, see [6]) that the theory might not
be consistent in the ultraviolet (UV), and after the advent
of the renormalization group (RG), it was suggested (7]
that the theory be free for non-perturbative reasons.
The conjecture, that the continuum limit of the (Euclidean)
Schwinger functions, determined by the action
I 2 I 2 2 SA(¢) = -Z(A)f(V¢) + -Z(fl.)m (A)f¢ +
2 2 0
(l)
where the field ¢ has some UV cutoff A, are those of the
(generalized) free field, no matter how the bare para-2 meters Z(l\.)>0, A.(A) >0, m0 (fl.) E:Dl are chosen, is "almost"
proven in the context of lattice cutoff with V the nearest
138
neighbour lattice gradient, [8], [9] These negative re-
sults still leave open the question of the meaning of the
formal perturbative construction. In fact, the same per
turbative RG which suggests the triviality of (1) for
A(A)>O indicates, that a non-trivial continuum limit might
exist, if A(A) is taken small negative: A(A)+-0 as A+oo 1
the theory then being asymptotically free in the UV (see
[1~ ). However, taking A(A)<O we run into the problem of
how to stabilize the action (1): the functional integral
doesn't make sense even for the cutoff theory.
In this talk I would like to discuss some investigations on
the negative coupling ¢ 4 theory done in collaboration with
K. Gawedzki. We difine the cutoff negative coupling ¢4 mo
del by analytic continuation in terms of a stable (conver
gent) functional integral and show, that the continuum li
mit of the Schwinger functions exist and are non-gaussian.
The resulting theory is UVAF. Moreover, we expect the renor
malized perturbation theory to be asymptotic and Borel
summable to our construction. This result thus answers to
the question concerning the meaning of the formal construc
tion. Our non-perturbative construction provides euclidean
Schwinger functions. However, they probably have no analy
tic continuation to Minkowski space since the crucial pro
perty of Reflection Positivity is most likely lacking, due
to the analytic continuation involved in the construction.
Nevertheless, we consider our result showing, how the
continuum limit of an asymptotically free theory may be
rigorously established. The Reflection Positivity problem
is peculiar to ¢: and is essentially an ~infrared'' problem,
already present in the unit lattice negative A theory.
The method used in the construction is very similar to the
one used by us in the proof of infrared AF property of the
positive coupling critical lattice ¢ 4 theory, Dq. It
consists of a rigorous application of the Kadanoff-Wilson
block spin transformation to perform the functional inte
gral, originally over field fluctuations in all distance
scales, each dealing with fluctuations in a fixed scale.
These are performed iteratively, each step of lowering the
UV-cutoff producing an effective theory of approximately
139
the ~ 4 -type, but the coupling replaced by a running
coupling constant, in accordance with the perturbative RG.
Each of these fixed scale integrations has, in proper
dimensionless variables, both an infrared and ultraviolet
cutoff of order unity and we evaluate them by performing
perturbation theory in the running coupling constant to
third order and bounding rigorously the remainder.
We sould like to stress then, that the question of renorma
lizability and existence of AF theories may be reduced to
an IR and UV finite perturbation theory, up to only a low
order, and bounding rigorously the remainder. Thus we
believe, that AF is a sufficient condition for the con
struction of continuum limits, contrary to some recent
speculations (12].
140
2. Negative A theory
We consider the action (1) where ~ is taken to be on a
hypercubic lattice of spacing A-l (f is replaced by a
Riemann sum) and restricted to a finite box V, to be
removed in the end. The correlations are given by
(2)
where D~= II d~ ( x). Both N and D in ( 2) are analytic in 2 X£V . 2 Z, m0 , A, in the reg1on Z, m0 £( , Re Z A>O. Moreover,
ia . 4ia 2 we may rotate the contour ~~e ~. prov1ded Re(e Z A)>O,
O<a<a (take a>O), obtaining
(3)
which furnishes an analytic continuation of the left hand
side to the whole region e given by Re(e 4 iaz 2A)>O. . a Obviously we may continue rotating, (3) giving an analytic
continuation to an arbitary 0 . In particular, for Z, a
m~£IR and A negative N is given in terms of a functional
integral (a= TI/4) with imaginary mass and (V~) 2 -term and
a positive ~ 4 coupling. We take this stable integral as
our definition of the negative A theory: a different one TI
(complex conjugate of the above) is obtained by a= - 4• If the partition function Din 2 doesn't have zeros, (as
we will show) the negative A correlations are given by
Nia 2ia 2 e GN(x. ,e Z,m ,A)
1 0 (4)
141
for Re(Z 2Ae 4 ia)> 0.
Remarks 1. Due to complex action in (4), reflection positi
vity is lost. This occurs already on unit lattice and is
not related to the UV-limit.
2. Formal perturbation theory obviously satisfies (4): it
corresponds to multiplying each propagator by e-Zia, each 4ia ia vertex by e and external legs by e . Thus our theory
has the same formal perturbation theory as the negative
coupling theory. In particular the complex nature of the
measure is a non-perturbative effect.
142
3. The result
We take as our bare parameters in (1)
(6)
where B2 and s3 are computable numbers (the first two
coefficients of the B-function),E is .1, say. We orove the
Theorem For the renormalized coupling g small,negative and
mz= 0(1), positive, there exists a bare mass m6(A)=
m~(m 2 , A(A)) so that the continuum (and V+DR 4 ) limit of
GN exist (eg. in 5'(R4 )N) and are non-gaussian with m the
physical mass.
Remarks !.The wave function renormalization Z in (5)
is taken finite. This is indeed what perturbative RG pre
dicts too, see below.
2. The construction in fact works for g complex in the
region depicted in picture 1 where near the positive real
axis (Reg) 2<g 0 !Imgl, g0 >0 small.
3. We get detailed information on correlations. E.g. the
usual physical coupling is
(7)
and the theory is UVAF:
r 4 (tpi) ~<logt)-l t+oo
(8)
143
Img
Reg
Picture 1. The analyticity region in g
4. We expect to prove analyticity in g in the region of
Picture 1 and combined with (7) to prove asymptoticity and
Borel-summability of the standard renormalized perturbati
on series to our solution.
5. E in (5) is for technical reasons, in order to have a
covariance with positive real part.
144
4. The Block spin RG
It is convenient to reformulate the continuum limit in
terms
nics.
of a scaling limit of a system of Statistical Mecha
For ~ a !-lattice field, introduce the dimensionless fl.
field ~ on unit lattice
~ ( X ) = fl.¢ ( fl. X ) ( 9)
and the (bare) "Hamiltonian"
(lO)
whence the continuum limit is given as a scaling limit
( 11)
2 -2 2 HI\. has the same form as (1), except m 0 (11.)~11. m0 (11.).
We study the continuum limit of (1) by "integrating out"
fluctuations on scales l~A~fl.. In terms of (11) this
corresponds to the block spin transformation. Explicitly,
consider the "effective theory" for "low momenta" resulting h . . h . h . SEFF d from sue an 1ntegrat1on, t at 1s, t e act1on fl. eter-
mining the distribution of the average continuum fields in
unit cubes O(x) centered at xEz4 • It is given by
EFF exp(-SA ('¥)) fexp(-Sfi.(~))IT8('¥(x)- !¢)0~ X 0 (X)
( 12)
'+ where x runs through V07l.. In terms of the Hamiltonian (10):
5EFF('I') [\
145
where Rfl is the block spin transformation mapping the
"space of unit lattice Hamiltionians" to itself:
the Cfl¢ being the block spins:
fl(fl- 4 l.: ¢(1\x+y)) IY l<fl
\l
(13)
(14)
(15)
The existence of the scaling limit (11) may thus be re
formulated in terms of Rfl: find a sequence of unit lattice
Hamiltonians Hfl' which after an infinite (as fl+oo) flow
under R have a limit
limR H = SEFF [\+oo [\ [\
(16)
the effective unit scale action of the field theory. In
particular, Hfl will approach the critical manifold in the
" space of Hs" .
146
5. Perturbation theory for RA
Th"e idea is to perform RA in many steps. RA has a semigroup
property
(17)
for A= LN. We tak~ L= 0(1) (i.e. N~oo) and analyze RL. This
is given in terms of a functional integral with UV cutoff -1 1, IR cutoff L and is thus well suited for a rigorous
analysis. Explicity, we split the gaussian measure d~z-lG
where the covariance is z- 1G= z- 1 (-~)-l to the block spin
and fluctuation parts:
(18)
(19)
(here ~'is linearly and locally related to~ and G'= G in
the IR). (18) and (19) now give, that for
H(q,)
const.exp(-!Z(~,G·- 1~)-TV(~)) 2
and the interaction is transformed to
(20)
(21)
(22)
147
Crucially, the fluctuation covariance r has exponential
falloff (with mass 0(!)). L
The rigorous analysis of (22) is built upon its perturba-
tive analysis in powers of V. For V local as in (1), TV
is given in terms of connected graphs with z- 1r, as lines
L-l~• on legs and as vertices the vertices of V. Each term
in this perturbation series is well defined due to the
massiveness of f(the series itself of course is divergent).
This also assures that TV, although not local, still is
approximately local, up to exponential tails. For V as
(we Wick order with G)
V(<!>) 1 2 1 2 4 -ZrL:<I>(x) : + -,Z AL:<I>(x) : 2 4.
( 2 3)
the leading part of TV will be given by a similar expressi
on, with r and A replaced by
(24)
( 25)
where the first term in (24) reflects the "relevant"
nature of the <1> 2 term whereas by (25) <1> 4 is "mariginal" in
the leading order. The second term in (25) is due to the
graph ')0( and for A negative, we see that I A I increases
in the iteration. There are of course plenty of other
terms in TV, all of "irrelevant" nature upon subsequent
iteration, except for a mariginal L(V ~1 2 term of order
A 2 (from the graph -6- ) , leading to
2 Z ~ z1 = (l+O(A ))Z ( 26)
Already this heuristic second order analysis suggests,
that if we succeed "fine turning" the bare mass r so as
to counter its increase (see below), the bare coupling
A(A) should be chosen approximatively as
148
to counter its expansion. Then (26) would lead to
N-1 ZIT (l+O(A~)) ~ ZIT(l+O(M- 2 ))
M=O
which stays bounded in ~ as claimed.
( 27)
(28)
To fix A(~) completely a third order analysis exhibiting
the loglog~ dependence is required. It may be shown, still
perturbatively, that the iteration of T quickly stabilizes
TnV to a fixed scaling form where A flows as
(29)
Our choise of the bare coupling is dictated by (29). The
reader may verify, that for the differential equation
dA (7\) /dlogA (30)
with our choise of A(~) as the initial value, A(l) will
be g+o(g 2 ), no matter what the 0(A 4 ) in (30) is. This is
essential for our rigorous analysis, where (29) will be
established rigorously.
Finally, the fine tuning of the bare mass is achieved by
iteratively restricting its range so that the running mass
flows as desired, at the end becoming m2 This succeeds,
since the expansion of rm at each step gives us room so
readjust r. The idea is from [1~
149
6. The rigorous analysis
Our rigorous analysis is based on the heuristic picture
presented above. Crudely speaking, we wish to establish
bounds for the corrections to (24)-(26) as well as to the
irrelevant terms in R~ • These bounds need to be such as
to allow us to iterate the transformation (22). Since the
covariance r in (22) has exponential falloff and we
expect the "couplings" in V to be small, it is natural to
analyze the non-locality of the Z integration using a high
temperature (cluster) expansion. The problem is not stan
dard, however, since the external field 'I'' in ( 22) may take
arbitary large values, for which V is not a small pertur
bation of a gaussian. Also, we need a very detailed know
ledge of TV for iteration, in particular a knowledge of the
"couplings"
(31)
The solution to these problems is the following. A natural
devise to go from bounds on TV to properties such as
bounds of (31) is analyticity. Indeed, it is easily seen
that T preserves analyticity of e-V in a large region, a
natural one being a (poly-) strip around 'I'' real. Thus in
particular we expect analyticity in a neighbourhood of
'1' 1 = O, which allows to study (31). The size of the analyti
city domain is determined by asking for which '1'1, V in (22)
is a small perturbation of d~z-lr: essentially, this
occurs for
( 3 2)
a large region for I A I small.
Given a field configuration r', we expect the cluster
150
expansion to be effective in the region Y of our lattice
where o/ 1 satisfies (32) allowing to obtain TV there as
TV(o/'l_l y
(33)
where the multibody potential Vy depends on o/ 1 lyand has
exponential falloff in Y. We use now the analyticity idea
to extract from (33) the quadratica and quartic pieces and
prove bounds for the remainder. Moreover, it is possible to
show that the third order perturbative computation of these
objects is asymptotic, thus rigorously establishing the
estimates of section 5.
What remains, is the control of TV in the large field
region
(34)
Note, that the large field part of V enters to our small
field analysis above: even for o/ 1 small, L -lo/ 1 +Z in (22)
may take large values (34). This, however, involves large
Z, having a neglible probability
exp(-0( IAI- 112 )) ( 3 5)
due to the gaussian d~z-lr' (here the E in (5) is used)
provided exp{-V(~)) won't get too big for~ large. If this
is the case, (33) makes the above small field discussion
feasible.
Thus, some stability bound for exp(-TMV(~)) is needed. A
good one turns out to be
I 112 12 I 4 ex p 0: ( - AM I I ~ ( x ) + A MI(I m ~ ( x ) ) + C ) X (34)
(36) is easily verified for (23) and its iteration succeeds
since the relevant 1~1 2 term tends to increase, part of
which increase may be used to kill dangerous constants.
151
Of course, the small and large field regions are actually coupled by d~z-lr and we need a representation for a general ~ configuration, a representation which involves the Hamiltonian (33) in the small field region and only the Gibbs factor (36) in the large field region. This naturally emerges from the cluster expansion and is the following. 1 Let DM be the region where 1~1> <IA"I-4>. T~ is analytic in IIJm~lloo< <IAMI- 114 > and has the representation
M exp(-T V(~))
ex p ( - E VyM ( ~ ) ) vcxc
g':j depends on
and XjnDM is a union
( 37)
are where Xj are deisjoint sets and
"islands" around D M: X= UXj:JD M of components of D M"
V~ have a representation in terms of the 3rd order perturbation theory in AM and a remainder with proper bounds. g~ satisfy a bound analogous to (36). The form (37) is preserved by the iteration ofT.
EFF This analysis establishes the limit lim RAHA= S as a well behaved unit lattice theory. The analysis of the effective action may be extended to a full analysis of the continuum limit correlations GN.
References
[11 [2]
[31
J.Glimm,A.Jaffe, Quantum Physics, Springer 1981
V.Rivasseau, Preprint, Ecole Polytechnique, Jan.l984 G.'t Hooft, Commun.Math.Phys.86,449 (1982), Commun.Math Phys.~,l (1983) -
[~ L.D.Landau, Coll.Papers, Gordon and Breach 1965 ~1 I.Pomeranchuk,V.Sudakov,K.Ter Martirosyan, Phys.Rev.
103,784 (1956)
~] A.Sokal, Ann.Inst.H.Poincare,12,317 (1982)
[71 K. Wilson, J. Kogut, Phys. Rep .12C, 75 (1974) l~ J.Frohlich,Nucl.Phys.B200,281 (1982)
152
[~ M.Aizenman, Phys.Rev.Lett.47 (1981), Commun.Math.Phys. ~,1 (1982)
~~ K.Symanzik, Lett.Nuovo Cim.~,no.2 (1973)
~~ K.Gawedzki,A.Kupiainen, IHES preprint, July 1984
[12] G.'t Hooft, Phys.Rep.l04,129 (1984)
[13] P.M.Blecher,Ya.G.Sinai, Commun.Math.Phys.ll_,23 (1973)
A.Kupiainen
Helsinki University of Technology
Department of Technical Physics
Espoo 15 Finland
Abstract
153
THE DYNAMICS OF A PARTICLE INTERACTING
WITH A SEMI-INFINITE IDEAL GAS IS A
BERNOULLI FLOW
c. Boldrighini, A. De Masi, A. Nogueria,
E. Presutti
A one-dimensional semi-infinite system of point parti
cles moving in cO,m) is considered. The particles have mass
m and are neutral except for the first one which has mass
M>m and charge 1. The particles undergo elastic collisions
with each other and with the wall at the origin; between
collisions the m-particles move with constant velocity and
the M-particle with constant acceleration, E/M. A Gibbs
measure exists for those values of the density and the tem
perature for which the thermodynamical pressure P of the
free gas (of only m-particles) is larger than E.
The so defined dynamical system is a Bernoulli scheme
when E<2P. This extends a result of ClJ, namely that for
E=O the system is a k-system.
0. Introduction
The ergodic properties of an infinite one-dimensional
system of point particles moving on the half line(O,~)have
been studied in a recent paper, ClJ. The system is interacting but the interaction is localized, as it happens in
154
C5J, cf. also C4J. More precisely the system has infinitely
many point particles which only interact through elastic col
lisions among each other and against a wall at the origin.
All the particles have the same mass, m, except the first
one which has mass M, M>m. The Gibbs measures are of course
stationary and in ClJ it has been proven that the dynamical
system associated to any of them is a k system.
In this paper we consider a more general case, namely
when a constant force E acts on the first particle. We prove
that if E is not too large the system is Bernoulli, thereby
strengthening the result of ClJ. We hope that our paper will
provide more insight into the ideas and the techniques in
troduced in C1J which might be relevant in solving a number
of interesting open problems. We discuss some such problems
in the concluding section of the present paper.
Be shall now give a heuristic outline of our results.
Precise definitions and statements will be given in Section 1.
Let q 0 (t) denote the position of the first particle at
timet, such particle will be called the "heavy particle",
h.p., throughout the paper. Since all the particles with non
zero velocity will collide with the h.p. in the past or in
the future (with probability 1) then it is clear that the in
finite particle configuration x at time zero is (modulo zero)
identified by the history of the h.p. {q0 (t), te m}. This
implies that if we consider the partition r;; generated by the
"past" history of the h.p., i.e. by {q0 (t), t.::_O} then
v Tt r;; is the partition of the phase space into points t>O
(modulo zero) with respect to the Gibbs measure considered.
The main point in C1J for proving the k-poperty, was to show
that r;; = 1\ Ttr;; "" t<O
is, modulo zero, the trivial partition.
Proving triviality of s 00 amqunts to prove that the
"behaviour" of q 0 (t) for T.::_t.::_+T, conditioned to its past
(i.e. to {q0 (t), t.::_O}) is close to the unconditioned one for
any fixed -r>O and for "most" specifications of the past,
when T is large enough. We shall prove here that one can take
T="", namely the whole future becomes independent of the past
for T large. In other words we prove that the process q 0 (t)
155
is a-mixing, which implies the weak Bernoulli property, and
hence that the dynamical system is Bernoulli C8J (see Sec
tion 1 for the details).
A main point in C1J as well as in the present paper, is
the use of a "copying procedure" by which the behaviour of
the process q 0 (t), t.::_T taken in any "atom of the past", was
copied within "any other" atom of the past; i.e. it was
shown that a copy of it appears in "most" other atoms of the
past. This was used in C1J to prove that the conditional
probabilities for different past histories when restricted
to the tail a-algebra of the far future, are equivalent. By
a general result, proved in C1J, this implies the k-property.
The procedure used here is more straightforward. We can show
that the discrepancy between the conditional processes is
small, by making the copying measure preserving, and get
directly the a-mixing condition and the Bernoulli (B-)pro
perty.
The copying procedure, here as in C1J, is based on the
occurence of a "nice event", which "reduces" the memory of
the past history of the h.p. This event consists of a
"cluster of fast incoming particles" which collide with the
h.p. in such a way and for so long time that the h.p. never
collides again with the particles with which it collided be
fore the cluster arrived. So the future of the h.p. depends
only on its position and velocity at the times when it starts
colliding with the cluster ("cluster time", c.t.) and on the
incoming particles. In C1J the future behaviour, after a c.t.
of the h.p. with some given past was reproduced in an atom
corresponding to a different past by suitably inserting a
finite number of particles, and leaving the incoming parti
cles (at c.t.) unchanged.
In this paper we show that we can choose such cluster
that completely "cancels" the past. Namely, if the particle
configuration in CO,LJ is in a compact set C, there is a
configuration is CL,+oo) (cluster) such that the motion of
the h.p. for large times is "focalized", i.e. is driven to
wards a fixed attractive cycle, independent of the choice
of the configuration in c. This is the loss of memory re-
156
required in the proof of the B-property. We think that this
property is interesting by itself and we present it separate
ly in Sect.2.
Some modifications have to be introduced when the h.p.
is subject to an external force E, which we assume for sim
plicity to be constant. First of all for the Gibbs measure
to be invariant we must have E<P, where P is the thermody
namic pressure of the gas of light particles. If density and
temperature are such that this inequality holds, it is easy
to show that there are "focalizing clusters" similar to that
for E=O. Troubles arises with the last part of the proof
which we have not mentioned yet. Namely we need to prove
that after focalization the h.p. is very unlikely to catch
up with the outgoing particles of its past. We need an ex
plicit estimate for that, since that equilibrium estimate on
the displacement of the h.p. cannot be used in the untypical
situation created by the cluster. This can be done only if
E<2P. In fact our bound comes from an estimate on the mo
mentum trasferred by the incoming (negative velocity) par
ticles to the "moving barrier" of the h.p •• We know that
their velocities eventually turn positive (after collision
with the h.p.), but we do not know much about their outgoing
values. So we know only that the momentum transfer due to a
particle of velocity v is mJvJ, and hence that the average
momentum transfer per unit time is larger than 2P. After
this paper was completed Carlo Boldrighini and Anna De Masi
have obtained a proof by a different method which extends
the result to any E<P.
1. Definitions and main results
The one particle phase space is~~ = {(q,v)SR2 :q~O} where q denotes the position of a particle and v its veloci
ty. Let X denote the phase space of the locally finite con
figurations in~~, and M the a-algebra of the Borel sets of
X • If xeX and A_~~ is a measurable set we denote by xA
157
the configuration xnA. MA denotes the a-algebra generated
by xA. By XA we denote the phase space of a system of par
ticles in A. Sometimes we shall not distinguish between XA A
and the subset XA = {xeX :x 2 = Q} and we shall identify JR+/A
the subsets of XA with the corresponding subsets of XA.
A point xeX can be identified by a sequence qk,vk , kEN,
where the paticles are labelled in order of increasing po
sition and, for equal position, of increasing velocity.
M will be the mass of the first particle (with coordinate
q 0 ) and m<M the mass of any other particle.
We fix the inverse temperature S and the chemical poten
tial A and we denote by v0 the Gibbs measure (with the above
parameters) for a gas of free identical particles in O,oo all
having mass m. We deonte by P the corresponding thermody
namical pressure, namely
SP
SP
l/2 (~ ) exp(SA)
Sm
p
where p is the density of the gas.
(l. la)
( l • lb)
We then fix E<P and we denote by v the Gibbs measure defined
as follows. The position q 0 of the h.p. is distributed with
Poisson law of parameter S(P-E), i.e.
CS(P-E)J-l f dy expC-S(P-E)yJ X
( l. 2)
Given q 0 =x the distribution of the positions of the other
particles is Poisson with parameter SP, as in v0 • Velocities
are independently distributed with Maxwellian law.
The time evolution is defined as follows. Light particles
cross each other without collision. The h.p. moves with con
stant acceleration E/M, until a collision occurs. When it
158
collides with the wall at the origin it just inverts its
velocity, when it collides with a light particle the out
going velocities V' of the h.p. and u' of the light particle
are given in terms of the corresponding incoming velocities
V,u by the elastic collision law,
with
(V, u)
V' aV + (1-a)u
u' = -au+ (1+a)V
M-m a = M+m
(V',u')
The main result in this paper is
( 1. 3a)
( 1. 3b)
Theorem 1.1. If E<2P the dynamical system ( X,~,Tt) is
Bernoulli.
The proof of the theorem is based on the a-mixing pro
perty for the process q 0 (t). Namely let s_CsTJ denote the
partition generated by q 0 (t), t<O Ct>TJ and consider the
quantity
a(T)
(if IT is a partition by AeiT we mean that A is
IT-measurable)
The following results holds.
Theorem 1.2. The process q 0 (t) is a-mixing, i.e.
lim a(T) 0
( 1.4a)
( 1. 4b)
( 1. 4c)
159
Proof of Theorem 1.1. The proof follows easily from Theorem
1. 2. As mentioned in the introduction { q 0 ( t), telR} gener
ates the partition into points, cf. also C1J. Let n be any
finite partition coarser than that generated by q 0 (t) for
+oo te(-1,0J. Then by (1.4) the v
n=-co weakly Bernoulli. From this the B-property follows, C8J,
since we can choose a sequence of increasing partitions nn
which generate the whole q 0 (t), te(-1,0J and the corres
ponding factors are all B-shifts.
Proof of Theorem 1.2. The rest of this section will contain
an outline of the proof of the Theorem.
To prove eq. (1.4c), we introduce another coefficient, na
mely y(L,T), as follows.
For L>O let
lR2 + lR+xlR ( 1. Sa)
~ 2 { (q,v)elR 1 :q~L} { (q,v)elR ~ :v?_O} ( 1. Sb)
CL lR~\~ ( 1. Sc)
and let ~L be the partition of X into points of ~' namely
the atom of ~L containing y is
~L(y)
We define
y(L,·r) svp ill(AnB)-ll(A)].t(B)I ~ .;JB
( 1. 6)
( 1. 7)
160
We will prove in Section 4 that ~L' for L large refines
on a subset of the phase space of measure close to 1.
Lemma 1.1. For any e>O there is aLE such that for any L>L£
and any T
a(T) < y(L,T) +£ (1.8)
The proof of Theorem 1.2. is completed by
Lemma 1.2. For any e>O there is a L£ such that for any L>L£
lim sup y(L,T)<e ( 1. 9) T+OO
Proof. The proof is based on coupling arguments and the
copying procedure of C1J, which we now outline.
Definition 1.1. Let L be fixed. We denote by aL the class of
the measures Q on (Xx Xl with the following properties.
1) the marginals of Q are u, namely if f is a bounded
measurable function on X let f(x,x') = f(x) then Q(f)=~Cf) and if g(x,x')=f(x') then Q(g)=u(f). We say that Q is a
joint representation of (X , 11), (X , 11).
2) For Q almost all atoms of ~Lx ~L the following
happens: fix ~L(x)x ~L(x') Q almost surely, then the margi
nals of Q ( .J f;L(x)xf;L(x')) are 11C .J f;L(x)) and 11( .J f;L(x')),
namely, using the same notation as above
161
3) The relativization of Q to ~LX ~Lis (~ sL)x(~ sL),
It is easy to see that if QeaL then
(1. 9)
In fact call ~A' 1B the characteristic functions of A and B.
If QeaL then from (1.7) we get
The proof of Lemma 1,2 now follows from the following re
sult proved in Section 4 (by Statement 4.4)
Lemma 1.3. For any E>O there is an LE such that for L>LE
there is a QeaL for which for soma T>O
(1.10)
2. Loss of memory
In this section we show that any "reasonable" configura
tion of particles in~' cf. eq.(1.5b), can be completed by
adding a special configuration y of particles in CL' cf. eq.
(1.5,c), in such a way that q0 (Tt(x y)) approaches asympto
tically a periodic motion independently of x. This is a key
point in our proof and we believe that such a property can
be found for many systems consisting of a "localized" inter
acting subsystems in a free particle bath, to which our re
sult can be extended.
162
To prove Thm. 1.1 we will have to consider small random
perturbations of y. We will first examine the case E=O then
EjO. Reasonable configurations are those in
V(L,N,V) =
{x:x=xR ,g0 (x)<L, E 1(g<2L)<N, max 1(g<2L)JvJ< V}, L (g,v)ex (g,v)ex -
(2.1)
where L,N,V are positive numbers, we will be interested in
the case when they are all "very large".
For sake of notational simplicity in the next theorem we
shall agree to have fixed a rule for the collisions which is
compatible with energy and momentum conservation and which
allows to define the evolution even triple or more collisions
occur. This will not generate ambiguities since such events
occur with probability zero.
Theorem 2.1. (E=O). For any positive L,N,V there is yeXc , !..
'!" eco,1), v L - so that for any xeV (L,N,V) '!"
lim Jq0 (Tt(xuy))-q0 (t)l t-HO
0, (2.2)
q (t) is a periodic function of period T. It describes a 0
motion with constant speed v which takes place between 0
and 2L, such that q 0 (;) = 2L.
Proof. The configuration y. The first particle in y is the
w1-particle which is at Land has velocity -w1 .w1 will be
chosen to be much larger than V. After the w1-particle there
is a "cluster" of n-particles, the n-cluster, each particle
in the n-cluster has velocity -1. All the particles in the
n-cluster are at time zero in an interval of length ~ with
right end point 3/2L-1. n is much larger than N and ~ is very
small. After the n-cluster there is the w2-particle which,
at time zero, is to the right of 3/2 L (its actual position
163
will be specified later). To the right of the w2-particle
there is the w3-cluster, made of infinitely many particles:
they have velocity -w3 and they are equally spaced, the
spacing is (l-a)-1 (l+a)L and the first particle of the w3-
cluster is at some suitable distance from the w2-particle,
as we shall see below.
Choice of the parameters. After the collision with the w1-
particle, the h.p. takes a velocity between -(l-a)w1-av and
-(1 -a)w1+aV. We choose w1 so large that the velocity is al
ways negative.
The h.p. then collides with the wall at the origin, ta
kes a positive velocity and then starts colliding with the
light particles of x. After u collisions its speed will be
larger than
u-1 au(l-a)w1 - E ai(l-a)V > au(l-a)w.-V
i=O 1
We chose w1 so that
aN+n(l-a)w. > V l.
(2.4)
As a consequence the h.p. will travel without being stopped
by the particles of x and by those which are in the n-clus
ter. We set
T'
so that in any case at T' the h.p. has already interacted
with all the particle of the n-cluster. We choose w1 so
large that
164
VT' < L/4 (2.5)
and
N+h (1+a)[a (1-a)w1-VJ-a(V+1)>V ( 2. 6)
Therefore all the particles of x which were initially in
CO,LJ have at time T' positive velocity larger than V. We
then take n large and 2 small so that
n ~L-1 + T'a (1-a)w1 < ~L (2.7)
hence the h.p. at time T' is still to the left of 3/2 L. We
place the w2-particle in such a way that at time T' it is at
3/2 L. At that time the h.p. is in (3/2L - 2, 3/2 L) with
. N+n n veloc1ty between a (l-a)w1-v and a (l-a)w1 • We chall com-
pare the actual evolution of the h.p. with that corresponding
to the "ideal case" when at time T' its position equals 3/2L
and its speed is zero. By choosing w2 very large we can make
the actual and the ideal coordinates very "close" to each
other.
Let v and T be the parameters which define the periodic
motion q (t). We impose that T'/~ k, k being a positive 0
integer. We choose w2=a so that in the ideal case at time
T'+2~ the h.p. will be at L/2 with positive velocity v. The
first w3-particle is at time T' at L/2+2~w3 > 5/2 L, since
_.!_ v
w3 = (1+a)(1-a) and VT = L.
In the ideal case therefore the h.p. moves like q 0 (t)
after time T'+2~.
Actually this is not the case because of the "small"
error with respect to the ideal case which is present at
timeT'. We shall now see that the collisions with the w3
particles are of contractive nature and lead the h.p. toward
a periodic motion, i.e. q0 (t).
165
To understand this let us first consider the case where
there are only two particles, the h.p. and a light particle,
both on the line. There are two cases we consider: (a) their
velocity and position are respectively (q,v) and (q',w),
with v>O and w<O, q>q'. (b) position and velocity are
(q+dq, v+dv) and (q' ,w), with v+dv>O and q+dq<q'. In (a)
the collision occurs at time T and in (b) at s. At time
t>max{T,S} we have, (q,v) and (q+dq, -i+dv) denote, respec
tively, the coordinates of the h.p. at timet in (a) and (b),
adv ( 2. 8)
dq a ( dg+tdv )
We now compare our case with the ideal one, i.e. when at <'
the h.p. is at 3/2 L with zero speed. We can apply the above
formulas if the h.p. does not recollide with the light par
ticles and it collides when it is moving with positive velo
city. In this case the collisions against the barrier at the
origin do not change our analysis. We shall now set condi
tions for this to hold, we proceed by assuming we can apply
eq. (2.8), we will then see that the result is indeed con
sistent with such assumption.
We will look at the coordinates of the h.p. when, in the
ideal motion, it is at L/4, i.e., at the times
3 - -Tk = <' + 4 T+k<, k=l,l, ••• At a timet, Tk<t<Tk+l' the
position and velocity of the h.p. are (L/4, V) in the ideal
motion and (L/4 + dq(k), V+dV(k)) in the actual motion. By
eq.(2.8) we have that for any t>max{Tk,Sk}' t<Tk+l
dq(k+l) a(t dv(k) + dg(k)) (2.9a)
d V(k+l) -adv (k). ( 2 • 9b)
166
As a consequence
n ldq(n+1)1 < T l: an+ 1-ildv(i)l+anldq(l)l.
i=1
Therefore
(2.10)
To apply eg. (2.8) we need that
ldvCl)l <V
I dq( n) I < L /4, n =1,2, ••••
By choosing w3 very large we make T very small. This takes
care of the first term on the r.h.s. of eq. (2.10), since
the sequence nan, n=1,2, ••• , is bounded. For the second
term it is enough to have
ldqCUI < L/4.
Definition 2.1. We define here small perturbations of the
configuration y introduced in Thm. 2.1. We order the par
ticles of xeXC in terms of the time they take to arrive at L
L under free motion. So xeXC is described by the sequence L
(Ti(x), vi(x)) , i~1, where , 1 (x)~T 2 (x/~ ••• and
(Tilvii+L,vi)i> 1 is the set of he coordinates of all the
particles of x. We choose 6 very small, for instance 6< lo• and we define, given L,N,V hence y,
167
N(y,6)={xecx:,i(y)~•i<x>~•i(y)+6, L
lvi(x)-vi(y)l<6, Yi.::_l}. (2 .11)
As a corollary of Thm.2.1, (and of its proof), we have that
given any £ there is n£
xeX (L,N,V), zeN(y,6)
T and 6£<T0 so that for any
v t>n < - £
-V n>n - E
where v0 (t) denotes the derivative of q0 (t). We choose
L - -£ < 10 and fix 6<6£, so that at time (n-l/4h, n.::_n£, the h.p.
has already collided against the wall, but not yet with the
next incoming particle of N(y,6). We denote by (~ ,v ) the n n
position and velocity of the h.p. at time (n-l/4)T. We call
!: the probability on N(y,6) and Fx the law of (gn,vn)
induced by P if xeV (L,N,V) is a configuration in XR • The L
process (qn,vn)' n>n£, is Markoffian since the law of
(qn+l'vn+l) is completely specified by (qn,vn). Actually
we have
~ x (dqn+ldvn+ll qn 'vn)
where p is a bounded function of its arguments uniformly in
n and x. Furthermore there are positive constants p and a
such that
where 11·11
168
is the Euclidean norm in JR 2 and r ( g ,v ) is the n n
value of (qn+ 1 ,vn+ 1 ) when the first light particle which in
teracts after time (n-1/4)~ has parameters as in y. We have
(~ + dq(n+1), ij+dv(n+1)), 4
where dq(n+1) and dv(n+1) are given by eq.(2.9).
(2.12)
The following is a classical theroem, see for instance
C 2 J, which applies directly to our case: let U c JR n be an
open bounded neighborhood of the origin. Let T: U ->-U, with
TU U, be a continuous map having the origin as an attrac
tive fixed point. Assume that for each xeTU there is a
measurable function y ->- p(y,Tx) with support in U and that
there exist positive constants p and a such that:
(i) p(y,Tx)_::.p, if lly-Txll.:::_a, (ii) lP(x,dy)=p(y,Tx)dy. Then P
defines the transition probability of a Harkov chain with
state space u. The chain has only one stationary measure v
and this is absolutely continuous with respect to the
Lebesgue measure and furthermore
lim IIPn(x,dy)-v(dy) II 0 , n
uniformly in x and exponentially fast. A similar argument
was used in c 3 J. In our case the state space is in lR 2 , i.e.
the position and velocity of the h.p. observed at the times
(n-1/4)~, n>n . The map T refers to the transformation (2.12). - £
Thm.2.1 and the above result extend easily to the case
EjO, In fact the action of the electric field can be regarded
as a small perturbation if we choose the velocity of the in
coming particles large enough. In this way the contractive
properties of the map T remain unchanged and the previous
conclusions keep their validity. We report the above consi
derations in Thm. 2.2. below in a form which will be suitable
169
for the construction of the coupling.
Theorem 2.2. (EjO). For any choice of the positive numbers
L,N,V there is a configuration y in CL and o>O so that for
any zeN(y,o), cf. eq.(2.11), and xeV(L,N,V) the path
q 0 (Tt(x z)) has the following properties:
(l) there is T'<1 such that q (T (xuz))9(3/2L-2, 3/2 L) and 0 Tr
for any t<T', q (Tt(xUz)) q (T (xuz)). Furthermore all the 0 0 T 1
particles of x which at time zero are to the left of L have
at time T' positive velocity larger than 1.
(2) After T' and before reaching 3/2 L the h.p. undergoes a
collision. Because of that the velocity of the h.p. becomes
negative and without further collisions the h.p. reaches
position 3/4 L at time T*~T'+~, ~<1.
( 3) After T'\ there are collisiors with new light particles
which keep the h.p. in the interval co, 3/4 LJ with speed
less than £V. The particles which collide with the h.p.
after T* interact only once with the h.p. and after that
they get some positive velocity larger than 1.
(4) Given any e>O there exists n and for all n>n the fol-e e
lowing holds. For S=(n-1/4)~ let
N(y,o,S) =
={zeCL(O,S): 3z'eN(y,o), z'nCL(O,S)
Let P(S) be the measure u conditioned to N(y,o,S). Then for 0
any x and x' both in V(L,N,V) there is an isomorphism
¢8 , of N(y,o,S) onto itself such that the probability ,x,x
P , on N(y,o,S) 2 which has support on x,x
{z, 4s,x,x'(z)}
170
and with marginals equal to P(S), is such that
P ,({(z,z'):q (T (xUz)) 1- q (T (x'uz'))})<E (2.13) x,x 0 s 0 s
3. Construction of the coupling
In this section we construct a coupling QeaL, cf. Def.
1.1, which depends on the parameters L,N,V, •.• , introduced
in Section 2. In the next section we will determine the
values of the parameters so that Q will enjoy the properties
required by Lemma 1.3 of Sect.1.
Definition 3.1. The sets ; 1 , ; 2 , ; 3 , L,N,V, ,Tare fixed,
cf. Def.2.1. for notation. We introduce the sets
F 2 {(q,v)e CL(O,T): -2 < v < 0},
where
CL(s,t) 1-1
{(q,v)ecL: lv (q-L)e(s,t)} s<t ( 3. 1 )
and CL is defined in (1.5).
One should think that L is very
small. Here T denotes the length of
T=(n-1/4)~ for a certain fixed n.
We set
-1 large and T L is very
{xeX xF e N( y, li, T)} , 1
the "nice cluster", hence
(3.2)
171
c.f. Theorem 2.2 for the notation used here. At time T the
cluster is over so we need to slow down the h.p. this is
accomplished by putting particles in F 2 • We define the set
J 2 as follows. It has a large number of particles in F2 with velocity in C-2,-1J. They would be in C~L,+ooJ at time
zero and in CfL,L) at time T, if the motion were free. By
(2) of Theorem 2.2 the h.p. will interact with the above
cluster only after time T. In fact at time T + T' the first
particle will be to the right of 7/4 L- 2(T'+2T) > 3/2 L
(at least for large L). The h.p. will not cross 3/4 L before
time n T (remember that T = (n-1/4)T because of condition
(3) of Theorem 2.2. We also require that
E mjvj > 4(2Mv + EL) (q,v)exF
2
Notice that by Theorem 2.2 rv is an upper bound for the
speed of the h.p. The motivation for the above requirement
is the following: if no new light particle arrives, then
the h.p. will cross L only after it has gone back to the
origin from where it bounces off with speed less than 2.
To have the situation described in theorems 2.1 and 2.2 we
must require that the only particles in the interval L,2L
which have non-positive velocity are those introduced so
far. Such condition implies that no new particle interacts
with the h.p. in the time interval O,T In fact the other
particles are both in CL(T,oo), cf. eq. (3.1), and to the
right of 2L. A collision might occur before time T only if
a particle reaches 3/2 L before time T'+T, cf. Condition
(2) of Theorem 2.2. Since at time zero such particle was to
the right of 2L this means that under free motion it would
reach L at time 2(T'+T). This is less than 4, we choose T
large, certainly larger than 4, so the particle can only
stay in CL(O,T). Besides the sets J 1 and J 2 introduced
above, we will need to consider in Section 4 also the sets:
We set
We define
2 { (q,v)e:JR +
{xeX
172
Q}
0,1,2, .••
where T~ denotes the free evolution, and tk= kT.
Analogously we define
T0 J i=l,2,3. -tk i '
We then shorthand
so that
(k) c
k n
i=O
( i) (X\ J )
G(O) G(s) C(s+l) . t't' f , ••• , , 1s a par 1 1on o
(3.3a)
(3.3b)
(3.4a)
(3.4b)
X •
A final remark on the notation we are employing: in the
definition of the previous sets the position of the h.p. is
not mentioned. So, in principle, it is not correct to say
that in xeJ , for instance, a cluster is arriving, because
the first particle in XF might be the h.p. itself. We will l
173
consider, however, J intersected with V(L,N,V) so, in
particular, the condition q0 (x)<L is fulfilled.
Definition 3.2. The coupling Q.Q is the product measure
ll r Xp'L XIJ ~X Et on X~ X X Et cf. Def. 1. 1. So it remains to
define the conditional probabilities of Q given
We set
whenever either q (xR_ )>L or q (xi_ )>L. For notation sim-o --r. - 0 --r. -
plicity IJ(. I xR ) denotes the conditional probability to the L
atom of F; L which contains xFt. So we have to define Q when
q 0 (xFt)<L and q0 (x~)<L. We will define a map ~=XcL +XCL
in such a way that Q(dxcL dxcL/xFt, xR£ will have support
on {(xCL' ~(xCL))} The map~ depends on xRL'x~. Assume
(k) xc e G , see (3.4), k~S. Then
L
There are two cases:
(2) its complement.
In case (2) we pose xc L
In case (1) we pose
~(XC )n F(k) L 1
~(XC ). L
174
where ~ is the map defined in Thm. 2.2. Finally
n c \F(k) L 1
C \F(k) L 1 •
( s+l) If xc ec , see (3.4), we set 'l'(xc) = xc therefore 'I'
L L L defines an isomoprhism of ( X c , ll ) onto itself. We
L o complete the definition of Q(dxcLdxcL/x~,x~), by requiring
that its marginals are )l 0 which is compatible with the above
position because 'I' is an isomoprhism. By its very definition
Q€ a L ( see De f. 1. 1 ) •
4. Proof of Theorem 1.2
In this section we prove Lemma 1.1, Lemma 1.3 and then
Theorem 1.2. We use extensively estimates proven in C1J
which extend essentially unchanged to the case EjO (in
such instances we will omit reporting the corresponding
proofs).
We proceed as follows: we first choose the actual values
of the parameters L,N,Vo,T,s introduced in the previous sec
tions. At this stage we just want to make explicit their
mutual interdependence. After that we define the coupling Q
and the reason for the above choice will become clear. We
first fix E>O, we are interested in the limit when E goes to
zero. We define L so that for all L>L E - E
(4.la)
(4.lb)
175
The sets appearing in eq. (4.1) will be introduced later on
and, at that time, we will also prove the existence of such
Le:. In the following we should think of e: as fixed and cor
respondingly L will always be taken larger than Le: • We
then fix N and V so that
u( V (L,N,V)) > 1-e: ( 4. 2)
where V(L,N,V) has been define in eq.(2.1). It is easily
seen that eq.(4.2) is satisfied inN and V are chosen to be
large enough.
Given L,N.V as above we define y as in the proof of
Theorem 2.1 and then 6, which depends on L,N,V,y, like
stated in Theorem 2.2.
After that we will fix T which denotes the time-length of the cluster, and then, finally, the time we are going to
wait for the cluster to arrive, this will be the time T to
which referred in Section 1.
The fist condition concerns the atoms of the past, we
pose, as in ClJ,
~L
A L
(4.3a)
- -(1r;+l) {(q,v)€CL:q~L, O~v~-ILe } (4.3b)
{x ~} (4.3c)
CL and~ have been define in eq.(l.S). The following sta
tement easily follows from the stationarity of the Gibbs
measure, its proof is similar to that when E=O, which is
reported in ClJ, so we do not give it
176
Statement 4.1. Let xe1;LnAL then the path q 0 (Ttx), t.::_O, is
such that the h.p. has never collided with any particle which
at time zero was in CL. Hence q0 (Ttx~) = q 0 (Tt x) for all
t<O.
Furthermore for any c>O there is L£ so that eq. (4.1) holds.
Proof of Lemma 1.1. Let A be /;-measurable and take L>L • - £
Define
A {x:x~eA}
hence A is i;L-measurable. Then, n below denotes symmetric
difference,
On the other hand by Statement 4.1 (AnA) ( 1; L n 1.. L) C),
hence Lemma 1.1. is proven.
Given L,N,V we take y and 6 according to Theorems 2.1
and 2.2. We then define T=(n-1/4)~ with n so large that (1)
(4.4a)
where J 3 has been define in eq.(3.3). Notice in fact that
by the defintion of F 3 it easily follows that
lim T+oo
1-£
The second requirement on T is ( 2)
177
( 4 0 4b)
where p x,x' is the coupling defined in Theorem 2o2, cfo eqo
(2ol3), and both x and x' are in V(L,N,V)o
We will consider xeJ (k) for some given k, cfo Defo3ol
for notation~ then everything is "prepared" for the nice
cluster to arrive at time tko However for this to really
occur we need to impose the following:
Requirement 1 The particles specified by J(k) should not
interact with the hoPo before time tko
Requirement 2 They should find a configuration in ~ which
belongs to V(L,N,V)o
To fulfill the requirements we pose:
~(k) (4o5a)
~;(k) L {x:qo(Tt+tk x~(k))~/L log+JtJ,
Yt~O~ Tt x~(k) c ~} k
( 4 o Sb)
r (k) L
To -tk rL {(q,v)elR~:(q+ tk,v )e rL} (4o5c)
A (k) L
{ x: x n r i,k) C/>} ( 4 o Sd)
Statement 4o2 (k) (k) 0
Let x€t;L n AL then Tt x = Tt k k
and Ttk x~ (k) c ~ o Furthermore eq 0 ( 4 ol) holds
for any L.::_L£ o
Statement 4o2 is proven in ClJ, in fact the proof does
not differ from the one when E=Oo
178
We define
{x eX: Ttk ~(k)evCL,N,V)}
and we have that
(k) A (k) )> 1-£ L n L
since by Statement 4.2
S. T_t V(L,N,V) k
and then eq.(4.6b) follows from eq.(4.2).
(4.6a)
(4.6b)
We remark that the above requirements, (1) and (2),
are verified in the set ~;i,k)n Ai,k)n olk).
We will now specify the value of s, sT will then equal
the time , introduced in Lemma 1.3. s will be a positive in
teger, large enough so that the probability that a cluster
arrives within sT will be close enough to one. It is possible
and convenient to assume that the choice of our parameters
is such that
s is chosen so that
0 _ JJ ( J ) ) s+ 1 < £ 0
( 4. 7)
179
Our last concern refers to the recollisions of the h.p. with
the light particles which had interacted before the arrival
of the cluster.
We first define
We then have the following
Statement 4.3 For xe A(k) c(k)nA (k) L n "L L
Tt x n r< +) (/J k L
(/J}
The proof of the above satement easily follows from - (k)
Statment 4.2 and the definition of AL •
Notice that by using Theorem 2.2 we also have
Definition 4.1 Let N L be thet set of configurations in RL
such that: (1) the speed of the incoming particles of xR L
is less than 2, (2) if xe NL then TtxR evolves for t>O in L
such a way that the h.p. will cross L only after having gone
on the origin, from where it bounced off with speed less
than 2.
180
Notice that by Def. 3.1. at time tk' after the cluster has
passed away, the configuration is in NL.
\\le need then
Proposition 4.1. Let E<2P. We pose
-(k) ~L
0 z = y U T t XC ( t "") } '
k L k'
where NL is defined in Def.4.1. Then for all L>L£
We postpone the proof Prop .. 4. 1. , which is similar to the
proof of Prof.A.2 in C1J, with the assumption that E<2P
(this is the only part in the paper where we need E<2P
rather than E<P • The choice of ~~k) is such that together
A ~k), it will imply that the h.p. does not collide after
tk with outgoing particles on the right of L.
Statement 4.4 Given any c>O, take any L>L 3 so that (4.1)
holds. Take then N,V, according to (4.2), 6 as in Def.2.1,
T as in (4.4), s as in (4.7). For such values of L,N,V,6,T,s
let Qe aL be the coupling define in Def. 3.2. Then there is
a function ~(c) which vanishes as E goes to zero such that
for T=ST
Q({ 3 t>T
This proves Lemma 1.3 of Section 1, which completes the
proof of Theorem 1.2.
181
Proof of Statement 4.4. We will call x<x,x') the characte
ristic function of the set {3t~T:q0 (Ttx)jq0 (Ttx')}, By (4,2),
we have
JdQx(x,x' ).::_2E:+!dQx(x,x' )l(q0 (x)<L)l(q0 (x') > L)
<2e:+ ~ JdQx(x,x') l(q (x)<L) 1(q (x' )<L) l(xeG(k)) k=O o o
+ fdQ (q0 (x)<L)l(q0 (x')<L)l(xe (s+ll),
where s<kl and c<s+ll are defined in C3.4l.
WE use the following shorthand notation for O.::_k.::_s, where s
is given by (4.7),
Let ~(k)(x) be the characteristic function of
~ ( k ln A ( k )n "' ( k+ 1 l,.,:r ( k) ( k) ( k) h 1 ( k) sL L c; L , "' L n V fl J 3 , w ere 3 is defined
in (3.3) and the other sets in (4.5) and (4,6). Then
We have that
x<x,x' ll <xeG<k l lx (k l (xlx(kl (x' l
Q -a.s. (4.8)
182
In fact by the choice of t:i,k), Ai.k), V(k), J(k), we know
both in x and x' that at time tk the cluster is arriving
and that it leaves a configuration in NL
the choice of -(k+l) d t;L an
-(k) AL the h .p.
at time tk+l' By
does not interact
after time tk+l with outgoing particles which at time tk+l
are on the right of L, as well as with the particles in Fik),
cf. De f. 3 .l.
The argument is completed by noticing that in F(k) and 2
CL(tk+l'oo) the configurations x and x' are the same with
Q-probability one,
This r.h,s. term of (4.8) is bounded by dropping the
condition that x and x' are in ~(k+l) A(k) • We then take "L n L
the conditional expectation with respect to fixing the con-
figurations x and x' in RL(k) (4.5a), The condition
q0 (Tt x) 4 q (Tt x') becomes the condition (cf.(4) of k+l 0 k+l
Thm. 2. 2)
where x = Tt ( XR (k)) and x' = Tt (x~ (k)) (we have used k L k L (k)
here that the remaining particles in F2 are the same in
x and x' and that "fast" particles in F 1 do not recollide
with the h.p. until time tk+l). By using ( 4 ) of Thm. 2.2, we
conclude that
l'l"e have
183
~fdp1
{ 1 (x~ ~ i,k)) + 1 cxeAi,k) )+
+ 1 (x~ ~i,k+l))+l(x~ Ai,k)) + l(x~tfk))+1(X~ 1~k))}
p( 1(k) )p( C (k-l)n[~ i,k)c u-~i,k)c A i.k+l)c u
Where we have used that
c ~ p( 1 (k) )p( A(k) )
(k) (k)c and analogously for 1 n 13 • So we get
( c _ c ( c c )c ( )c p(C(k-1) <; k) u<;Ck+l) uA k) uv<k> u A(k u 1 k ) < L L L L 3
k-1
~ 6/£(1 -p( 1) )-2-
and the proof is concluded by remarking that
p(1)121£ Cs+l) fdQ X(X 1 X 1 )~ 2£ + E + .::;...:...---'-"~-=-- + p ( C • ) 1-/I-)J( 1)
184
Proof of Proposition 4.1 The proof only requires a few
modifications with respect to the case E=O, which is treated
in ClJ. For xeK, qelR+' t~O, s>O, let
m z: [vJ (q,v)ec (t,t+s) x
q
which is the absolute value of the total momentum of the
particles of x which cross q between times t and t+s. We
will simply write in the following n (t,t+s) instead of q nq(t,t+s;x). We want to estimate the probability that within
timeT the h.p. reaches q (~L). For this to happen it is ne
cessary that before T the h.p. collides against the wall at
the origin; this is so because we know that the configura
tion in ~ is, at time zero, in RL' cf. Def. 4.1. Let t be
the time of such collision and v the speed of the h.p. at
that time.
Let t+s be the time when the h.p. reaches q, then, neces
sarily
MV + Es > n (t,t+s) q
(4.9)
Since the average value of n (t,t+s) is 2Ps and E<2P, q then the event in eq.(4.9) occurs only when there is a large
fluctuation either of V or of n (t,t+s). One then under-q
stands why the event defined by the inequality (4.9) has
small probability: to prove Prop. 4.1. we simply need to
make the above argument. more precise. We start fixing L
and then we consider the positions L, L+1, L+2, L+3, •••
For notational simplicity let us suppose that L is an in
teger. Given k>L we set
exp((k+1(l/ 4 ))
and
so that
We define
where
-(o) t;L c
R' k
n k>L
185
2 {(q,v)elR+: L.::_q.::_k}
k3/8
It is easy to see that
lim u0 ( n R ' ) 1 k ~00 k>k k
0 - 0
N
We then introduce sk so that
186
and we define
We have that Rk_ n Rk n R k and it is a simple computation
the proof that
c n R "> k>k k
- 0
1
cf. C1J. Proposition 4.1 is therefore proven.
5. Concluding remarks, open problems
In this section we discuss some open problems. The first
and the most interesting one, in our opinion, is to prove
that a particle which interacts with a large system at
equilibrium approaches a Brownian motion in the "usual"
space-time scaling, without the limiting assumption that
the mass ratio M/m should diverge. The one dimensional ver
sion of this problem (the large system is an ideal gas at
equilibrium which interacts with the test particle by elas
tic collisions) has been solved in the case when all masses
are equal, cf. C9J for the equilibirum case and C7J for the
analysis of more general situations.
The problem is still open when M4m. Even the ergodic
properties are not known. Notice that a dynamical system can
be defined also in the infinite case: the Palm measure which
describes the ideal gas as seen from the h.p. is stationary.
Such system is conjectured to be k. In fact in the infinite
case we do not expect the process q 0 (t), taR to be a-mixing,
as it is in the semi-infinite case, cf. Thm.1.1. The reason
is that the analogue of Lemma 1.1 does not hold. If the h.p.
187
moves like t 1 12 then it will eventually collide with par
ticles of the remote past. More precisely consider the par
ticles which have coordinate (q,v) in the region
{q~L, O>V> -!} U {q~-L, q O<V< 1
T<if}
the space coordinate are chosen in the frame when the h.p.
is at the origin.
Then for any choice of L the above region will contain
infinitely many particles, since its Lebesgue measure is
unbounded. Such particles will interact in the future with
the h.p. and if the h.p. moves like ltl 1 / 2 there will be
interactions also in the very remote past, if L is large.
The copying procedure we have used in the present paper as
well as that employed in C1J would therefore be destroyed
by collisions with such very slow particles which are spe
cified by the past history of the h.p.
Even assuming the k-property holds for the "real" in
finite case, then, even in that case, the last step (i.e.
to prove convergence to Brownian motion) might be troub
lesome.
There is an analogous problem also in the model we have
considered here. In such case of course t- 1 / 2 q 0 (t) _,. 0.
There are however other variables for which a central li
mit theorem could be stated. For instance the fluctuations
on the positions of the h.p. at the times when it gets a
negative velocity (we are thinking now of the case E=O).
Convergence to a Wiener process could be proven by obtaining
a fast decay of the a-mixing coefficient C6J. Our estimate
are too rough for such a purpose. We discussed this problem
with Magda Peligrad, and we agreed on the following strate
gy. The p-mixing coefficient is defined as, C6J,
SUI? I)J(A B)-!iA))J(B)ICL]l(A)lJ(B)J- 1 12 =: p(L) Ae<: B8~ T
188
Convergence to the normal law holds if the p coefficient
goes to zero. No way to prove that in our case, at least
with the techniques we used. So we tried the following.
(1) to prove that if the supremum is taken over all subsets
of a large set then it vanishes as T+oo (2) to prove that
in the remaining the a coefficient has fast decay properties;
finally (3) to prove that when (1) and (2) both hold then
there is a convergence to the normal law. We attacked this
problem with Magda Peligrad, she actually accomplished her
duty and solved her part (3); we did not do ours, unfortu
nately.
Acknowledgements
We are indebted to A. Pellegrinotti, Ya. G. Sinai and
R.M. Soloveitchik for many helpful discussions and comments.
We are also indebted to J.L. Lebowitz for proposing the
problem and for many helpful comments. We are indebted to
M. Peligrad for many helpful comments (c.f. the remarks in
Setion 5). One of us (A.N.) would like to thank Francesco
Guerra for the kind hospitality at the Dept. of Mathematics
of the University of Rome. Three of us (C.B., A.N. and E.P.)
also acknowledge the kind hospitality of the University of
L'Aquila, during the workshop "Hydrodynamical Behavior of
Many Particles Systems" in October of 1983.
References
ClJ Boldrighini, c., Pellegrinotti, A., Presutti, E.,
Sinai, Ya.G., Soloveitchik, R.M. Ergodic Properties
of a One Dimensional Semi-Infinite System of Statis
tical Hechanics, Preprint (1984).
189
C2J Doob, J.L. Stochastic Processes. Joh Wiley & Sons
(1953).
C3J Goldstein, s., Ianiro,N., Kipnis, c. Stationary
States for a Mechanical System with Stochastic Boun
dary Conditions (in preparation).
C4J Goldstein, s., Lebowitz,J.L., Presutti,E. Ergodic
Properties of Dynamical Systems Coupled to a Thermal
Reservoir. Colloquia Mathematica Societatis Janos
Bolyai 27, Random Fields, Esztergom, pp. 403-419 (1979).
C5J Goldstein,S., Lebowitz,J.L., Ravishankar,K. Ergodic
Properties of a System in Contact with a Heat Bath:
A One Dimensional Model. Commun. in Math. Phys. 85
p. 419 (1982).
c6J Ibragimov, T.A., Linnik, Yu.V. Independent and Statio
nary Sequences of Random Variables. Wolters-Noordhoff
Netherlands (1971).
C7J Major,P., Szasz,D. On the Effect of Collisions on
the Motion of an Atom in R1 , Annals of Probability~. p. 1068 (1980) •
C8J Ornstein, D.S. Ergodic Theory, Randomness and Dynamical Systems. Yale University Press, New Haven and
London (1974).
C9J Spitzer,F. Uniform Motion with Elastic Collision of
an Infinite Particle System, J. of Math. Mech. ~. p.
973 (1969}.
191
A GENERALIZATION OF CARATEODORY'S CONSTRUCTION FOR
DIMENSIONAL CHARACTERISTIC OF DYNAMIC SYSTEMS
Ya. B. Pesin
In this paper we propose a generalization of the clas
sical Carateodory's construction of various characteristics
of dimension type. Our approach allows us to obtain well
known dimensions (for example, the Hausdorff dimension) as
well as new ones. In particular, for invariant sets of dy
namical systems we define a class of dimensions which de
pend on the dynamics of the system. This class includes
such well-known characteristics as topological pressure and
topological entropy, and also a new notions the dimension
with respect to the map. It seems to me that the latter one
can be used for the description of the topological and geo
metrical structure of the invariant set as well as the Haus
dorff dimension. In multidimensional case we obtain the for
mulae, connecting this dimension with the characteristics
of trajectory instability of the dynamical system (such as
Lyapunov exponents). These results were obtained for the
two-dimensional case in [7]. Our construction deals with
continuous maps of non-compact subsets of compact metric
spaces. Therefore, we can also consider discontinuous maps
(they are continuous on the non-compact set which does not
contain preimages of the discontinuity set), for example,
one-dimensional piecewise monotonic maps, Lorentz attractor
(cf. [6]) and so on.
1°. Let X be a compact topological space, YcX . Let
F be a collection of open subsets in Y . Suppose that
there are three functions
192
(2Y is a set of all subsets of Y ) , satisfying the follow
ing conditions: for any zcy , UEF , a, B E lR
A1) E;(Z,U,a+B) ~ n(Z,U,B)E;(Z,U,a) ,
A2) for any E>O there is o>O such that for every
UEF with 'JI(U) ~ 0 we have
n(Z,U,B) ~ E if 8>0 and
n(Z,U,8) ~ -1 if 8<0 E
We put
M(F,a,Z,E) inf { L E; (Z,U,a) :'±'(U)~E, u U=>z} , GcF UEG UEG
where G is a finite or countable subset in F . It is
easy to see that M(F,a,Z,E) does not decrease when E
tends to 0 . Therefore there exists a limit
m(F,a,Z) = lim M(F,a,Z,E) . E->-0
We describe some properties of the function m(F,a,Z)
Theorem 1. The function m(F,a,·) is a regular Borel
outer measure on Y
1. m(F,a,<l>) 0
That is:
m(F,a,z 1 J ~ m(F,a,z 2 J , if z 1cz 2cY ( 00 I oo
mlF,a, U Z.J ~ L m(F,a,Z.) , Z.cY i=1 l i=1 l l
2.
3.
4. every Borel set is measurable, i.e.
m(F,a,Z) = m(F,a,ZnB) + m(F,a,Z,B)
for any set z and any Borel set BeY ; in addition
m(F,a,·) is a o -additive Borel measure on the o -algebra
of Borel sets;
5. for any ZcY there exists a Borel set B , ZcBcY
such that m(F,a,Z) = m(F,a,B)
The proof of this theorem is an easy modification of
the arguments given in [1]. The following assertion is the
direct consequence of our conditions A1 and A2 .
193
Theorem 2. The function m(F,·,Z) has the following
property: there exists an a 0 such that
roo 1 if M (F, a, Z) = 1
t 0 1 if
Let
dime Z = a 0 = sup{a:m(F,a,Z)=oo} = inf{a:m(F,a,Z)=O}
The value dime z is called the earateodory's dimen
sion of Z • It depends of course on the choice of the
family F and on the functions E;, n, lj! • Therefore, we
will sometimes use the notation dime,*Z , where one or
several parameters, which we would like to take into con
sideration, will be written instead of "*" .
2°. Now we formulate the basic properties of earateo
dory's dimension. Their proofs follow from the definitions
given above and Theorems 1 and 2.
Theorem 3.
1) dime ljJ = o ;
2) dime z 1 .!>dime z 2 , if z 1 cz 2 cY 1
3) dime[.~ zilJ = s~p dime zi , zicY • J.=1 J.
Let X' be a compact topological space, Y'cX' , F'
be a collection of open subsets in Y' , E;', n', l)!' be
three functions, satisfying conditions A1 and A2.
Theorem 4. Suppose that there exists a continuous map
x : X _.. X ' such that
1) x(Y) = Y' 1 2) for any Z'cY' , U'EF', aElR
we have
x- 1 (u') EF, E;'(Z',U',a) = Ux- 1 !Z'),x-1 (u'),a),
n'(Z',U',a) = n!x- 1 !Z'), x- 1 !U'),a),l)!'(U')=l)!(x-1 (u')l
Then
dime F I " I I ,,, I z I ;;:: dime F " ,,, x- 1 ( z I ) , ,s ,n to/ , ,s,n,'+'
194
Moreover, if x is a bijective map and x(U) EF' for
any UEF , then
dime F' ~· , ,,, z• =dime F ~ ,1, x- 1 (Z') , ,s ,n '"~' , ,s,n,'+'
3°. We introduce another general class of characteris
tics of dimension type which are also used in the applica
tions as well as the Carateodory's dimension. Let again X
be a compact metric space, YcX , F be some collection of
open sets in Y , [,, n, 1jJ be three functions, satisfying
conditions A 1 and A2 • We put
R (F, a, Z, E) = inf { I [, ( Z, U, a) : '¥ (U) = E , GcF UEG
ll U=>Z} UEG
where G is a finite or countable subset of F . We denote
r(F,a,Z)
E_(F,a,Z)
lim R(F,a,Z,E) E+O
lim R(F,a,Z,E) E+O
One can show that the functions r(F,·,Z) and
E_(F,·,Z) have the property described in Theorem 2 and de
fine the values which we call respectively upper and lower
Carateodory's capacity of Z and denote by
Cap Z c
inf{a:r(F,a,Z)=O}
Cap Z = inf{a:E_(F,a,Z)=O} . ---=-c
It is obvious that dime z ~ ~ z ~ Cape z .
4°. We consider some examples. Let X be a compact
metric space, Y=X , F be the collection of all open sets
in X . For ZcX , UEF alOlR we put
[,(Z,U,a) n(Z,U,a) (diam U)a , '¥(U) = diam U •
The dimension dim Z is the Hausdorff dimension C,F,[,,n,'¥ of Z (cf. [1]). We denote it by di~ Z •
5°. Let X be a compact metric space, YcX, f:Y+Y
be a continuous map, ¢: X+ lR be a continuous function.
Let also U be a finite open cover of X . Denote by
195
u = u .... u. ~0 ~m
a collection consisting of the elements of
the cover U ( m=m(U) is the length of the collection); W
(U) is the set of all such collections. We put
( 1 )
F+ = {y+ (!::!_) :UEW (U)} ,
E; (Z, y+ (!::!_),a) = exp [-am(!:!_)+ sup+ :I> (f<(x)) J (2)
xEzny (!::!_)
+ n(Z,Y (!:!_),a)~ exp(-am(!:!_)) ,
~(Y+(Qll = [m(Qll- 1 •
It is easy to see that the functions E;, n, ~ satisfy
conditions A1, A2. The dimension dime Z depends also on
the map f , the function ¢ and the cover U . We denote
it by
One can show that there exists the limit
+ + Pf,Z(¢) = lim Pf,Z(U,¢)
diam U->-0
It is called the topological pressure of the function
¢ on the set Z (with respect to f ) . Our definition of
the topological pressure includes the case of discontinuous
maps and also
Z • The value
entropy of Z
the case of a noncompact and noninvariant set
h~ (f) = P;,Z(O) is called the topological
(with respect to f ) .
6°. The definition of the topological pressure for non
compact subsets of compact metric spaces were introduced by
B. S. Pizkel and the author in [4]. In that paper the prob
lems of variational principle for the topological pressure
and the existence of equilibrium states were also consid
ered. We formulate some of the results obtained there.
Theorem 5. Let f be a continuous map of a compact
metric space X . Then the definition of the topological
196
pressure given above is equivalent to the "usual" one (cf.,
for example [2]), and the definition of the topological en
tropy is equivalent to Bowen's definition (cf. [3]).
Denote by Mf(X) the set of probability Borel f -in
variant measures in X by Hf (Y) the set of JJ EMf (X)
satisfying JJ(Y) = 1 and similarly Mf(Z) for an f -inva
riant subset zcy .
Theorem 6. For JJ EMf (Y)
h (fjY) + J ¢dJJ ~ P;,Y(¢) jJ y
Let n-1
xEY • Consider the sequence of measures JJx,n
n L o k , where o k=O f (x) Y
is measure concentrated on y
Denote by V(x) the set of limit points (in the weak
topology) of the sequence JJx,n It is easy to see that
V(x)cMf(X).
Theorem 7. Let ZcY be an f -invariant subset, z 1 = = {xEZ:V(x)nMf(Z)#~} . Then for any continuous function ¢
on X + sup (h tfiZJ+/¢dJJ) = Pf z (¢)
JJEMf(Z) lJ Z ' 1
Consequence 1. Suppose that for any xEY the intersec-
tion V(x) nMf (Y) -# ~ • Then
sup (h (f:Y)+!¢dJJ) JJEMf(Y) lJ Y
Let JJ E Hf (Y) be an ergodic measure. Denote by
the union of all forward generic points for lJ , GJJ
JJx,n weakly tend to ]J}
Consequence 2. For any ergodic lJ E Mf (Y)
+ h (f[Y) + !¢dJJ = Pf G (¢) . lJ X ' lJ
The measure )J=JJ¢ is called the equilibrium state for
the function ¢ if ]J¢ EMf (Y) and
h (f[Y)+f¢dJJ_. JJ¢ X "'
sup (h (fiYl+f¢dJJ) JJEMf(Y) lJ X
The next assertion is the generalization of Bowen's
197
criterion [2].
Theorem 8. Suppose that f satisfies the following
conditions:
1) f is a homeomorphism of Y
2) f separates the points of Y
3) the set Mf(Y) is closed in the weak topology in
Mf (X) •
Then for any continuous function ¢ on X there
exists the equilibrium state ~¢ .
7°. We put
Y-(.Q_) = {xEY:f-k(x) EU. , k=O, ••. ,m(.Q_)}, ~k
It is easy to see that the topological pressure
P; z(¢) constructed using the collection F and the I
functions ~' n, ~ (defined by (2) with -k instead of
k ) coincides with P+ (¢) f- 1 ,z
We put for !:!_ 1 ,!:!_2 E W (U)
{xEY: f-k(x) EU~ , k=1, ... ,m(U 1 ), ~k
fk(x) E u~ I k=O, ... ,m(.Q_2 )-1} ~k
Let also
1 2 1 2 F = {Y (!:!_ I!:!_ ) : !:!_ I!:!_ E w ( u) } •
Denote by Pf,Z(¢)
using the collection F
S. Pizkel has shown that
the topological pressure, defined
as pointed out above. Recently B.
p f 1 z ( ¢ ) ~ min { p; 1 z ( ¢ ) I p; 1 z ( ¢ ) }
and constructed an example of a noncompact f -·invariant
set (where f is a subshift of finite type) for which
these three values are different.
8°. We consider some more examples of Carateodory's
dimension. Let X be a compact metric space with the met
ric p , YcX and let f :Y + Y be a continuous map, v a
198
probability Borel measure in X , for which v (Y) = 1 . Let
WcY be some f -invariant subset. Fix o>O and put for
xOl
k k Y0 (n,x) = {yEY: p(f (x), f (y)) ~ o, k~O, ... ,n}
Suppose that the map f satisfies the following con-
dition A3) for every small y>O there exist the sets (y)
wk , k=O, 1 , 2, •••
a) Whl ~whl b) k ~ k+1 ,
having the following properties,
u w~Y) = vl , c) for any k~O
k.<:O ,
E>O there is N=N(k,E) such that for any xEWk and n>N
V (Y 0 (n, X) ) ~ E •
We introduce condition A3 because inequality (3) usually
does not hold uniformly in xEW for diffeomorphisms of
general type. We put
(3)
(y) s0 ,k(Z,Y 0 (n,x),a) = (y) a
n0 ,k(Z,Y 0 (n,x),a) =V(Y 0 (n,x)) ,
(y) -1 ~o,k(Y 0 (n,x)) = n
It is easy to see that the functions
~1~~ satisfy conditions A1, A2 and define
dimension which we denote by dime s k Z I lJ I tY
ZcW we call the value
s1~~ , n1~~ , Crateodory's
Z c ,,,(Y) , "k . For
dimf z =lim sup lim dimc,o,k,y(H~y) n Z) y-+O k<:O o-+o
the dlinension of Z with respect to f . It has the proper-·
ties formulated in Theorem 3.
In a similar way one can define upper and lower Cara
teodory's capacities of the set Z with respect to the map
f . We denote them by Cf(Z) and ~f(Z) respectively.
Now let ~ be a Borel measure on X for which
~ (W) = 1 (we do not suppose here that ~ , v are f -inva
riant, but later on we will consider only the case where ~
is f -invariant) . The values
199
dimf ~ = inf{dimfz : zcw, ~(Zl~1} ,
cf(~l lim inf{cf(Z) Zc\v, ~(zn1-o} o+o z
lim inf{~f (Z) o+o
ZcW, ~ ( Z ) ~ 1 - o }
are called respectively measure dimension with respect to
f , upper and lower capacity with respect to f
Theorem 9. Let ~, v be Borel measures in X , for
which ~(W)=1 , v(Y)=1 . Suppose that the topological dimen
sion of X is finite. Assume also that there is ZcW
~(Z)=1 having the following properties: for any
k~O there exist a
and x E w(Yl k n z
~ (o ,yl ~ lim n+oo
log ~ lim
n+oo log
o > 0 such that for any k,y
log ~ (V~y) (n,x)) ~
log v(V~y) (n,x))
~(v?l (n,x)) ~ ~ (o ,yJ
~(V~y) (n,x))
y>O ,
o~ok ,y
Let also ~ ~ a(o,y) ~ ~(o,y) fa for any o, y. Then
Consequence 3. If the hypotheses of Theorem 9 hold and
there exists the limit
lim lim ~ ( o , y) y+O o+O
lim lim a(o,y) y+O o+O
a ,
then dimf ~
10°. Let f be a diffeomorphism of class of a
smooth compact Riemannian p -dimensional manifold M ; let
~ be an f -invariant ergodic Borel measure with nonzero
Lyapunov p
~ ••• ~ X~
1 k k+1 exponents (cf. [8]) x~ ~ ... ~ x~ > 0 > x~ ~
We put X=H , Y=X • Let W
the sense of Lyapunov) points in 1 i . 1 (y) equa to X~ , 1= , ••• ,p, Wk
uniform estimates (cf. [8]); v
be a set of regular (in
X whose exponents are
be a set of points with
be a Riemannian volume in X .
200
Theorem 1 0. There is a set ZcW , satisfying the hypo-theses of Theorem 9 and Consequence 3, and
k i a = h ]J (f) I L X lJ , (4)
i=1
where h ]J(f) is the metric entropy of f
So, the measure dimension, upper and lower capacities
with respect to f are equal to the expression in the
right side of (4). If we consider the map f- 1 instead of
f then one can prove that
dim 1 lJ = h (fl I I I xi i f- ]J i=k+1 ]J
It is easy to show that dim _ 1 lJ coincides with the f
measure dimension constructed by means of the collection
F 0 , formed by the sets
-k -k Y0 (n,x) = {y Y:p(f (x), f (y)) ~ o, k=O, ... ,n}
-The differences between the collections F 0 and F 0 involve the differences between dimf lJ and dim _ 1 1J
f Therefore, it may be interesting to study the measure dimen
sion with respect to f constructed with the help of a sym
metric (with respect to forward and backward iterations)
collection of sets
However, I do not know any results for such a dim en-·
sion.
11°. In some cases it is interesting to know not the
dimensions of the set itself, but of its cross-section in
some direction. For example, in the case of a hyperbolic
attractor one calculates the dimension of its intersection
with the stable layer because it has Cantor structure in
this direction. The dimensions dimf lJ and dim _ 1 lJ meas-f
ure just the dimensions of the invariant sets in the direc-
tion of unstable or respectively stable layers (it is con
nected with the choice of F 0 and F 0 ) . Another approach
201
to the definition of the dimension with respect to a map in
the direction of unstable or stable layers is given in [5] .
In that paper there are formulae similar to (4) (and
also other results) for the calculation of this dimension
for locally maximal hyperbolic sets.
Results similar to (4) were obtained recently in [9]
for attractors of Lorenz type.
References
[1] Federer H. Geometric Measure Theory. Berlin, SpringerVerlag, 1969.
[2] Bowen R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lect. Notes Math. V. 470, p. 108, 1975.
[3] Bowen R. Topological Entropy for Noncompact Sets. Trans. Amer. Math. Soc. 184, 125-136 (1973).
[4] Pesin Ya. B., Pizkel B.S. Topological Pressure and Variational Principle for Noncompact Sets. Funct. Anal. and Its Appl. (in Russian), 1984 (to appear).
[ 5] Pes in Ya. B. On the Notion of the Dimension vJi th Respect to a Dynamical System. Ergod. Theory and Dyn. Syst. 1983 (to appear).
[6] Afraimovich V. A., Pesin Ya. B. Ergodic Properties and Dimension of Lorenz Type Attractors. Proc. of Intern. Symp. of Non-linear and Turbulence Processes in Phys. Kiev, 1983.
[7] Young L.-s. Dimension, Entropy and Lyapunov Exponents. Ergod. Theory and Dyn. Syst. N 2, 109-124 (1982).
[8] Pesin Ya. B. Characteristic Lyapunov Exponents and Smooth Ergodic Theory. Russ. Hath. Surveys N 32, 55-114 (1977).
[9] Afraimovich V. A., Pesin Ya. B. The Dimension of the Attractors of Lorenz Type (to appear) .
203
CONVERGENCE OF IMAGES OF CERTAIN MEASURES
M. Misiurewicz and A. Zdunik
Abstract. Let X be a compact metric space, let
f: X->- X be an expansive homeomorphism satisfying the speci
fication property and let ~ be an invariant probabilistic
totally ergodic measure with the whole X as a support. We
prove that then there exists a probabilistic measure K
with supp K = X such that f*K is absolutely continuous
with respect to K and the averages n-1 .
to ~ , but the averages -n L f 16 i=O * X
1 n-1 . - L f 1 K converge n i=O * are dense in the set
of all invariant probabilistic measures for K -almost every
X •
0. Introduction and statement of the result
We start with some motivations. For a diffeomorphism
<P of a compact manifold M (with or without a boundary)
into itself there are usually many invariant probabilistic
measures. Then one would like to know whether any of them
are of special interest from some point of view. Certainly,
the Lebesgue measure or any measure equivalent to it (at
least this equivalence class is well defined on M ; we
take one of its representants and denote it by A ) would
be such measure. If none of these measures is invariant or
even if it is not obvious that one of them is invariant, we
can use the following procedure, motivated by the ideas
from physics. We choose a point xEM "at random"and look
at the sequence of averages
n-1 . n: I <P 16
j=O * X ( 1 )
204
where ox is the probabilistic measure concentrated at x
and ~* is the transformation on measures induced by ~
i.e. for a measure ~ and a Borel set A , (~*~)(A) =
= ~(~- 1 A) If for A -almost every x (this explains what
"at random" means) this sequence converges in the weak -*
topology to some measure ~ , then ~ is our special meas
ure. Existence of such measures is known in some cases (see
e.g. [4)).
By integrating the averages with respect to A , we
get n-1 . I ~*~A~~ as n + oo •
n i=O (2)
This fact alone is also interesting. Such convergence is
often used, for instance for finding absolutely continuous
invariant measures for interval maps (see e.g. [3)). In such
a way, a problem arises: whether (under some reasonable con
ditions on the system (M,~,~) ) it can happen that (2)
holds but there is no convergence of (1) to ~ for A -al
most every x •
The conditions imposed on the system should eliminate
trivial examples, like the measure ~ concentrated at two
points y and z and the averages (1) approaching o and y o2 on subsequences. From this point of view, the assumption
(iii) below seems reasonable.
In this paper, we consider the problem similar to the
above one, but in a more general case. Instead of the mani
fold M , a diffeomorphism ~ and a Lebesgue measure A , we take a compact metric space X , a homeomorphism f and
any measure K • Since we have less assumptions, it is easier
to find an example. However, we obtain the result much
stronger than the example. Now we shall state this result
precisely.
Let (X,d) be a compact metric space, ~ a probabi
listic measure on X , f: X~ X a homeomorphism such that:
(i) f is expansive: there exists E>O such that if
x,y EX , x'ly , then d (fnx,fny) > E for some integer n
(ii) f satisfies the specification property (cf. [1)
Def. (21.1)): for every o>O there exists a positive in-
205
teger a(o) such that for any sequence of trajectories
and integers
point y of
a 1 ,a2 , .•. ,ak ~ a(o) there exists a periodic
period a 1+ ... +ak+t 1+ ••. +~ such that
i m p-1 d(f x ,f y) < o where m= i+ L (t.+a.) for Hp~k , O~i<tp
p j=1 J J
(we say that the orbit of y o -specifies the trajectories
[ . t -1)k
(f 1 xp) i~O p=1
(iii) ~ is f -invariant, totally ergodic (i.e. er
godic for all fn , n~1 ) and supp ~=X (i.e. ~(A)> 0 for
every non-empty open set A ) .
We consider the space ffi(X) of all probabilistic meas
ures on X and its subspace ffi(X,f) of all probabilistic
f -invariant measures on X , both with the weak -* topology.
Theorem. If X, f, ~ satisfy conditions (i)-(iii),
then there exists a measure K Em (X) such that
(iv) supp K X I
(v) f*K is absolutely continuous with respect
n-1 41K (vi) lim n L: = ~ I
n->-oo i=O
(vii) for K -almost every point xEX I the set
points of condensation of a sequence [ n-1 loo
.!_ I fio I n L * X i=O Jn=1
equal to ffi(X,f)
to
of
is
K I
all
To prove Theorem, we construct (using coding technique)
a Borel map g: X->- X such that the measure v = g * ~ has
properties (vi) and (vii) (with K replaced by v ) . Then
we show that the measure K = L 2-k- 1 ~v has properties k=O
206
(iv)-(vii).
We denote by N the set of all non-negative integers.
1. Decomposition of N
Suppose we have four sequences of positive integers
(cn):= 1 1 (in):=1 1 (p(k) )~= 1 and (r(k) )~= 1 and that the
sequence (in):=1 is increasing.
We shall call a finite set of consecutive integers a
block and denote by 1·1 its length (i.e. the number of
elements). We divide N into consecutive blocks T1 1T 21···
(which we write N = T 1 T 2 • . . ) 1 then we subdivide each
0 T: T1=R1R2 .•. Ri1 IT =CR. +1R. +2 ... R.
n n n ~n-1 ~n-1 ~n for n>1
and at last we subdivide each Rk
In such a way we obtain
P1c111 Pr(1lc11r(1) P1 ci1 11 Pr(i1)ci1 1r(i1) N = 1 1 . . . 1 1 ... i 1 1 ... i 1 1
1 in_1+1 11 co P~ +1cn
n ~n-1
1 i 1 1 P. c n ~ n
n
Ri +1 n-1
r(i ) i 1r(i ) P. n c n n ~ n n
We advise the reader to consult this scheme often
while reading the paper.
we require that lc~l = lc~~j I =en and \Pal =p(k)
207
Hence, the sequences (cn)~= 1 , (in)~= 1 , (p(k))~= 1 and
(r(k))~= 1 determine the whole division described above.
2. Sequences
Our four sequences have to satisfy certain conditions.
In order to state them, we set
k
v(k)
u(k) =I p(j) for k=1,2,3, ... , j=1
jR1 i+ ... +]1\-1 j if l\cT1
jT1 i+ ... +]Tn_1 j+jc~j+[Ri +1 ]+ .•. +]Ik_1 ! if IkcTn and n~ 2 n
for k=1,2,3, ..• ;
A(x,n) = {zEX d(f jx,fjz) < E f · 1 } or J = , ••• , n
for xEX and n>O ( E is the constant of expansiveness);
B(x,y) = {k>O: fu(k)yEA(x,p(k+1))} for x,yEX.
The conditions are the following:
(S 1) c ~ a(2-n- 1E) n for n=1,2, .•. ,
(S2) c n+1 ~ c n for n=1 ,2, ... ,
(S3) i n+1 ~ i n for n=1,2, ...
c (S4) n 0 lim (. )
n-+oo P 1 n-1
(SS) p (k+1) ~ p(k) for k=1,2, ... ,
(S6) lim p(k) = 00 , k-+oo
(S7) for ll -almost all yEX and all periodic points xEX
the set B(x,y) is infinite
(S8) lim v(k)
= 0 k-+oo p(k)r(k)
We have to show that such sequences exist. The existence
of a sequence (cn)~= 1 satisfying (S1) and (S2) is obvious.
Once sequences (cn)~= 1 and (p(k))~= 1 is satisfied, the existence of a sequence
are chosen and (S6)
(in)~= 1 satis-
208
fying (S3) and (S4) is also obvious. Then a sequence
(r(k))~= 1 satisfying (S8) can be defined easily by indue-
tion (notice that v(k) depends only on
Thus, the only problem is to show that a
satisfying (S5)-(S7) exists.
r(j) with j<k ) .
sequence (p(k) )~= 1
Since f is expansive, so is fn , and thus the set
of fixed points of fn is finite. For positive integers
m, n set D (n ,m) = {yEX: if fnx = x then there exists
jE{0,1, ... ,m-1} such that fjnyEA(x,n)
Since supp f.l =X, we have f.l(A(x,n)) >0. Since f.l
is totally ergodic, by the ergodic theorem for fn there -n exists mn>O such that f.l(D(n,mn)) ~ 1-2 . Now we set
n-1 n p(k) = n if }:m.<k~ }:m ..
j=1 J j=1 J
Clearly, (SS) and (S6) are satisfied. In order to prove (S7), -t.
fix n and x such that fnx = x • If y E n f J.D(in,m. ) , i=1 m
where t. = u[i~- 1 m.) , then for each J. j=1 J t.+k.in
tion of D(in,m. ) , we have f J. J. J.n
kiE{0,1, ..• ,min-1} (if fnx=x then
and u(k)
i~q , by the defini-
yEA(x,in)
finx = x
for some
for all i ).
we have t.+k.in = ]. ].
By the definition of p(k)
u[ii- 1m.J+k.in = u(q.) j=1 J ]. ].
and p (qi +1) =in for some
in-1 ~ L m.
j=1 J By the definition of B (x,y) , this belongs
in-1 to B(x,y) Since lim L m. = oo , this proves that for
i -+oo j = 1 J such y the set B(x,y) is infinite.
Since the measure f.l is invariant, we have for each n
lim inf f.l[.~ f-tiD(in,min)) ~ q-+oo J.=q
~ lim inf [ 1 - I ( 1 - f.l ( D (in , m. ) ) ) ) ~ . J.n q-+oo J.=q
~ lim inf [ 1-. I 2 -in J = 1 , q-+oo J.=1
and consequently (S7) is satisfied.
209
3. The map g
We shall define the map g as a limit of Borel maps,
attaining only finite number of values each. We proceed by
induction.
To define g 1 , for xEX consider a sequence of tra
jectories
.t: u (1) (X 1 LX, ••• ,f X)
u ( 1) (x,fx, ••. ,f x) l j
...,
I J
u(i 1-1)+1 u(i 1 ) {f x, ... ,f x)
~
I I I
u(i 1-1)+1 u(i 1 J j' (f x, ... ,f x)
r(1) times
r(2) times
r(i 1 )times
Denote this sequence by G1 {X) • Set u1 = X
k k -3 {yEX: d(f x,f y) <2 E: for all k with Os;k~u(i 1 J}
The family
There exists a finite subcover
(see e.g. [2], proof of Theorem 1 k1
Borel partition {Aj}j=1 of X
cover of a compact space X . 1 k1
{Uj}j=1 It is easy to show
4.2) that there exists a
such that A~c:U~ J J
and
]..1 1 a A~ 1 = o J
ary of A
for j=1,2, ..• ,k1 , where 3A denotes the bound
We may assume that all sets A~ are non-empty.
Then we can fix a point x 1 E A 1 for each j • From (S 1) it J J 1
follows that there exists a periodic point g 1 (x.) of period
IT 1 1 which orbit 2-2 r -specifies the sequence JG 1 (x~) we set for all By the definition of
u\, for each yEX the orbit of g 1y 2-1r -specifies G1 (y) xj
Note the connection between the above construction and
the structure of T1 •
210
k {A~- 1 }.n- 1 and the
J ]=1 Now we assume that the partition
map gn_ 1 are already defined, and that gn_ 1 is constant
n-1 on the sets Aj . We consider the open cover of the clos-
n-1 ure of Aj by the sets n k k -n-2 Ux= {yEX: d(f x,f y) < 2 £for
all k with 0 ~ k ~ u (in) } . Then we choose its finite sub
cover and take a finer partition of
with boundaries of measure zero. By
these partitions over j from
n-1 Aj into Borel sets
taking the union of
to kn_ 1 , we obtain a
partition k
{A~}i~1 k
finer than {A~- 1 }.n- 1 such that for J ]=1
each i 1 A~ c: un l. X for some x and J.l ( ClA~) = 0 • Again we can
assume that each A~ ].
n n is non-empty and fix a point xi E Ai
To define gn ,
jectories Gn(x) :
for xEX consider a sequence of tra-
( JT1 J+. · .+JTn_1 1 )
gn_1 x,f (gn_1x), ... ,f (gn_1x)
[ u(in-1)+1 u(in-1+1) I l f X1 ••• ,f XJ 1
.[ . ~ ~ ~~ ~ ~ ; : ~ ....... ~ ~ ~~ ~ ~ : ~ ; . ). Jl f x, ... ,f x
J
r(in_ 1+1) times
r(in) times
From (81) it follows that for each i there exists a
periodic point gn(x~) of period [T 1 J+ ... +:Tnl which
-n-1 n orbit 2 £-specifies the sequence Gn(xi) • We set gny=
gn (x~) for all yEA~ •
If yEA~ then by the definition of U~ we have
k k n -n-1 d(fy,fxi)<2 £ forall k with O~k~u(in) .Since
is finer than k
{A~- 1 }.n- 1 , we have J ]=1
211
Therefore for all yEX the orbit of gny 2-nE -specifies
Gn (y) •
Note again the connection between the above construe
tion and
Set
the structure of the blocks j
P = U Pk , C = N'P . For k, j
T1,T2, ... ,Tn .
mEP define s(m)
u(k-1)+i if m is the i -th element of Pa Let yEX
Since the orbit of gny 2-nE -specifies Gn(y) , we obtain:
if mET 1u ••• UTn_ 1 then d(fm(gny) ,fm(gn_ 1yJ) < 2-nE ,
if mETn and mEP then d(fm(gny),fs(m)y) < 2-nE .
We can draw two conclusions from this.
The first conclusion is that for all n and y ~ 00
d(gny,gn_ 1yJ < 2 E . Hence, the sequence (gny)n= 1 con-
verges. vie set gy = lim g y . The map g is Borel as a n-+oo n
limit of Borel maps.
The second conclusion is that if mEPnT. and jSn
then d(fm(gny) ,fs (m)y) ~ (2-j+2-j- 1+ ... +2-n) ~ . Therefore
4.
Jl
if mEPnT. then d(fm(gy), fs(m)y) ~ 2-j+1 E (3) J
The measures
Set
lim .l n
n-+oo
v = g*Jl . We are going to prove that
n-1 i L f*v •
i=O
Fix a Borel set A with Jl ( 8A) = 0 . For n>O denote
A+n = {xEX: dist(x,A)5n} ,
A-n = X'{XEX: dist(x,X,A)5n}
Fix y>O . Since Jl (<lA) = 0 , there exists o>O such that
11 (A+o J < 11 (A) +y and 11 (A-o) > 11 (A) -y
Now we choose n 0 EN such that
-n +1 2 O E ( 0 (4)
and
< y for all j~k0 (5)
whera
xEX
then
212
n 0 E Tk0
(this can be done by (S4} j •
It follows from (4} and (3) that if n~n0 , nEP
then d(fn(gx) ,fs(n)x) < o • Hence if
f-s(n} (A-o} cg-1 (f- 1A) cf-s(n) (A+o}
and
n~n0 and nEP
. Therefore for
n~n0 if nEP then
\1 (A) -y < \1 (A-o)
s \l(f-s(nl (A+o}) = \l(A+o) < \l(A) +y.
Since \l(g- 1 (f-nA)) = (f~(g*\1)) (A) = (f~v) (A) , we obtain
I (f~v) (A)-\l(A) i < y for n~n0 , nEP . (6)
We have to estimate the number of elements of C be-
tween n 0 and n-1 • The set
into blocks of elements of C
{n0 ,n0 +1, •.• ,n-1} is divided
and P , as in Section 1 (we
call them C -blocks and P -blocks respectively). If the
first or/and the last block is shorter than in Section 1
(i.e. if n 0 or/and n-1 is not the first (resp. last)
element of the corresponding block from Section 1), we
either take the whole Section 1 block - if it is a C -block,
or omit it - if it is a P -block. After such modification,
we shall have perhaps more elements of C and less elerents
of P . If n is sufficiently large, we have at least one
P -block. Then to each of considered C -blocks we assign one
of considered P -blocks (the closest one to the left or to
the right). If the length of a C -block is cj then the
length of the assigned P -block is p (m) for some m~i. 1 J-By (5) and (SS) their ratio is smaller than y • Since there
are no more than 2 consecutive C -blocks, to each P -block
we assigned at most 4 C -blocks.
Hence, we get for n sufficiently large
Card ({n0 ,n0 -r1, •.• ,n-1}nc)
Card ({n0 ,n0 +1, •.• ,n-1} n P) < 4Y '
and consequently
Card ({n0 ,n0 +1, ... ,n-1} n C)
n-n 0
< 4y •
213
Thus, in view of (6), 1n=1 . j 11 L (f!-v) (A) -nf.l (A)
we obtain for n sufficiently p,-1 .
~ l: I {f!-v> (A) -f.l (A) J ~ large i=O I i=O
n-1 . ~ n + 4y (n-n ) + L i (f!-v) (A) -f.l (A) [ .:; n0 + Sy (n-n0 )
o o i=n
iEP0
I n-1 . , lim sup
1i .L (f!"vl (A)-f.I(A) 1
1 ~ Sy • Since y>O was
n+oo ~=0
Hence,
chosen arbitrarily, we obtain
n-1 . lim ..l L (f!-v> (A) = f.l (A) • n+oo n i=O
Since A was an arbitrary Borel set with f1 (<lA) = 0 , we 1 n-1 .
obtain lim- L f!-v = f1 in the weak-* topology (see e.g. n+oo n i=O
[1], Proposition (2.7)).
n-1 . 1 l: f\s n i=O * Y
5. The measures
We start by proving that for every Lipschitz continuous
function ¢: X+ R and every yEX
( v(k+1)-1 lim l·-v~(~~~~1~1 m=LO (f~(g*oy)l (¢) -k+oo •
( 7)
1 u (k) . ) - p(kJ l: <f!-o J <<Pl = o •
i=u(k-1)+1 Y
Let ¢ satisfy the Lipschitz condition with a constant
L . Ive fix n>O and then take k 0
such that
c n < n if ~c:Tn and k>k (8) P (in-1) - 0
v(k) < n if k.l!k0 (9) p(k)r(k)
1 if k~k0 ( 10) P (k) < n . Such k
0 exists by (S4) ' (S8) and (S6). We have
m 0 (¢ofmog) = ¢ (fm (gy)} ( 11 ) (f* (g*oyl > (¢l y
and
214
( 12)
Consider a block Pj~ For all mE Pj we have by k n k
(3) J<P(fm(gy))-<jl(fs(m)y)j~2-n+ 1 £L, and hence
L J<P<fm(gy))-<jl(fs(m)y) i ~ 2-n+ 1£L·p(k) • (13)
rnEPj k
By arguments similar as in Section 4, we get in view of
(8} if k<::k0
Card ({v(k),v(k)+1, ••• ,v(k+1)-1}nc) v(k+1)-v(k) < 2n
(the estimate is sliqhtly better now because of the defini
tions of v(k) and v(k+1) • Thus,
v(k+1)-1 L [<P!fm(gy)) I .S 2nv(k+1)J<PI ( 14)
m=v(k) mEC
(where I ·I is the sup norm).
If m runs over the set {v(k),v(k)+1, ••• ,v(k+1)-1}np, r(k) .
U P~ • Therefore from (11)j=1
then s(m) runs over the set
(14) we obtain for k~k0
lv(k+1)-1 u(k)
1 I (f~(g*oy) l <<Pl - r (kl L (f;oy> <<Pl ~ m=O t=u(k-1)+1
,v(k+1)-1 u(k) , ~ ! I <P <fm(gyl > -r <k> I <P <tty> I +
1 m=v (k) t=u (k-1) +1 1
mEP
v(k+1)-1 v(k)-1 + I i <P < fm < gy > > i + I I ¢ < fm < gy > > I ~
m=v (k) m=O mEP
~ 2-n+1£·L p(k)r(k) + II<PII (2nv(k+1)+v(k))
Since p(k)r(k) <!~I sv(k+1) and by (9), we obtain
I 1 v(k+1)-1 m r(k) u(k) t ' v(k+1) L (f*(g*oy)) <<PJ -v(k+1l l (f*oyl<<t>>j s;
I m=O t=u (k-1) +1
-n+1 . v (k) -n+1 s 2 £L+II<PU<2n..- p(k)r(k))5 2 £L+3nii<PI •
215
Notice that in vi.ew: of (8) and (9)
p(k)r(k) = (v(k+1)-v(k)-p(k)r(k)+v(k) ::; - v (k+1) v (k+1)
cnr(k)+cn+J+v(k) 2cn+J ::; p(k)r(kJ < ----- + n < 3n
p(k) Hence,
I 1 v (k+1 ) ~1 m 1 u (k) t I v(k+J) L (f,;(gf,oy)) (q,) -p(k) I (ff,oyl (q,) ::;
ro=O t=u(k-1)+1
u(k) ::; 2-n+1 EL+3niiH +P~~) L jq,(fty) I :52-n+1 EL+6nUq,ll .
t=u(k-1)+1
Since n+oo as k+oo and n>O was arbitrary, we obtain (7).
Now we fix a point xEX such that fnx = x for some 1 n and set Ax= -(<5 +of + ... +o 1 ) n X X fn- X
We shall prove that
for J.l -almost all yEX the sequence [1 m-1 i loo
rn .I f*<g* 6y>Jm=1 ~=0
A X
has a subsequence convergent to
Take y for which the set B (x ,y) is infinite (see
Section 2). Let q, be as before. Fix n, 6 >0 . Since f is
expansive with the constant
teger t such that for each
for all i with lil<t then
lary 1, p. 109).
E , there exists a positive in
z1 ,z 2 EX if d(fiz1 ,fiz 2) < E
d(x,y) <n (see [1], Corol-
Assume that kEB(x,y) and
t p(k) ~ 6 . ( 1 5)
For all m with u(k-1)+t+1 :5m:5u(k)-t we have (by the definitions of B(x,y) and A(x,n) d(fm+jy,fm-u(k- 1 )+jx) <
< E for all j with ljl<t and consequently
( 1 6)
Thus, we have in view of (15) and (16) (if 6 is sufficient
ly small)
I 1 p (k) . 1 u (k) I p(k) ) (f,/ox) (q,) - p(k) L (f~oy) (rp)
~=1 m=u(k-1)+1
I u(k) u(k) ~ L <P (fm-u (k-1) x) - L <P (fmy) I ::;
PTkT m=u(k-1)+1 m=u(k-1)+1
216
[ u(k)-t
$ p/k) L L d(fm-u(k-1)x,fmy) + 2tU cpu) $
I1FU (k-J) +t+ J
1 $ p(k) (Lp(k) n + 2p(kl en H l = Ln + 2eu cpu •
Since e>O is arbitrary, the set B(x,y) is infinite and
lim p(k) = oo , we obtain k+oo
[ 1 p(k) i
lim p(k) .L (f*cx) (cp) k+oo 1=1
1 u(k) ) -p(k) L (f~cy)(cp) =O.
m=u(k-1)+1 kEB(x,y)
Together with (7) this gives
[ 1 v(k+1)-1 1 p(k) . ) lim v(k+1 ) l: (f~(g,.,cy)) (cp)-p(k) . (f~cx)(cp) k+oo m=O 1=1
kEB(x,y)
Since
therefore
lim k+oo
kEB(x,y)
v (k+1) "-x ( cp)
0 •
and
This equality holds for every Lipschitz continuous
function cp • But Lipschitz continuous functions are dense
in the space of all continuous functions on X (this fol-
lows for instance from Stone-Weierstrass' theorem) and it
follows easily that this equality holds also for every con
tinuous function cp: X + R . Therefore
lim k+oo
kEB(x,y)
v(k+1) A X
The space ffi(X,f) is metrizable. Since the maps
z ~cz and f are continuous, so is the map
n-1 .
( 1 7)
'- I f 1 c for each n • Therefore the sets E(x,n,m) = z r n L ,., z i=O
{zEX:
the set
sequence
dist[l ni1 f;cz "-x) < k} are open, and consequently n i=O
F(x) = {zEX: "-x is a point of condensation of the
ll 1 f~o = r n-1 . ) "' } n '-' z \ i=O !n=1
217
n n v E(x,n,m) m=1 k=O n=k
is a Borel set. By (17) and (S7) and since
we have v (F (x)) = 1 .
Denote F = {zEX: the set of all points r1 n-1 . ]"'
of the sequence l- L f 1 o is equal to n i=O z n=1
of condensation
ffi(X,f)}
It is clear that every point of condensation of such sequence belongs to ffi(X,f) . On the other hand, if z be-
longs to F(x) for all periodic points
set {>.x: x is periodic} is dense in
x then, since the
ffi(X,f) (see [ 1] ,
Proposition (21 .8)), z belongs also to F . Consequently,
F = n{F(x) x is periodic} . The map f is expansive and
therefore the set of all periodic points is at most countable. Hence, F is a Borel set and v (F) = 1 •
6. The measure K
In previous sections we have constructed the measure v Em (X) satisfying the properties (vi) and (vii) (with K replaced by v ) .
\ -k-1 k We set now K = L 2 f*v . Clearly, K Em (Xl • It k=O
is also clear that K satisfies (v) . We shall show that it satisfies (iv). If U X is an open non-empty set then there exists a continuous non-negative function ~: X..,. R
which is equal to 0 outside u , but is positive on some open non-empty set contained in U
have 11(~) > 0
[1 n-1 . I n .I f!vj <~>
1=0
1 n-1 . . Since lim n L f!v
i=O > 0 for some n , and
Since supp 11 =X , we
= 11 , we have
hence i (f 1,v) (~) > 0
for some i Therefore K ( ~) > 0 and consequently K (U) > 0 • Thus, supp K = X .
Now we have to show that K satisfies the conditions (vi) and (vii). If >.Em (X) and ~: X..,. R is a continuous
function then we have
218
I( n-1 ~ f n+k-1 l . !. I f~+kAJ (p) ~ ...L I f~A (p) I=
n i=O. " . J 'l.n+k j=O "
I k-1 1 . n+k-1 1 1 . 1
.I n+k(f~v)(cp)+ .I <n-n+k)(f~>.)(cp)l::s; ]=0 ]=k
::s; [k _1_ + n (.!- _1_)] B <PH < k+n+k-nll <PII ::s; 2kll "'II n+k n n+k - k+n n " '
where II<PII= sup I<P(xll • xEX
1 n-1 . Since lim - I f~v
n->-co n i=O ~ , also
(18)
lim y 2-k-1 1 n+k-1 k +k L f*v = ~ •
n j=O We have by (18), for all con-
n+co k=O
tinuous functions cp: X + R ,
I (1 n-1 · ( "" k 1 k ) ) n: . I f~ I 2- - f~,v <<Pl -~=0 k=O
- [ ~ 2-k-1 1 n+.t1 fiv) (cp) I ::s; k=O n+k J=O
I 2-k-1 2ku cpu k=O n -n+co 0 ,
and hence 1 n-1 .
lim- L f!K = v . This proves (vi). n+co n i=O
densation of the sequences
By (18) applied to >.=ox , the sets of points
r.! ni1 f!+ko )""- and ln i=O x n-1
[1 n-1 i r equal. Since fi+ko = f!o - f.o are n .L "x n=1 * X fkx ~=0
of con-
,
this means that the set F is f -invariant. Therefore k v(F) 1 for all k and consequently K(F)=1 (f,~v) (F) = = ,
This proves (vii).
.
219
References
[1] Denker M., Grillenberger c., Sigmund K. Ergodic theory on compact spaces. Lect. Notes in Math. 527, Springer, Berlin-Heidelberg-New York, 1976.
[2] Misiurewicz M. Topological conditional entropy. Studia Math. 55, 175-200 (1976).
[3] Misiurewicz M. Maps of an interval, in: Chaotic Behaviour of Deterministic Systems (Les Houches, Session XXXVI, 1981), North-Holland, 1983.
[4] Ruelle D. A measure associated with Axiom A attractors. Arner. J. Math. 98, 619-654 (1976).
221
CLUSTER EXPANSION FOR UNBOUNDED NON-FINITE ~
R. R. Akhmitzjanov, V. A. Malyshev and E. N. Petrova
Z" c R" , v~2 , be the v -dimensional lattice,
distance between t, t' E z" . Let
the
Let A c z" be a finite volume. We consider the Gibbs
measure jJ on lRA : A
( 1 )
where x={xt,tEA} ERA
and
dx = rr dxt , :>..>0 is small, m>O , tEA
K>O is an integer, £>0 , the sum runs over all pairs
t,t' from A •
The partition function is
Our main result is the following
(2)
Theorem. Let the pararreters v, m, £, K satisfy the inequality
mv + mE - 2vK ~ 0 •
Then there exists a
partition function
z = A
:>.. 0 >0 such that for each 0<:>..<:>.. 0
ZA has the cluster expansion
I A\ ur. 1 c ~ . .
K r . . . K r 1 n
(3)
the
(4)
where the sum runs over all collections {r 1 , •.. ,rn} of
222
pairwise nonintersecting subsets of
C depends on the paraneters A and m
below by an absolute constant:
A : riCA , i=1 , ..• ,n,
, but is bounded from
C = C(f.,m) ~ 2e- 1 (5)
We denote by I A I the cardinality of the set A c !It v
Moreover, the values of Kr satisfy the cluster es
timate: for each N
l: I Kr I r:rso lri=N
The sum in ( 6) runs over all sets r c !It v such that r
contains the origin and has fixed cardinality N , and
o(f.) + 0 when f.+O
(6)
The potential Utt' in consideration is non-finite
and unbounded. In the case of finite unbounded potential
the only condition for the existence of the cluster expan
sions is the boundedness of the potential from below. For
non-finite, but bounded from above potential condition
c>O is sufficient. Both these results were obtained in
[2]. In both of these cases the initial independent measure
is arbitrary and not necessarily has the density
exp{- L lxtlm} as we have in (1). Cluster expansion for tEA
non-finite unbounded potential is also established in [1]
In terms of our paper conditions in [1] are as follows:
We improve these conditions.
The main idea of the expansion is the following. \"i'e
choose some barrier B and in the case when the values of
the random field in consideration are less than B we use
the known techniques (see [2]) of expansion and estimation.
We build some neighbourhoods of such tE!Itv , for which
I xt I > B , and unite them into clusters. In this case we get
the cluster estimate because of the smallness of
exp(-lxtlml.
Note that because of (4)
223
and hence the standard cluster techniques can be used only
in the presence of an estimate:
L !Krlc-lrl::; (o(>.))N r:r3o !r!=N
but since we have (5) it is sufficient to prove (6).
Proof of the theorem. Cluster expansion
We need to formulate some definitions. We call a set
A c Z v connected iff for each t 1 t' E A there exists a se-
such that tiEA 1 i=1, ... ,n and
=rt t = ... =rt t =rt t•=1 · 1 2 n-1 n n
We say that a collection of sets T = { A1 , ... ,An} ,
Aiczv, i=1, ... ,n is connected iff for each l,m E
E {1 , ... ,n} there exists a sequence i 1 , ... 1 iK; ij E
such that A 0 n A. o:f 0 1 E {1 1 ••• 1 n} 1 j=1 1 ••• 1 K 1 "- ~1
for all j and Ai n Am o:f ~ We call K
n r = u A. the support of the collection T = { A1 , ... ,An} .
i=1 ~ Now we shall describe the construction of the expan-
sion (4) . First we fix an arbitrary configuration x =
={xt,tEA} and construct clusters r 1 , ... ,rn corresponding
to the fixed configuration.
Let us put
B B(>.) = ~.-1/8K . (7)
For each tEA with !xti>B we construct the v -dimen
sional neighbourhood Ot having the center t and radius
Rt :
Denote
(8) Rt = (lxti·B-1)2K/(v+£)
M = {tEA: lxti>B} . Let V 1 , ... 1 V p be the maximal con
u Ot . We shall refer to a tEM
nected components of the set
vi as a drop.
224
Let G be a graph with vertices 1 , ... ,p (note that
p is the number of constructed drops), a line connecting
i and j , i;fj , exists iff there exist t E Vi n M and
t 1 E V. n M such that J
( 9)
In general G is not a connected graph. For each maxi
mal connected component G of G consider the union
u v. iEG ~
with i running o~er all vertices of G • Changing the
p components we get the sets A1 I ••• ,At , u A. = u v . .
i=1 ~ i=1 ~ We will refer to as fragments. So, the number
of constructed fragments is equal to the number of connec
ted components of G
Let us denote by T=T(x) the collection of such pairs
(t,t 1 ) , that t and t 1 do not belong simultaneously to
one and the same fragment. Note that for each (t,t 1 ) E T
( 1 0)
In fact, if lxtl < B and lxtl I< B ( 1 0) follows from
( 7) • If lxti>B and lxtl I>B , then since t and tl be-
long to different fragments, (9) is not fulfilled and hence
( 1 0) is true. If lxti>B and lxtl I<B then since t 1 t%Ot
rtt 1 > Rt , so
2K 2K -(v+E) < '·B2K 2K R-(v+E) < '·B4K<_ li i,xt xtl rttl - ~ xt t - ~ " .
The following identity will be useful for us:
exp{-1, L U 1 } = L n a 1
(t,t 1 )ET tt QCT (t,t 1 )EQ tt ( 11 )
where the sum runs over all subsets QCT (includingthe
empty set) and
( 1 2)
If Q=0 we put the corresponding term equal to 1 .
We call each pair (t, t 1 ) a Zink. Let us fix an arbit
rary QCT . Let T be the collection of sets, consisting of
225
all constructed fragments A1 , ... ,Al and all links belong
ing to Q Let T 1 , •.. , T n be the maximal connected subcollec
tions of T, and respectively r 1 , ••• ,rn be their sup
ports. We call each of r 1 , .•• ,rn a cluster, corresponding
to fixed configuration x and QCT(x) and define
fy (x) = IT exp{-!tUA(x)} IT att, IT exp{-lxtlm} i J!FTi - (t,t')ETi t€fi
( 13)
where the product IT AET.
l.
runs over all fragments belonging
to T. , IT 1 (t,t')ET.
l.
is meant over all links (t,t I )EQ be-
longing to Ti , and
So, we have constructed a collection of clusters r 1 , ••• , r n
which corresponds to the fixed configuration x and fixed
QCT(x) , and defined the "weights" fT. of these clusters. l.
Consider an arbitrary collection r 1 , •.• ,rn of pair-
wise disjoint subsets ric A , i=1, •.. ,n (r i is not ne
cessarily connected) • Let X ( r 1 , ••• , r n) c RA be the set,
consisting of configurations x with the following proper
ty: there exists QCT(x) such that r 1 , ..• ,rn is just the
collection of clusters corresponding to (x,Q) Note that n
restriction of any xEX(r 1 , ... ,rn) on A\ u r. belongs to i=1 l.
A\ ur. . l.
[-B,B] 1 , hence, the set X ( r 1 , ••. , r n) can be represen-
ted as a direct product
[ -B,B]
A\ ur. i l.
where xr. is exactly the set of configurations r.
xER 1 such l.
that for any xEXr. there exists l.
pair (x,Qi) generates ri
For any finite r c z v denote
K = f r x~
Qi CT(x) such that the
( 14)
226
r where Xr c R is the set of such configurations xr that
there exists QcT(xr) such that the pair (xr,Q) gener
ates just the cluster r , the sum I runs over all such Q
Q , and fT(xrl is defined in (13), dxr = IT dxt • tEr
we obtain the expansion (4)
B c = J exp(-!y[m)dy ~
-B
Indeed,
Taking into account (14)
with
1 -1 f exp(-IYimldy ~ 2e .
-1
The sum is meant over all collections of pairwise disjoint
(not necessarily connected) sets A1 , ... ,Af, AicA , i=
=1, ... ,f and integration is over the set of configurations
xERA such that A1 , ... ,Af fare exactly all fragments, gen
erated by x, T = (AxA)-.... U (A.xA.). . 1 l l
Using now (11) for e~= tA I Utt'IJ we obtain (t,t') T
Since a collection A1 , ... ,Af together with Q de
termines uniquely the clusters {r 1 , ... ,rn} , we may repre
sent a summation as follows:
Here the summation runs over all collections {r 1 , ... ,rn}
227
of pairwise disjoint clusters r 1 , ... ,rn, then over all
fragments ~ 1 , ... ,Ai and over all collections
Qc (Axil)' U (A.xA.) of links, such that the collection i=1 l l
of supports of maximal connected subcollections of the col-
lection {A1 , ... ,Ai, Q} coincides with U 1 , ... ,rnl .
Performing now the integration over xt , tEA"- U r. i=1 l
and taking into account the definition of Xr , we get:
jA-..Vr.l n C l l II
i=1 J
Proof of the theorem. Cluster estimate
First of all we obtain the cluster estimate for clus
ters, consisting only of fragments but not of links.
Let us fix the cardinality of r : ]ri=N . We regard
only the clusters r containing the origin: rso
Moreover, let us assume at first that r is a v -di-
mensional spherical neighbourhood of some
ously, ixt I> B . We will show now that
1 1 ,m , N exp(-},xtl ) s (rA)
t . Then, obvi-
( 15)
Let 0 be a v -dimensional sphere having radius R ,
and N the number of points in 0: N= izvnol . There
exist some constants and (depending on
that
and therefore it is sufficient to show that
c Rv
exp(-~lxtlml s (/I) 1 t
or, taking the logarithm and using (8)
\) ) such
228
Together with (7) it leads to
m(V+E)-2KV ( V+E) ~ 3 c A v/4 (v+E). ( -ln A)
2 1
If (3) holds and A is sufficiently small, the last
inequality holds, too. Consequently, if r is a sphere and
lri=N , then
iKri s; (/I)N Jexp{--} L lxtlm} II dxt ~ (C/I)N tET t
(17)
where the constant C depends on m .
Now, let r be a drop, i.e. r = UO t t
spherical neighbourhood of t • Note that
where ot is a
r is connected
in this case. Let us choose Mer a subset of the centers
of spherical neighbourhoods for which r = u Ot . (It can tEM
be done in at most 2N different ways.) For any tEM sim-
ilarly to (15) we have
and since L JOt I;:: Jri = N we have (17) with another contEM
stant C . (C denotes different constants not depending on
A • )
Note that r is connected, Jri=N is fixed and rso
The number of different sets r c z v with these properties
does not exceed CN for some constant C , depending on
v . Therefore, taking into account (17), we get (6) for
connected r .
Now we are going to prove the cluster estimate in the
case when r is not connected. Then, according to our con
struction, if r contains no links it consists only of one
fragment. We regard at first fragments containing only two
drops, i.e. r is the union of two connected sets. Let us
denote these dregs by v1 fix V1 and let [V 1 [=N 1
and V2 and let V1so • Let us
We fix also t 1EV 1 such that
229
Since the drops V1 and V2 form one fragment, there
exists t 2cv 2 such that
vAb21Kt1K r-(V+E:)?; -:2 t1t2
where b 2 = [xt [ . That is, if t 1 , b 1 and b 2 are fixed, 2
t 2 belongs to the spherical neighbourhood of t 1 having 2K/(V+E:) , radius (b 1b 2 ) , so that the number of d1fferent
ways of fixing t 2 does not exceed c 1 (b 1b 2) 2Kv/(v+£)
Denoting N 2 = N-N 1 and summing through all connected V 2 ,
having cardinality N2 and containing fixed t 2 , similar
ly to (17) we have:
L Jexp<-Wv lexp[- L lxtlmJ .. IT dxt :5 V2:V2st2 2 tEV2 tcV2,t2
;v21=N2 ( 18)
N ( 2 bm) . ( c ") 2 :5 exp - 3 2 v A
The left hand side integral is over the set of such con-
figuratiOnS On v2 1 that v2 fixed.
Similarly,
2: V1:V1s0
IV11 =N1
is a drop and xt =b 2 is 2
where the sum runs over all connected V 1 having cardinality
N1 and containing the origin, integration is over the set
of such configurations on v1 that v1 is a drop and
xt =b 1 is fixed. 1
Consequently,
230
· explr_ L !x 1m) rr dx J tEV2 t tEV2,t2 t
( l·Jexp(-AU )·
v2
Substituting (18) and (19) into the last inequality we
obtain
L IKrl s (C/I)N J b12Kv/(v+E) exp(-fb~) r:r9o B lri=N
J b2KV/(V+E) (-~bm)db (C/I)N 2 exp 3 2 2 s B
Now we shall examine the case when a fragment contains
an arbitrary number s of drops. Let us fix V190 and
t 1EV 1 . For each fragment there exists a tree Y with verv tices t 1 ,t2 , ... ,ts, ti EZ such that tiEVi and a line
between t. and t. meaning that l J
~ 2K 2K r-(V+E) VAXt. Xt. t t ~ 1 •
l J i j (20)
Let us denote bi=xt. , i=1, ... ,s Let us fix t 2 , ... , l
,ts , b 1 , ... ,bs , bi 2:B for each i , and a tree Y , satis-
fying (20). Let us fix integers N1 , ... ,Ns , such that
LNi = N (i.e. we fix the cardinalities of the drops V1 , ... ,
,Vs , of a fragment). Obviously,
[ \' 1 , m I s ( ( , I ,mJ I
exp(-.\Ur)exp- L 1 xt1 JS .rr lexp(-.\Uv.l·expl- L xtl j· (21) tH l=1 l tE:Vi
Using (18) for each i=2, ... ,s we have
231
(22)
2 N. s exp(-3 b~) (C/I) l
Let us keep b 1 , ... ,bs fixed and estimate the number
of trees Y , containing a fixed vertex t 1 and satisfy
ing (20). We shall describe an algorithm, which enumerates
all such trees Y •
An algorithm, enumerating the trees
1. step. Fix a vector of
.,ns) such that n 110 , ns=O
it can be done in at most 4s
nonnegative integers (n1 , ..
and Ini=s-1 . Evidently,
different ways.
v 2. step. Choose n 1 vectors v 1 , .•. ,v , viE Z , n1
i=1 , ... ,n1 satisfying the following conditions: vi be-
longs to the spherical neighbourhood of the origin having
radius (b1bi+1 >2K/(v+£) . Construct:
t1 + v n1
Construct the lines between t 1 and each of the t 2 , ... ,
,t +1 . n1
3. step. Choose the first (in lexicographic orden of
the constructed vertices, excluding t 1 be tj . If n 210 , choose n 2 vectors
. Let this vertex
v +1 , ... , v + n 1 n 1 n 2 v vi E :It for each
cal neighbourhood (bjbi+1)2K/(v+£)
t n 1+2
i , such that vi belongs to the spheri
of the origin having radius
• Construct
t. + v +1 J n1
Construct the lines between and each of
232
t +2 , .•. ,t + +1 . If n 1=o , pass to the next step with-n1 n 1 n 2
out any construction.
We proceed by induction. Let p steps be already per-
formed and vertices be constructed.
(p+1). step. Choose the first (in lexicographic order)
of all vertices having been constructed during the previous
steps excluding those which have been already chosen ear
lier. Let this vertex be tt . If np=O , pass to the next
step without any construction.
If n to p , choose vectors v + + +1'"""1 n 1 ... np
for each i such that vi belongs
neighbourhood of the origin having radius
Construct
t = t +v n 1+ •.. +np_ 1+2 t n 1+ ... +np_1+1
Construct the lines between tt and each of the ver
tices constructed in this step.
After s steps the construction is finished.
Thus, if b 1 , ... ,bs, n 1 , ... ,ns_1 and v 1 , ..• ,vs_1 are chosen, the graph constructed by our algorithm is
unique. Choosing all possible b 1 , ... ,bs , n 1 , ... ,ns_1 ,
v 1 , ..• ,vs_1 we can construct among other graphs all pos
sible trees Y . Moreover, we construct each tree moretlioos.
Indeed, let T be a tree having s vertices and a root
t 1 . Let us denote by n 1 the number of lines in T in
cident with t 1 , (ni+1) the number of lines in T in-
cident with ti , i=2, ... ,s
ted by our algorithm at last
Then the tree T s rr (ni!)
i=1 times.
is construe-
In fact, let b 1 , ... ,bs and v 1 , .•. ,vs_1 be the vec
tors, generating the tree T . Let us divide a collection
(b1 , ... ,bs) into subcollections in the following way. The
first subcollection consists of only one element, namely
233
S1 . The second subcollection consists of n 1 vectors:
S2 , ... ,bn 1+1 ; the third one consists of the next n 2 vee-
tors, and so on. Consider now a new collection
(b1 , ... ,bs) , obtained from (b1 , ... ,bs) by arbitrary
permutations in each of the described subcollections (but
without any permutations between the subcollections). Let - -v 1 , ... ,vs_1 be a collection, obtained from v 1 , ... ,vs_1
by the same permutations. Obviously, the tree generated by
b 1 , ... ,bs' v 1 , ... ,vs_ 1 is T , and there are exactly
TI(ni!) such permutations.
The number of different ways of choosing v 1 , ... ,vs_1 ,
having and n1 '· · · ,ns-1 fixed, is
2K(n.+1}/(v+E) b. J ~.
(23) J
where we have denoted by i. the number of the vertex, J
which is chosen as the first one in lexicographic order in
the (j+1)th step of the algorithm, j=2, ... ,s-1 , and
used (16).
Note that
n. ~
(Cni) (24)
Taking into account (21), (22), (23) and (24) finally
we have:
L I K I r:r30 r I r I =N
::; (C/\)N I I s::;N (n1 , ... ,ns_ 1 )
s [ n. n (Cn.) ~ ·
(Tini!) i=1 ~
Now we must consider the case when r contains links.
Since
'B4K -(v+E) A rtt' ::; cfi (26)
the case when r consists only of links is trivial. The
234
general case follows from (26) and (25) by induction on the
number of links and fragments in the cluster.
References
[1) Cammarota Camilla. Decay of correlations for infiniterange interactions in unbounded spin systems. Cornrnun. Math. Phys. 85, no. 4, 517-528 (1982).
[2) Malyshev V. A. Serniinvariants of the unlocal functionals of Gibbs random fields. Matern. Zarnetky 34, no. 3, 443-452 (1983, September).
235
Thermal Layer Solutions of the Boltzmann Equation
Claude Bardos
Russel Caflisch
Basil Nicolaenko
Abstract
The steady Boltzmann equation in one space dimension with a small
mean free path is solved for a gas contained between two plates at
different temperatures. The solution consists of the Chapman-Enskog
expansion together with boundary layer expansions near each plate.
These expansions are truncated at a finite order, and an additional
error term is added, so that the sum is an exact solution of the
Boltzmann equation. The analysis of the boundary layer equations
requires a solution of the Milne problem.
1. Introduction
In this paper we give a kinetic theory description of a gas
contained between two plates at different temperatures. The state of
the gas is described by the steady nonlinear Boltzmann equation in
which the mean free path is assumed to be very small. The boundary
conditions are diffuse reflection, i.e. a particle hitting the
boundary is absorbed and reemitted with a Maxwellian distribution at
the prescribed temperature.
In this problem gravity is excluded and uniformity in the lateral
spatial directions is imposed. It follows that the average velocity of
the gas is identically zero. The corresponding Navier-Stokes equations
then reduce to the equations of constant pressure and Fourier's law for
heat conduction. We show that the Boltzmann solution agrees with these
two macroscopic equations, thus giving a basic derivation of Fourier's
236
law. This continues a program [2] of relating solutions of the fluid
equations with solutions of Boltzmann's equation.
As a future problem, we plan to include gravity and allow three
dimensional spatial variation. A bifurcation in the steady solution,
corresponding to the Rayleigh-Benard instability, will then be sought.
Note however that this instability is fully understood only for an
incompressible fluid in the Boussinesq approximation [5]. A study of
the Rayleigh-Benard problem for a dense gas was made recently in [7].
The present solution for the steady thermal layer problem is found
as a decomposition into several parts. The first part is the Chapman
Enskog expansion (modified to omit the Burnett and super-Burnett
equations), which is valid in the interior of the region. The leading
order term in this expansion is a Maxwellian distribution, which can be
chosen to satisfy the boundary conditions. However the higher order
terms in the expansion do not satisfy the boundary conditions, and so
boundary layer terms must be added on. Since these terms are required
only at high order, the resulting boundary layer equations are linear.
The Chapman-Enskog and boundary layer expansions may be only
asymptotic, not convergent. In order to obtain an exact solution,
these expansions are truncated at finite order and an error term is
added. The error term is chosen so that the total sum gives an exact
solution of Boltzmann's equation. The equation for the error term is
simpler than the original Boltzmann equation because it is only weakly
nonlinear.
The analysis of the boundary layer equations utilizes a solution
of the linear Milne problem for the Boltzmann equation. This solution
uses energy estimates similar to those in [1] and will be only briefly
described here.
Some mathematical details in the analysis of the remainder
equation and of the Milne problem are not yet completed; so the results
here are not completely rigorous. The purpose of this paper is to set
out the motivation and formulation of the thermal layer problem and its
solution.
The plan of this paper is as follows: After the Boltzmann equation
is summarized in section 2, the thermal layer problem and its solution
are described in section 3. The Chapman-Enskog expansion is developed
in section 4, the boundary layer expansion in section 5, and the error
term in section 6. Finally in section 7 the Milne problem is
discussed.
237
2. The Boltzmann Equation
The steady Boltzmann equation in one space dimension is
a 1 ~ 1 ax F = £ _g_CF ,F). (2.1)
in which the molecular distribution function is F = F(x,i) with x € R1
and ~ (~ 1 ,~ 2 .~ 3) € R3 • The mean free path E is assumed to be small
here.
For the collision operator _g_ we take the one corresponding to hard
spheres orto the Krook model.
properties:
In general _g_ has the following
(i) Q_(F,F) - 0 iff F- M, in which the Maxwellian distribution M
has the form
(2.2)
(ii) f W Q_(F,G) d~ = 0 for all F,G iff w is a linear combination
of the summational invariants wi given by
with
w0Cp = 1
wiC~)=~i
w4C~) = ~2.
(2.3)
Corresponding to these orthogonality relations, we decompose F as
(2.4)
(2.5)
and
238
3 {a(x) + E bi(x)~i + c(x)~ 2}M(p
1
fori= 0, ••• ,4.
(2.6)
(2. 7)
The linearized collision operator ~ is defined, relative to a
Maxwellian M, by
LF 2 Q_(M,F) (2.8)
It has the property that
f M F LF d~ ~ o f M ~~ d~ (2.9)
for some positive constant o depending only on M.
1. ThE!__T_12.~.E.ma:l,_ Layer Problem
The thermal layer problem is to solve the steady Boltzmann
equation in a slab 0 < x < 1 with specified temperature
1 and with specified total mass density p There are thermal boundary conditions and the condition
TL at x = 0 and 1
= J J F d~ dx. 0 -
of no mass flux
at x 0 and x 1. We seek a solution for which the macroscopic
velocity is zero.
The Boltzmann equation for the thermal layer is
for () < x < 1 (3 .1)
with boundary conditions
and normalization
1
J J F d~ dx = p. 0
239
for I; 1 > 0
for I; 1 < 0
The Maxwellian distriputions at the boundaries are
(3. 2)
(3. 3)
(3.4)
(3. 5)
(3.6)
(3.7)
In these equations T1 , TR and p are given positive constants, while p1
and pR are unknown. In addition we ask that
(3.8)
for all x and for i = 1,2,3. Fori= 1, (3.8) is implied by (3.4).
We shall find the solution F = FE for the thermal layer problem
(3.1)-(3.5) forE small by decomposing FE into three parts: an interior
solution Fe given by the Chapman-Enskog expansion, a boundary layer
expansion FB at x = 0 and at x = 1, and an error term FE. The form of
FE is
(3. 9)
These three components of FE are analyzed in sections 4-6.
240
The main character of the solution F£ is its agreement with the
Navier-Stokes equations, which reduce to Fourier's law for heat
conduction for this simple problem. 1o/e shall show that
(3.10)
in which Po• T0 , u0 satisfy the steady one-dimensional Navier-Stokes
equations with u0 = 0.
The Navier-Stokes equations are
a -PU ax
0 (3.11)
~ (pu(.!.. u2+e) +up)=£ a (llu ~ax u) + £ ~ (>.(T) ~ T) (3.13) ax 2 ax a ax ax
with the equation of state
p p T ' e=l..T 2
(3.14)
The viscosity ll(T) and the heat conductivity >.(T) are those which
result from the Chapman-Enskog expansion. In the thermal layer problem
with u = 0, these equations reduce to the hydrostatic equation
~pT ax
0
and Fourier's law
~ (>.(T) ~ T) ax ax
(3.15)
0 • (3.16)
241
In addition we specify the total density p and the temperature boundary
values TL and TR , i.e.
1 J p dx = p 0
T(O)
(3.17)
T(l) (3.18)
The problem (3.15)-(3.18) has a unique solution for any p, TL , TR
which are positive.
Our main result for the thermal layer problem is the following
theorem.
Theorem. If c, TL , TR and p are positive and if E is
sufficiently small, there is a unique solution FE of the Boltzmann
thermal layer equations (3.1)-(3.5). Moreover ~ satisfies the
stationarity condition (3.8) and
(3.19)
in which cis a constant (independent of c), u0 = 0, Po and To solve
the Navier-Stokes equations (3.15)-(3.18), and H•U is an appropriate
norm.
4. The Chapman-Enskog Expansion
Away from the boundaries x = 0 and x = 1, the solution FE of the
thermal layer problem is given primarily by the Chapman-Enskog solution
Fe. The ehapman-Enskog expansion is discussed in detail in (3] and
[4]. Here we summarize the results for the steady problem with zero
velocity.
The solution Fe has an expansion
(4.1)
242
in which Mo is the Maxwellian
with Po , To solving (3.15)-(3.18). The higher order terms FT , F~ can
be decomposed as
F~ = tn +'I' n (4.3)
such that 'l'n is the hydrodynamic part and tn is the nonhydrodynamic
part, i.e.
(4.4)
0 for i 0, ••• ,4. (4.5)
The fluid dynamic variable Pn(x), Tn(x) satisfy (forced)
linearized Navier-Stokes equations. For n = 1 these equations are
a ax (pOT1 +p1T0) 0 (4.6)
aax (A (To) a ax T1 +A '(To)T1 aax To) 0 (4. 7)
1 f p 1 dx = P'1 (4.8)
0
T1 (0) TLl ' T1(1) = TRl • (4.9)
243
The constants p 1 , TL 1 and TR1 determine p 1 , T1 uniquely and will be
found in section 5 using the boundary conditions, The equations for n
= 2 are similar but contain forcing terms.
The nonhydrodynamic components ~n are given by
(4.10)
(4.11)
in which ~ = -2~(M0 ,•) and P is the projection onto the null space of
L+. Note that ~ 1 is completely determined by Mo , and ~ 2 is determined
once o/ 1 is known.
The truncated expansion (4.1) does not exactly solve the Boltzmann
equation (3.1) or the boundary conditions (3.2)-(3.3), but the
expansion does satisfy (3,4). The boundary conditions (3.2), (3,3) can
be rewritten for FE - M0 as
F£-MO = l>pL '1L
F£-MO = l>pR MR
for x
for x =
0 (4.12)
(4.13)
in which l>pL and l>pR are undetermined as yet. The right hand sides of
(4,12), (4.13) are purely hydrodynamic, In order to accommodate the
nonhydrodynamic components ~n , which enter in the Chapman-Enskog
expansion, we must introduce boundary layer terms. However since the
nonhydrodynamic part of FC is size Q(E), we need boundary layer terms
only of size Q(E). It is useful to expand lip L and /:;p r as
(4.14)
(4.15)
244
5. The Boundary Layer Expansion
The boundary layer expansion FB provides rapid transition (in
distance £)from the imposed boundary conditions at x = 0 and x = 1 to
the ehapman-Enskog expansion in the interior. There are actually two
boundary expansions: one at x = 0 denoted FL and one at x = 1 denoted
FR. These are written in terms of scaled variables as follows
X 1-x (5.1) - ·-
£ £
The Boltzmann equation for FL is
(5.2)
for 0 < y < ®• In addition FL is required to satisfy a boundary
condition at y = 0 and to not extend into the interior, i.e.
for I; 1 > 0 (5.3)
at y 0 (5.4)
as y + ® (5. 5)
In (5.2) we have replaced Fe by its truncated power serise Fe around y
= 0, given by
Fe(y) = ML + £{Fy(x=O) + y :x M0(x=O)}
2 a e 1 2 a2 } + £ {F~(x=O) + y- F1(x=O) +- y - M0(x=O) , (5.6) a X 2 a X2
245
so that Fe is defined for 0 < y < oo. Similar equations are found for
FR, and FL is coupled with FR by the normalization condition
1
~ J J FL ds dy + ~ J J FR ds dz + J J Fe ds dx = P (5. 7)
0 0 0
We ask only that (5.7) hold with an error size ~ 2 • The boundar layer equations (5.2)-(5.5) and (5.7) are solved by
expanding FL as
( 5. 8)
The resulting equations for Ft are
a L L I; 1 - F1 = - L F for y ) 0 ay - 1 (5.10)
(5.11)
at y = 0 (5.12)
as y + oo (5.13)
There is a similar expansion for FR and a set of equations for
F}. In addition (5.7) is rewritten as
1
J J Ft + Ff ds dy + J J F~ ds dx = o (5.14) 0 0
246
Equations (5.10)-(5.14) will be solved in several steps.
First solve the following Milne problem:
I; ~GL 1 a Y
= - L GL for y~O (5.15)
cL(o,p -~1(o,p , for I; 1 > 0 (5.16)
I F;; 1 cL d~ 0 , for y ~ 0 (5.17)
in which l = -2Q(ML ,•). The solution cL is found (formally) in
section 7. In particular it is shown that
as y + ~, with aL and cL constants which are determined once ~ 1 is
known. Recall that ~ 1 is determined by knowing M0 •
Now for FT we choose
which satisfies (5.10), (5.12), (5.13). Similarly we choose F~ to be
(5.20)
I; ~ cR - L cR for z > 0 1 a z
(5.21)
247
for ~ 1 < 0 (5.22)
f ~ 1 GR d~ 0 , for z > 0 (5.23)
in which ~ = - 2~(· ,MR) and aR , cR constants.
The conditions (5.11), (5.14), and a condition for F~ analogous to
(5.11) will be satisfied by the proper choice of the constants TL 1 ,
TR1 , P1 , P2 , 6pL1 , 6pR1 • Evaluate (5.11) using (4.3), (4.4),
(5.19), (5.16) to get -(aL+cL~ 2 )ML + {p 1(0)/p 0(0) + (TL 1 /2TL)(~ 2 /TL
-3)}ML = 6pLlML for ~ 1 > 0 , or
(5.24)
(5.25)
Similarly we find that
(5.26)
(5.27)
The condition (5.14) is evaluated to obtain
P2 = - f f FR + FL d~ dy 1 1 -
(5.28) 0
This completes the determination of Ft and F~ and of the constants
TL1 • TR1 • P1 • Pz • 6PL1 • 6PRI • ~s seen above, the netermination of the Chapman-Enskog and
248
boundary layer expansions are coupled through the choice of the
constants TLn , TRn , Pn , L\pLn , L\pRn and the boundary values of ol>n •
The logical order to determine these two expansions is as follows:
First ~O is found as in (4.2). Then ~ 1 is found from (4.10). Third Fy and F~ are given by (5.19), (5.20). Fourth, TL 1 , TR1 , P1 , P2 ,
L\pLl , L\pR1 are found by (5.24)-(5.28), while simultaneously ~ 1 is
found through (4.4) and (4.6)-(4.9). Then the process is started again
to determine ~2 ' etc. There are equations similar to (5.10)-(5.14)
for FL and FR 2 2 The combination F = Fe + F8 is not an exact solution of the
Boltzmann thermal layer problem (3.1)-(3.5). Although Fe satisfies
(1.4), the errors in the other equations are
Since
E2 ' £2.
a - _ ~ -1 < ) t; 1 a;{F ~ 9_F,F
1 E4 J J F df dx - P = £ J _1 J (FY + F~)(y,f) d~ dy
0 £
+ £ 2 f j (F~+F~)d~ dy. 0
FL and FR are found to decay exponentially for large
E3 and E4 are exponentially small in -1 £ • The error - -We have denoted PL = Po(O) + L\pL ' PR = Po< 1) +Lip R •
6. Th~<:rr~J~ .. 9 . .'~ation
(5.29)
(5.30)
(5.31)
(5.32)
y (or z),
E1 is size
The equation for FE is found by asking that p£ =Fe+ FB + £ 2FE be
an exact solution of the thermal layer Boltzmann equation (3.1)-(3.5).
It must make up for the error terms E1 , E2 , E3 , E4 in (5.29)-(5.32).
It follows that FE satisfies
(6.1)
249
for x > 0 with boundary conditions
E E -2 F (x=O ,f) = llpL ML - c E2 for I; 1 ) 0
and normalization condition
1 I I FE d~ dx = - c-2 E4 0
in which lip~ and lip~ are constants to be determined.
(6.2)
(6.3)
(6.4)
The analysis of (6.1)-(6.4) proceeds as in the analysis of the
similar error equation in [2].
7. The Milne Problem
The Milne problem consists of solving the linearized Boltzmann
equation with a prescribed incoming velocity distribution and
prescribed average normal velocity, along with a condition of
asymptotic behavior at~. The resulting equations are
0 , for x > 0
for I; 1 > 0
I I o + I~ I )~ ~ d~ dx < ~ 0
(7.1)
(7.2)
(7. 3)
(7.4)
250
in which the constant average normal velocity u and the incoming
distribution g are to be given, 1n this section we use the
decomposition F = ~F + fF = ~F + ~F • ~described in section 2.
The Milne problem for neutron transport was analyzed in [1] and
that solution has motivated this study. For the Krook model of the
Boltzmann equation, the Milne problem has been partly analyzed in [4]
and [8]. Partial results for the full Boltzmann equation are contained
in [6] and [9]. The results presented here give a complete solution of
the Milne problem for the Krook model; for the full Boltzmann equation
there are still a number of unfinished technical details.
Theorem l (Existence). Suppose that it; 1>0 (l +II; I ) 2 g(~) 2 dlf. < oo.
Then there exists a solution F of the Milne problem (7.1)-(7.4) with 2 + ~FE Lloc( Rx).
Theorem 2 (Uniqueness). Let F1 and F2 , both in
Lioc< Ri ,L2( Rt)), be solutions of the Milne problem (7.1)-(7.4). 1f
F1 (0,~) = F 2 (0,~) for I; 1 > 0 and if J I; 1F1dlf. = J I; 2F2dlf. at x = 0, then
F1 = F2 for all x) 0, ~ R3.
~~ (Orthogonality and Asymptotic
FE Lioc< Ri ,L2( Rt)) solve (7.1)-(7.4). Then F
properties:
Properties). Let
has the following
(7.5)
(7.6)
(7. 7)
Moreover b1 - J ~; 1 F dl; is constant in x and equal to u. As x goes to
infinity, a(x), b2(x), b3(x), and c(x) converge exponentially to
constants a"", bz, b3, c"" while ~F converges exponentially to zero. To
be precise if o <a, with a defined by (2.9), there is a constant K
such that
251
(7.8)
Finally there is the following symmetry property: If g(~ 1 .~ 2 .~ 3 )
g(~ 1 ,-~ 2 .~3) = g(~ 1 .~ 2 ,-~3) for all i = (~ 1 .~ 2 .~ 3 ), then b2 = b3 = 0.
8. References
[ 1] Bardos, Santos and Sent is. Diffusion approximation and computation of the critical size. Preprint.
[ 2] Caflisch. The fluid dynamic limit of the nonlinear Boltzmann
[ 3]
equation. Comm. Pure Appl. Math. 1l (1980) 651-666.
Caflisch. Fluid dynamics Nonequilibrium Phenomena, eds.
and the Boltzmann equation, in Lebowitz and Montroll, (1983)
North-Holland, 193-223.
[4] Cercignani. Theory and A£Plication of the Boltzmann Equation (1975) Elsevie-r-.~- --
[5] Drazin and Reid. Hydrodynamic Stability (1981) Cambridge.
[6] Guiraud. Equation de Boltzmann lineaire dans un demi-space.
[ 7]
Comptes-Rendus 274 (1974) 417-419.
Kirkpatrick and Cohen. convective instability.
Kinetic J. Stat.
theory of fluctuations near a Phys. 11 (1983) 639-694.
[8] Pao. Application of kinetic theory to the problem of evaporation and condensation. Phys. Fl. ~ (1971) 306-312.
[9] Rigolot-Turbat. Probleme de Kramers pour l'equation de Boltzmann en theorie cinetique des gaz. Comptes-Rendus 3Zl (1971) 58-61.
Claude Bardos Ecole Normale Superieure 75005 Paris
Russel Caflisch Courant Institute of Mathematical Sciences New York, NY 10012
Basil Nicolaenko Los Alamos National Laboratories Los Alamos, NM 87544
Research supported by the National Science Foundation and the Department of Energy.
253
HYPERBOLICITY AND MOLLER-MORPHISM FOR A MODEL OF
CLASSICAL STATISTICAL MECHANICS
E. Presutti, Ya. G. Sinai and M.R. Soloviechik
Abstract. We consider a gas of point particles in E+ . The first particle has mass M , the others m and M>m .
The particles interact by elastic collisions (among them
selves and with the wall at the origin). Let X be the
phase space and ~ a Gibbs measure for the system, St de
notes the time flow and (X,~,St) is a dynamical system.
We identify the m -particles during their evolution so
that they keep the same velocity until they collide with the
M-particle. Hence the motion is free, asymptotically far
from the origin: free particles come from +00 , interact
with the M-particle and then move back free to +00 • We
prove that the Moller wave operators n± exist, asymptotic
completeness holds and that n:1n+ defines a non-trivial scattering matrix for the system. n+ define isomorphisms
between the dynamical system (X0 ,~0~S~) and (X,~,St) , 0 0 0 I* ,~ ,St) refers to the case when all the particles have
mass m and ~0 has the same thermodynamical parameters
as ~
An independent generating partition is explicitely 0 0 0 known for the system (X ,~ ,St) and n+ transform it in
an independent generating partition for (X,~,St) , thereby
proving that this is a Bernoulli flow.
The proof of the existence of the wave operator is
based on the (almost everywhere) existence of contractive
manifolds. Namely we prove that for almost all configuratkns
xEX the following holds. Fix any finite subset I of par
ticles in x and consider all the configurations y ob
tained by changing the coordinates of the particles in I
while leaving all the others fixed. Then if the change is
254
small enough Stx and Sty become (locally) exponentially
close.
0. Introduction
Very little is known of the ergodic properties of sys
tems with infinitely many degrees of freedom which describe
the behavior of a gas in classical statistical mechanics.
Results have been obtained in particular cases: the infi
nite ideal gas [25], [1], [12], the one dimensionalhaxd rods
system [10], [22], [2], [5], the infinite chain of harmonic
oscillators [8], [11]. The analysis in the above models is
based on a more or less complete knowledge of the time flow.
For more complex systems it seems hopeless to look for ex
plicit solutions of the equations of motion. It is not even
completely clear what one would like to find, i.e. which
features are responsible for the good ergodic properties of
the system. Here we find a difference with the finite dimen
sional case where it is generally believed that the main
mechanism of chaos relies on the existence of foliations of
the phase space into stable and unstable manifolds [3] • The
construction of Markov partitions [26], [27] frames then
the problem within the classical theory of stochastic proc
esses. It is conceivable that some analogue of such hyper
bolic structure extends to the infinite systems. Besides it
another mechanism, peculiar of the unboundedness of the sys
tem, is expected to play an important role. This is very
well described in [16]: " ••• A subsystem undergoes all kinds
of interactions with other parts of the system [due to the
infinite size of the system] ••• [and] owing to the compleK
ity of [such] interactions [it] will pass sufficiently of
ten through all its possible states ••• ". In the ideal gas
local perturbations move freely away and in the other mod
els mentioned above similar mechanisms come into play. In a
model considered in [14] the analysis does not exploit know
ledge of the time flow. The authors consider a semi-infinite
point particle system. The first particle, h.p., has mass
M , all the others m and M>m • The h.p. is confined be
tween 0 and L by two elastic walls, the latter being
transparent to any other particle. The only interaction con-
255
sists in the elastic collisions between the h.p. and the
light ones. The states of the system are determined by the
corresponding Gibbs measure. The mechanism responsible for
the ergodic properties of the model (which in [14] has been
proven to be a Bernoulli flow) is the following. whenever
the h.p. bounces off from the wall at L , then the informa
tion of the whole past is only the velocity of the h.p. at
that time. The process of the velocity at such times is then
proven to be a Markov Doblin chain and from this the ergodic
properites of the system follow. An extension of the result
to the "infinite" case when light particles can also arrive
from the left has been obtained in [13] (the analysis covers
the case when the gas on the right has different temperature
from that on the left) ; An extension to the two dimensional
case has been recently worked out [15].
The main feature of the semi-infinite case considered
in [14] is that the interaction is strictly localized, the
system behaves like free in the interval L, oo • If one
takes off the wall at L then the h.p. is free to move fur
ther to the right and it goes eventually past any fixed
point in the line, since the Gibbs equilibrium measure gives
a non-zero probability to such event. For such reason the
Markov property exploited in [14] is lost. The analysis re
quires then a deeper study of the dynamical structure of the
system and, as we shall see, it singles out both mechanisms
we have outlined before: the existence of local contractive
"manifolds" (hyperbolic structure) and the occurrence of
clusters of particles which "come from infinity" and play a
role analogous to that predicted in [16].
This model, as well as some of its variants, has been
recently studied in [7], [6], [19], [4]. It has been shown
to be a Bernoulli flow. The intuition one has of the system
is that the incoming particles exert a pressure on the h.p.
which is consequently kept close to the wall at 0 • Only
rare fluctuations of the pressure allow the h.p. to escape
from its confinement and to travel far away. Hence one ex
pects that the dynamics is asymptotically (far away from the
origin) free: namely free particles come from +oo , arrive,
interact with the h.p., then flow away to gain back asymptotic
256
freedom at +00 • Such picture calls to mind the Moller wave
operators in scattering theory [21]. In the present paper
we will prove their existence. Asymptotic completeness is
also shown: the wave operators provide then isomorphisms be
tween the dynamical system !*,~,St) , i.e. the system hav
ing the h.p., and (*0 ,~0 ,5~) , the system with only the
light particles. (Here and in the following ~ denotes a
Gibbs measure at some fixed temperature and density for the
system having the h.p •• ~0 has the same thermodynamic
parameters and refers to the system where all particles have
mass m • ) Independent generating partitions are explicitely
known for the discrete-time dynamical systems (*0 ,~0 ,s(n)),
S (n) = s,n , for any T>O • Hence our results provide an ex
plicit construction of the Bernoulli isomorphism for the
system !*,~,St) . In [6] its existence was only implicitely
granted by the general theorem of [20] via the estimates ob
tained in [6]. [The scattering matrix in infinite systems of
Statistical Mechanics has been recently considered in [17]
for a quantum model. Its relation with the existence of
quasi-particle excitations has been established in a very
general setup in [18] .]
The existence of the wave operator in our system ex
ploits essentially three characteristics of the time flow:
(1) th.e existence of a contractive manifold, similar to a
"leaf" of the stable foliation in the hyperbolic systems
(we shall see that there are several analogies with the
dispersed billiard problems) (2) the cluster structure of
the dynamics [23], i.e. the h.p. interacts with a finite set
of light particles and when such interaction finishes, anew
finite group of particles arrives, when also this is over
the same phenomenon starts again, and so on (3) the occur
rence of rare, very large fluctuations in the flow of the
incoming particles: they cause a traumatic change in the be
havior of the h.p. which in turns determines an almost com
plete loss of the memory of the past history of the h.p ..
Points (2) and (3) provide an example of the mechanism
proposed by [16] and described before. Point (1) offers an
analogy with the hyperbolic structure assumed to be relevant
257
in the analysis of chaotic behavior of finite-dimensional
systems. We think such remarks might come useful in the un
derstanding of the approach to stochasticity in the infinite
systems.
Let us consider a configuration xE* • The first par
ticle (q0 ,v0 ) in x is the h.p., the coordinates of the
other particles are denoted by (qi,vi) • We denote by ~
a fixed Gibbs measure for the system and by St the time
flow, hence Stx is the configuration at time t starting
from x at time 0 • Let us consider a light particle
(qi,vi) in x and denote by (qi(t),vi(t)) its coordi
nates at time t in Stx . It is convenient to assume that
whenever two light particles collide they pass through each
other with unchanged velocities. It is easy to see, cf. [7],
that almost surely each particle interacts in a finite in
terval of time, say t , t + • Then the
0 lim s+t (qi (-t) 'vi (-t)) t-Too
exists, S~ being the free evolution with elastic colli
sions at 0 • By repeating the argument for each light par
ticle in x we construct a configuration $+x in * 0
which can be written as
~+X lim so t E S -tx ( 0. 1 )
E is the map from * onto *0 obtained by "erasing" the
h .p •• The problem is then to reconstruct X out of ~+X This looks a very difficult task since the coordinates of
the h.p. in ~+x are hidden in the requirement of compati
bility which states that the asymptotic velocities of the
light particles should be those given in ~+x • If one can
solve this problem then he can invert ~+ to obtain the
wave operator ~+ which maps *0 into * It turns out
that a direct construction of n+ is easier than trying to
invert ~+ • Namely we start considering a x 0 E*0 and then
we "try" to define
(0. 2)
258
where I is a map from X0 into X which consists in in
serting, in some way still to be specified, the h.p. in the
available space. Obviously the existence of the limit in eq.
(0.2) is linked to proving that the indeterminacy in the in
sertion of the h.p. is irrelevant in the limit when t goes 0 -to infinity. Let X be the set where ~+ is defined and
assume that X0 is S~ invariant. Then as a consequence of
this and of eq. (0.2)
(0.3)
Namely n+ carries trajectories into trajectories. With the
help of eq. (0.3) one can then try to prove that the inverse
of n+ is ~+ . Notice that if X~ is measurable and 0 0 0 0 0
1.1. (.it+) > 0 , then, by the ergodicity of (.it ,1.1. ,St) ,
1.1. 0 (.it~) = 1 . If we can also prove that fi+ is an isomorphism
between the probability spaces !*0 ,1.1. 0 ) and !*,IJ.l then
defines an isomorphism also between the dynamical syso 0 0
!*,IJ.,St) and !* ,1.1. ,St)
Unfortunately we have not been able to proceed in strict
agreement to the above scheme. The main problem arises from
the request that the limit in eq. (0.2) should exist with
1.1. 0 probability one. The way out is to modify the r.h.s. of
eq. (0.2): we define a family It of insertion maps from
* 0 into * and we pose
(0.4)
It is clear that greater freedom in the choice of the inser
tion map makes it easier the proof of the limit in eq. (0.4).
At this point, however, it is not any longer evident that
St~+ = ~+s~ • Such property will be regained by choosing the
It 's so that they are constant in long time intervals which
become infinitely long when t goes to infinity. We in fact
prove the following
Theorem 0.1. One can define It in such a way, see
Def. 3.5, that there exists a set * 0 c * 0 1.1. 0 !*~l = 1 , + I T 0 0 0
stx+ =X+ , such that
a measurable map from
the limit in eq. (0.4) exists. ~+ is
*~ onto ~+*~ which establishes a
259
0 0 0 v modulo zero isomorphism between l* ,~ ,St) and (~,~,St)
Analogous construction holds for t<O . Denote by n the
corresponding wave operator. Then n=1n+ is a non-trivial
isomorphism of (~0 ,~0 ,S~) into itself (modulo zero).
n=1n+ is the scattering matrix of (~'~'St) .
In Section 1 we prove the existence of contractive mani
folds through each configuration in a set of full measure.
In Section 2 we discuss the large fluctuation - nice
event - which causes an almost complete loss of the memory
of the past. At this stage we recall results obtained in [7],
[6], [4].
In Section 3 we prove the existence of the wave opera-
tors.
In Section 4 we conclude the proof of Theorem 0.1.
Detailed proofs of our results require more space than
what is available in the present Proceedings. So we did not
reproduce proofs "essentially" existent in the literature.
Horeover we took the attitude to give for granted all those
probability estimates which origin from properties of Pois
son processes and which are extensively considered in the
analysis of models describing the systems of equilibrium
statistical mechanics.
1. Contractive manifolds
The contractive nature of the system (~'~'St) is
stated in Thm. 1.2, first we pose the following:
1.1 Notation. Let xE* and let L and r be positive
numbers. V(x,L,r) denotes the set of all configurations
which can be obtained by shifting all coordinates (both posi
tion and velocity) of any particle of x which is in O,L
by less than r .
If y is in V(x,L,r) there is a natural labelling of
the particles of y which is inherited from that of x We
consider the collisions of the h.p. both with the light par
ticles and with the wall at the origin. i(n) is defined as
the label of the light particle at the n -th collision. If
the collision is against the wall at the origin we set
i (n) ~ 0 . We will then say that the h.p. has the same order
260
of collisions in x and y E V(x,L,r) if i(n) is the
same for each n both in x and y •
1.2 Theorem. There exists a set .lEcd.L(Xc) = 1 , such that
for every xE.lE the following holds. Let xE.lEc and L>O ,
then there is r>O such that for every y E V (x,L,r)
( 1 • 1 )
exponentially fast. [ q0 {x) denotes the position of the
h.p. in the configuration x .] Furthermore the h.p. has the
same order of collisions in x and y •
Given e and L (positive) there is r and a set
Xc (€ ,L) such that for every x E Xc (€ ,L) and y E V (x,L,r)
eq. ( 1 • 1 ) holds and
( 1. 2a)
( 1. 2b)
1.3 Proof of the "infinitesimal version of Thm. 1.2. We as-
sume that the coordinates (qi,vi)
changed by infinitesimal amounts,
other coordinates are kept fixed.
of x for qi<L are
dqi dvi , while all the
We will see that for x
in a set of full measure dq0 (t) dv0 (t) will tend exponen
tially to zero. Since we consider infinitesimal changes the
collisions of the h.p. with the light particles and the ori
gin occur at the same times in x and in the varied config
uration, here we assume that x is chosen so that there is
no triple collision. Due to the exponential decay of dq0 (t)
and dv0 (t) we will then be able to show that for finite,
small enough changes of the qi , vi analogous property
holds: namely the h.p. has the same order of collisions in
x and y and this will be enough to extend the result to
the finite variation case.
The main point in the proof is the reduction of the
estimate to that concerning a finite number of particles.
we call cluster time a time t when the h.p. collides
with a light particle for the first time and after t the
h.p. does not collide with it anymore and furthermore no
other collision occurs with any of the particles involved in
261
the previous interactions. Cluster times exist with positive
probability [7], and since the system is ergodic, they have
non-zero frequency. Hence we introduce the set
( 1. 3a)
where N(t) denotes the (random) number of cluster times
up to time t . Then given any E>O there are c 1>o and
such that
J.!({N(t) ~ c 1t- c 2 'v't~O}) > 1-E • ( 1 .3b)
In the following we consider xE {N(t)~c 1 t-c2 'v't~O} . We
also take x so that no triple collision occurs. Let us
assume that the particles which collide with the h.p. up to
the first positive cluster time are (q 1 ,v1 ), ..• , (qn,vn)
and that the coordinates of the h.p. are (q0 ,v0 ) • We de
note by (dq0 ,dv0 ), ••• ,(dqn,dvn) the infinitesimal changes
in the coordinates of such particles, not necessarily being
all different from zero. The evolution of such systems is
reduced to the motion of a particle in a n+1 -dimensional
billiard with planar boundaries [24]. Since the kinetic
energy is conserved it is natural to consider
which the square length of (dv0 , ••• ,dvn) is
+mdv~ . We then change variables: y 0 = M1/ 2q 0
a metric for 2 Mdv0 + ... +
w =M1 / 2v , 0 0
- 1/2 - 1/2 • . -Yi-m qi, wi-m vi, 1.-L .. n. In such variables we
consider the usual Euclidean metric. We assume to have la-
belled the particles in such a way that yi~yi+ 1 • The time
evolution read in the new coordinates induces a motion in
the n+1 -dimensional billiard 0 .S:y0 ~ (M/m) 112y 1 , yi ~yi+1 for i=1, •.• ,n-1 • The dynamics is such that when the point
gets to the boundary of the billiard it is elastically re
flected. From this it follows that
( 1. 4a)
262
There will be a k such that after the k -th cluster and
for the first time the only indeterminacy concerns the h.p ••
The "error" will be {dq~,dv~) and in general will be
larger than the initial one. Let us call t- the interval
of time between the k -th and the (k+1) -th cluster times.
t means that we are considering what happens just before
the last collision. Using eq. (1.4b) we have
We can compute explicitely the effect of the last collision,
which, by definition of cluster time, involves a "new" par
ticle {which is colliding for the first time) . Hence we get
I +I I I 1 dv 0 ( t ) $ a dv ~
( 1 • 5)
a= (M+m)- 1 (M-m) < 1 .
Iterating the argument for each other cluster we have an ex
ponential decrease of dq0 (t) , dv0 (t) once we notice that
xE{N(t)~c 1 t-c 2 , 'v't~O}
1.4 Proof of Thm. 1.2. In subsection 1.3 we have seen that
the system behaves like a billiard with planar faces. Con
cave billiard have hyperbolic structure in the finite dimen
sional case. Concavity is here replaced by the infinite size
of the system and the cluster structure of the dynamics.
Like for concave billiards troubles arise from discontin
uities. In our case they appear as changes in the order of
the collisions of the h.p. and were disregarded in 1.3 under
the assumption of infinitesimal perturbations. Our argument
goes like this. We assume that the collisions of the h.p.
with the light particles and with the wall at the origin
have the same order both in x and y . In such case the
bounds of the infinitesimal case extend to the present case
due to the locally linear character of the dynamics. We then
need a consistency check to lift the above assumption. We
can assume that with probability 1 there are c 1 >0 and
c 2 so that x E {N(tnc 1t-c 2 'v't~O} [7]. Supposing that in
y the order of collisions is the same as in x , we get
263
Here y and a 0 are some suitable positive coefficients
and !oq! !ot! are the Euclidean norms of oq0 , ••• ,oqn
ov0 , ••• ,ovn respectively, these being the only varied co
ordinates in going from x to y •
For the light particles we have the following estimate:
let ti denote the time of the last collision of the par
ticle with label i then
The consistency of eqs. (1.6) and (1.7) with the assumption
that x and y have the same order of collisions is based
on some restrictions on the choice of x and on making the _,. _,.
initial error oq + ov small enough. We first give the con-
ditions on x , we then show that the consistency check
holds and, in the next subsection, that the above conditmns
hold in a set of full measure.
Eq. ( 1 • 7) shows that !oqi (t) 1 increases in time, hence
the only way out is to show that in Stx the i -th par
ticle gets so far away from the h.p. that the "error"
oqi(t) becomes unimportant. The first requirement for X
is therefore that
'v't~O ( 1 • 8)
Then we need a lower bound on the speed of the particles
(the bounds given below are not optimal)
( 1 • 9)
ti being the last time that particle i interacts with
the h.p. From eqs. (1.7) and (1.9), assuming that !oq!+!o~! is small enough, it follows that 2!ovi(t) l~!vi(t) I , there
fore
Then there exists k 2 so that
for t>t. - l
264
(1.10)
This shows that for any i after k2 +t~ the i -th par
ticle stays strictly to the right of the h.p. both in x
and y : we only have to control the collisions of each par
ticle for a finite interval of time. Anyway the collisions
of the h.p. need to be "well separated" in time. We there
fore require that for some constant k 3
~t(x) ~ (t2+k3)-1 (1.11)
ot(x) ~ (t2 +k3)- 1 (1.12)
By ~t(x) and 6t(x) we mean the following. Given t let
t_~t be the time of the last collision before t of the
h.p. (at 0 or with a light particle) and t+ the next
collision (strictly after t ) . Then ~t (x) = t+-t- . Let q
be the position of the h.p. at t If q=O then ot(x)
denotes the position of the next light particle. If q=O
6t(x) is the minimum between q and the distance from q
of the next particle (of course disregarding the particle
which is involved in the collision at q ) .
Another request on x is motivated by the following
fact. Assume the h.p. has a collision with a light particle,
the two velocities being almost the same: then if the co
ordinates are even slightly changed the resulting delay in
the collision might be large. Hence we require that there is
k 4 such that at any collision time of the i -th particle
with the h.p.
( 1 . 13)
Finally we need an upper bound on the velocities of the par
ticles. x is chosen so that if t is a collision time the
outgoing velocities v 0 (t) and vi(t) of the h.p. and the
light particle, respectively, satisfy the inequality
It is now easy to see that under conditions (1.6) •.. (1.14)
265
the order of the collisions in x and y is the same, pro
vided that [oq[+ioti is small enough.
Assume in fact that at time t there is a collision in
x with parameters (q0 ,v0 );(q0 ,v) • Assume that in y the
parameters of the same particles are (q0 +oq, v0 +ov)
(q0 +Oq , v+ov) • The delay time for such collision in y is
( 1.15)
Since t,Sti , by definition, by using eqs. ( 1 • 6) , ( 1. 7) and
(1.13) we get a bound which becomes exponentially small in
time. By eq. (1.11) and taking [oql+lotl small enough we
prove the required consistency for any such collision.
Assume now that at some time t we have (q0 +oq0 ,
v 0 +ovo) and (q+oq , v+ov) with q+oq = qo +oqo , namely a
collision in the varied configuration. We then need to show
that (1) also in
ticles, let t:.T
x there is a collision between these par
be the corresponding delay; (2) the pre-
vious and the next collision in x occur with a delay
larger than t:.T . The argument now goes as follows. t:.T is
again given by eq. (1.15). By eq. (1.10) t~k2 +t~ ( i be
ing the particle involved in the collision). Hence t:.T is
exponentially small with t . In principle there are two
possibilities. (a) the collision corresponding to (q0 ,v0 )
and (q,v) is one of those occurring in the history of x
(b) this is not the case and in the x history another light
particle collides with the h.p. preventing the (q0 ,v0 )
-(q,v) collision. In case (a) everything goes like before
and the consistency check is verified. Case (b) contradicts
our conditions and cannot occur. In fact using eq. (1.14)
we bound the velocities v 0 and v so that the light par
ticle for !t'I.St:.T will be at a distance from the h.p.
which is less than 2 (c 5log+t + c 6 )t:.T • This is exponentially
small for t large and it is incompatible with eq. (1.12)
if in the meantime another collision (case (b)) occurs. By
choosing loql+lo~l small enough the same reasoning leads
to contradiction whatever the time t of the collision is.
The collisions of the h.p. against the origin can be
analysed in the same way, so we are left with the proof that
266
the conditions on x we have stated in this subsection are
satisfied in a set of full measure.
1.5 Probability estimates. We just outline the proof that
the estimates used in subsection 1.4 are verified with prob
ability 1. The estimate relative to eq. (1.8) is proven in
[ 7 J • Eqs. (1.9) .•• (1.14). We describe the dynamical system
(~'~'St) by introducing the following special flow repnesen
tation [9) , [23). The basis of the flow is the set of all
configurations where the h.p. is colliding with a light par
ticle. Hence the base is described by (1) a configuration
xE~ where the colliding light particle is ignored and (2)
a velocity v 1 which represents the outgoing velocity of
the colliding light particle. (The position of the collision
is of course q 0 (y) .) The special flow representation is a
dynamical system in the space { (y,v 1 ,tJ} where t ranges
from 0 to t(y,v1 J , which denotes the next time the h.p.
collides again. (y,v1 ,tl is the configuration evolved for
a time t starting from the configuration corresponding to
(y,v1 J • The measure on the base is
1 2 -1 - 1/ 2 d~(y,v 1 ,tJ =exp(-"2"13mv1 J (13 m) (v1-v0 Jx(O.S:t.S: t(y,v1JJ
( 1.16)
~ (dy)dv1 dt
and the flow is represented as an upward lift and a trans
formation s on the base.
From eq. (1.16) it is easy to see that the number of
collisions of the h.p. with the light particles and the
origin grows linearly in time with probability 1 • The es
timates given in subsection 1.4 then easily follow from the
analogous estimates for the transformation S on the base.
2. Large fluctuations (the nive event)
We report here some results obtained in [6) and [4]
which prove the existence of the nice event described in
the introduction.
2.1 Definitions. Assume L,N,V be given positive numbers.
267
The set V(L,N,Vl is the set of all configurations x
which have the following properties:
c .1 The particles of x are in ~ , namely if (q,v) Ex
and v50 then q~L ;
C.2
by N
c.3 than
The number of particles of
and q0 (x) < L/2 ;
The speed of any particle of
v
X
X
in [0,2L] is bounded
in [ 0, 2L] is less
In our applications we will consider the limit L go
ing to infinity and we consequently choose N and V so
that {x: x n RL E V (L ,N, V)} has large probability.
The aim is to show that there•is a set of configura
tions of incoming particles in (L,oo) so that after some
time T the new configuration in [O,L) is very close to
some prescribed configuration xL independently of
x E V (L ,N, V) • The incoming particles are specified in a
bounded region of the one particle phase space, the region
as well as the time T depend on the target xL and the
accuracy with which it should be reached.
2.2 Theorem [6], [4]. Let L,N,V,E,xL be given. Here xL
is a configuration of particles in [O,L) , q0 (xL) <L , and
for notational simplicity we assume that if y E V (xL,L, E) ,
cf. Notation 1.1, then also y is a configuration in [O,L).
Then there exist T, y and o>O which depend on L,N,V,E,
xL so that the following holds. Let
CL(O,T) = { (q,v): q~L, v<O, q+vT ~L} (2.1)
Then there exists a set of configurations C(L,T,E,xL) (for
ease of reference we only explicit the dependence of these
parameters) which is cylindrical with base in CL(O,T)
such that:
p .1 C(L,T,E,XL) in CL(O,T) is of the form v (y, 0) where
y is a configuration in CL(O,T) and v (y, 0) is the set
of all configurations in CL(O,T) obtained by shifting the
coordinates of any particle of y by less than 0
P.2 Let xEV(L,N,V) and let zEC(L,T,E,XL) be a con
figuration of particles with negative velocities such that
268
the only particle in [L,2L] are those specified by the
condition that zE C(L,T,s,xL) • Then ST(xUz) E V(xL,L,E)
P.3 Let x and z be as in P.2. Then the h.p. does not
collide with the light particles of z in CL(T,oo)
= { (q,v): q~L, v<O, q+vT > L} in the time interval [O,T]
P.4 Let x and z be as in P.2. Then all the particles of
x which are in [O,L] as well as those which have collided
with the h.p. before time T have at time T positive ve
locity larger than 1 . The same holds for all the particles
of z which at time T have positive velocity and are to
the right of L
It is possible and convenient for the sequel to arrange
the parameters so that
We shall refer to the set C(L,T,E,xL) as "the nice event",
the specification of the nice event being the set of param
eters: L,N,V,s,xL,T,y,o
3. The wave operator
In this section we prove the existence of the wave
operator, i.e. of the limit in eq. (0.4). Given x 0 we con
sider S0 x 0 as a function of tE~ Special values of t -t + • are when s~tx0 has a set of incoming particles like those
specified by the nice event described in Thm. 2.2: for such
t 's we define It by inserting the h.p. so that, at such
times, the conditions of Thm. 2.2 are all satisfied and the
nice event actually occurs. The problem arises for other
t 's. Assume t is a time for which the nice event occurs
and take t'>t • Then we would like that St'-tit,s~t,x0 is
still a configuration for which the conditions of Thm. 2.2
apply. We do not know if this is true uniformly in t'>t •
So we proceeded as follows. We fix a sequence of different
nice events: the parameters L,N,V being chosen so that the
probability that the configuration is not in V(L,N,V) be
comes very small along the sequence. We denote by t~ the
first time when has a set of incoming particles vfuich
269
satisfy the conditions for the first
t*>t* is the first time after t* 2 1 1
nice event to occur.
when the second nice
event's conditions are verified and so on. We define
* * for t E [tn , tn+ 1 ) 0 0 so to force ItS-tx to have the nice
0 0 event occurring at st-t*ItS -tx . Then we need to show
n that this holds also when t E [t~+p, t~+p+ 1 ) , p>O . After
each nice event the state will be in some contractive leaf
so it will be sufficient to check such property only for
p=1 and for all n larger than some N . We will first de
fine the sequence of nice events, then the stopping times
t~ , the map It , the set ~~ , for which the above can be
proven, and finally the wave operator.
3.1 Notation. For ease of reference we report here some
notation which will be frequently employed in the sequel.
For L>O
rL {(q,v): q~L, O~v~-L 1 / 2 exp(-qL- 112+1)}
r~ {(q,v): q~L, O,Sv,!!;L 1/ 2 exp(-qL- 112+1)}
RL = { (q,v): either v~O or v<O and q,!!;L} .
For O,Ss<t CL(s,t) = {(q,v): q~L, v<O, q+vs~L~q+vt}
3. 2 Definition. We choose an increasing sequence Ln, Ln-~ oo ,
so that
IJ. 0 ( { x n r ~ = 0 } l > 1 - 2 -n n
Given Ln we define Nn and Vn so that
cf. Def. 2.1 and Notation 3.1.
(3. 1)
(3.2)
(3.3)
3.3 Remarks. We will prove the existence of the wave opera-
270
tor by means of an iterative procedure, the "allowed" error
at the n -th step will be 2-n • Notice that in
+ 1/2 {x n rL=!.O n {q0 (Stx)~L log+ t, 'v't~O} the h.p. never col-
lides with any of the particles which at time 0 have po
sitive velocity and are to the right of L • Eqs. (3.1) and
(3.3) are easily proven since ~ and ~0 are Poisson meas
ures. Eq. (3.2) is proven in [7].
3.4 Definition. Given L as in De f. 3.2 we define a con-n figuration XL of particles in [O,Ln]
n time Tn > 2T so that n
c. 1 -n ~ UL ( 2 'L ) I XL )
1. n n > 1_2-1/2n
fined in Thm. 1.2.
rn is defined in such a way that if
y E V(x,Ln,rn) then eq. (1.2) holds.
where
' a time T >0 ' a n
Jti(E,L) is de-
and
C. 2 For any set A measurable w. r. t. the a -algebra gener
ated by {q0 (t), t~Tn}
I~ (A ixL ) -~(A) I < 2-n n
(3. 4)
C.3 ~({there is a cluster time in (T ,2T l}lxL) > 1-2-n • n n n
The definition of cluster time is given in subsection 1.3.
C.4 Let J(s,t) , O<s<t , be the set of configurations
where the h.p. does not interact with any of the particles
in CL(t,oo) during the time interval [O,s] . Then
- A , -n ~(J(2T ,T) lxL) > 1-2 •
n n n (3 .5)
Remarks. We need to prove that the above definition is
well posed, namely that xL , T , T actually exist. By eq. n n n
(1.2) C.1 holds in a set of measure larger than 1-2-112 n
By choosing Tn large enough eq. (3.4) holds in a set of
measure larger than 1-2-n , cf. [6]. The same holds for
condition C.3. Once T is fixed J(2T ,t) has a measure n n
which goes to as t diverges, as it can be easily seen.
Hence the XL n
for which eq.
271
(3.5} holds have measure
larger than 1-2-n for T large enough. Eq. (3.6) holds
in a set of measure larger n than 1-2-1/ 2 n as it can be seen
using eq. {3.2). The intersection of all the above sets has
positive measure hence it is not empty.
The motivation for the above conditions is the follow-
ing: xL is (essentially) the configuration left in n
[O,Ln] after the n -th nice event. Tn gives the rate to
which equilibrium is reestablished after the nice event. The
other conditions ensure that particles far away from the
origin behave like free, so that the "next" nice cluster of
particles responsible for the new nice event is not "per
turbed" before its arrival.
3.5 Definition of It • We first decompose the positive
real axis into intervals [Tn,Tn+1 ) whose length increases
very fast with n . In fact the Tn 's are chosen according
to the following requirements:
Given n let C(L ,T ,r ,xL ) be as in Thm. 2.2 n n n n
(rn is defined in C.1 of Def. 3.4 and Ln,Nn,Vn in Def.
3.2). For notational simplicity we assume to have chosen the
parameters so that
If this were not the case we would choose
respondingly TE as in Thm. 2.2. By making
would then have the above inequality.
(3. 7)
(3. 8)
E < r and corn
E small we
Let now tn be the infimum of the times t>T - n such
that
0 0 S tx E C (L ,T ,r ,xr ) - n n n ~n
We define t*=t n n if
t~ ="" . The Tn 's are chosen so that
otherwise we put
272
We require that 1n+1 - 1n is much larger than the
other times introduced so far. For instance 1n+1 - 1n > > 10(Tn+1+Tn+1+Tn+1) Tn in C.1 above and
as follows. Let
largest integer for which
is defined in C.2
in C.4 of Def. 3.4.
of Def. 3.4,
It acts on
t>t* . Then - m
Let m~n be the
0 0 ItS-tx is obtained
from S~tx0 by (1) erasing all the particles of s~tx0
which are not in CL (t-t*,oo) , (2) inserting the h.p. at m m
with zero speed. If there is no t* which is less than m
0 0 t then ItS-tx only h.p. at 1
in the sequel. )
is the configuration consisting of the
with zero speed. (Such case will not occur
Next we introduce the set *~ which will be the domain
of definition of the wave operator
of rather technical origin specify
Several conditions
We first give some
qualitative description of these conditions. The basic re
quest is that {t~<oo} from some N on. Denote by zn the
0 0 configuration s_t*+T x n CL (O,oo) n n n
-n then require that yn E *c(2 ,Ln)
and
so that a contractive
leaf through Yn exists. Furthermore we ask that Styn
satisfy the conditions for applying Thm. 2.2 at t
= t*-T -t* . We also need that the particles of n n n-1
- * * S- y , t = t -T -t 1+T 1 , which have positive velocity t n n n n- n-
and are to the right of Ln_ 1 will not interact with the
h.p .. In this way s_ Y t n
is in the same contractive leaf
of and an iterative procedure can be applied.
3.6 Definition. The set is the set of all con-
figurations x 0 for which there are N and c>O with the
following properties. For each n~N
c. 1 {t* < oo} n
C.2 We denote by {tnEI} the event { s0tx0 ~ C (L ,T ,r ,xr. )'v'tEI} n n n 11
and by {tnEI} its complement. Then
273
{t ~I } , I = {t: Jt-1 J<cn} U {t: Jt-1 --21 (1 +1-1 ) !<en} • n n n n n n n -
(Condition C.2 will be used when proving that
invariant.)
0 0 { } S-t*x n (q,v): q E [0 ,Ln], v~O, q+vTn ~ 0 n
C.4 We write here and in the following Yn
0 0 Zn S_t*+T x n CL (O,oo)
n n n
0 0 (Notice that zn = ST It*S-t x ncL , hence by C.4 after
n n n n the n-th nice event the configuration is in the contractive
leaf of yn • )
c.s q (S y ) _< Ln1/ 2 log,_ t, 'v't~O o t n ,
C.6 For there is a cluster time in the interval
[Tn,2Tn]
C.7 ynEJ(2Tn,Tn) , cf. C.4 of Def. 3.4.
(Condition C.7 will ensure that the h.p.
interact with the light particles of the
in the initial time [0,2Tn] .)
C.8 Let =z(1) +z(2) Yn+1 n+1 n+1 where z (2) =
n+1
in Yn (n-1)
does not
nice event
0 = s_t* _ +T +t*-T zn . Then the h.p.
n-t-1 n+1 n n interact with the light particles of
in yn+1 does not
(2) zn+1 in the time in-
* * terval [0, tn+ 1-Tn+1-tn]
(By C.8 we have that St* -T -t*Yn+1 has the light parn+1 n n
ticles required for the n -th nice event. we need however
274
to impose that (1) the configuration is in
(2) that the particles initially in XL n+1
V(Ln,Nn,Vn)
U z (1 J and which n+1
at t* -T -t* have negative velocity are in [O,Ln] . Non+1 n n tice that for (1) and (2) we need not to worry about the
light particles of x U zn(1)1 which have interacted with Ln+1
the h.p. before Tn+ 1 . In fact we know that in [Tn+1'2Tn+1J
there is a cluster time and after that they will not inter
act anymore with the h.p. Finally we want to prove that
St* -T -t*+T Yn+ 1 is in the same contractive leaf of n+1 n+1 n n
yn . Since yn does not have light particles with non-nega-
tive velocity to the right of Ln we need to require that
the h.p. in St* -T -t*+T yn+ 1 does not interact with n+1 n+1 n n
the non-negative particles which are to the right of Ln .)
C.9 S * * (XL U z ( 1 ) ) c R tn+1-Tn+1-tn n+1 n+ 1 Ln
c. 10 Let nn+1
without the light
h.p. before Tn+ 1
be the configuration St* -T -t*Yn+ 1 n+1 n+1 n
particles which have interacted with the
Then
nn+ 1 n RL E V(Ln,Nn,Vn) n
Furthermore the h.p. in nn+1 does not collide for all t~O
with any of the particles of nn+ 1 which (in nn+1 are to the right of Ln and have non-negative velocity.
3.7 Theorem. The set *~ defined in Def. 3.6 is ~0 meas
urable. ~ 0 (* 0+) ~ 1 . 5°*0 = *0 for any t . The limit t + +
exists. rl+
range rl+*~
is a measurable map from
and
into with
Proof. *0 is a countable intersection of measurable + sets, hence it is measurable. By C.2 of Def. 3.6 t~(S~x0 )
= t~(x0 )+t for all n~N , where N might depend on t . It
275
is easy to see that C.2 .•. C.10 of Def. 3.6 also hold for
s 0 x 0 , hence ~0 is s 0 invariant. t + t
Proof that J,L 0 ~~~) = 1 • We use Borel-Cantelli, so we
need to estimate the probability of each of the events which
specify ~~.We denote by xi(n) the characteristic func
tion of the events C.i , i=1, ••• ,10 of Def. 3.6.
C.1 J,L 0 (x 1 !n)) > 1-2-n by C.1 of Def. 3.5.
C.2 We will use the notation en for C(L ,T ,r ,xL) n n n n
We have by eq. (3.8):
where k ranges over the integers such that [k-T I < lk-(T +_l(T -T )} l
n < cn+1 and < cn+1
1 n 2 n+1 n
C.3 We condition on the event {t~=t} . Yn and z n are
defined in C.4 of De f. 3.6. We have k,.,
J.L 0 (1-x 3 (n)lt*=t) ~ IJ. 0 ({t >t})- 1J.L 0 [ n {t ~[t-(2k+2)T , ' n n k=O n n
( 3. 9)
t-(2k+1)Tnl} n {s~tx0 n {C.3}=~:n)
where {C.3} denotes the event in C.3 with t*=t and k* n
is 0 if t-Tn < 3Tn otherwise it is the largest integer
k such that
We have just relaxed the condition {t~>t} so that the
events in the r.h.s. of eq. (3.9) become mutually indepen
dent. We shorthand
(3. 10)
we use Cauchy-Schwarz in the last term on the r.h.s. of eq.
(3.9), then by eqs. (3.1) and (3.7) we get
IJ.o(1-x (n) lt*=t) ~ ,o({t*>t})-1 (1-o T )1/2k*.3·2-1/2n 3 1 n ~ · n · n n
Notice that by P.5 of Thm.2.2. For k>2T n
T >1 n we have (t s=t*-T ) n n n
~ (1-o T )-1 (1-38 T ) n n n n
276
( 3 • 11 )
By summing eq. (3.11) over all the positive k 's and using
the above estimates we get
The procedure for proving C.4, ... ,C.7 is similar so we only
outline it for the condition C.4. We use the same approach
as for C.3, the only difference arises from the fact that
here we need the estimate for ~0 ({y ~* (2-n,L )}) . We n c n write
( Acompl. U 8compl. , + ~ IXL
n
since in AnB the h.p. does not interact with the light par
ticles which have positive velocity and are to the right of
Ln . By C.1 and C.S of Def. 3.4 and eq. (3.1) we have a
bound which goes like 2- 1/ 2n .
277
C.8 We condition to the evep.t {t~+ 1 =t, t~=s, z~~~} where,
using the same notation as in C.8 of Def. 3.6,
(2) zn+1 s 0 z -t-T -s n n+1
Proceeding like in the proof of C.3 and writing x =
= xL U z 11 l U z( 2 1 we come to the estimate of n+1 n+1 n+1
(1} } I (2) 1J.({zn+1 : o0 <t-s 1xLn+1'zn+1 ) (3.12)
where o0 is the first time that in Stx the h.p. collides
with a particle of z~~~ ; ot, is the first time after t'
when this happens. We can reduce the estimate of eq. (3.12)
to the following
(3. 13)
the "error" being as in C.7. The event to estimate is now
measurable w.r.t. the history of the h.p. after
we can disregard, by C.1, the conditioning on
[7] we have that
iJ.({x:o <t-s}iz 1+2 )1 Jx({z n fL =ofb}) S o n n n
T hence n+1 xLn+1 . As in
5 IJ.({x: q (St t'x) ~ L1/ 2 log+lt'l,'v't'<O}lx!{z nrr. =fb}) o -s- n n --n
which is exponentially small, cf. eq. (3.2). We are then
left with the estimate of x ({z n rL =(3}) which can be n n
estimated as in C.3, we omit the details.
C. 9 We proceed like for C .8. vle notice that the event
{ St-T -s (xL U z ( 1) ) E RL } if {there is a cluster in Tn+1, n+1 n+1 n+1 n
2Tn+1}
becomes measurable w.r.t. the history of the h.p. after
Tn+ 1 . In fact particles which have interacted with the
h.p. before Tn+1 have positive velocity after 2Tn+ 1 (because of the existence of the cluster time) , hence they
are in RL . After such remark the proof is reduced by n
278
C.2 of Def. 3.4 to the one without the conditioning on
x and an exponential bound can be obtained, cf. [7]. Ln+1
Proof of c. 10. The first condition in C.10 concerns
the behavior of the h.p. after Tn+1 and can be easily re
duced to an equilibrium estimate, the proof goes like in
[7] and we omit giving the details. The other condition re
quires more care because just after the nice event the state
is "atypical" and we cannot use (directly) equilibrium es
timates • However, by C. 5 of Def. 3 • 6 q0 (StY n) .$ lib log+ t ,
Vt~O . If is in the same contractive leaf of
IT -n then q0 (St nn+ 1 ) ~ vLn log+ t+ 2 Vt~O • We require that -+-nn+1 n r L- 13 ,
n
f~n = {(q,v): q~Ln' O.s;v.s;exp[-q~-n -1)}
n
so that the last condition in C.10 holds. On the other hand -+ the condition {nn+1 n rL =13} can be proven to hold with
n large probability when n~oo , just by using equilibrium es-
timates.
We will now prove the existence of the limit which de
fines the wave operator ~+ • We shall actually prove that
0 ~+X =lim St*-T yn (3.14)
n n
For notational simplicity let us consider 'n+2 >t~'n+ 1 + 1
+2('n+2-'n+1 > where n~N , cf. Def. 3.6. By definition of
It we have that
where yn+ 1 is in the same contractive leaf of yn+ 1 •
Therefore the h.p. in yn+ 1 will remain 2-(n+1 ) close
to the h.p. in yn+1 at all later times. We then consider
the evolution of Yn+1 . At time
the configuration SEyn+1 is such that:
279
(i) Styn+1 n CL (O,Tn} n
definition of t~ .
E C (L ,T ,r ,xL ) n n n n
by C .8 and the
(ii) There is no particle in { (q,v): q E [Ln, 2Ln] ,
v<O, q+vT > L } n n because of C.3 and C.8, as far as the par-
ticles of (2) zn+1 are concerned, and by C.9 with regard to
the particles of z ( 1 ) n+1
(iii) In RL the configuration is in V(Ln,Nn,Vn) n
except for particles which will never interact in the future,
by C.10 of Def. 3.6.
Therefore by Thm. 2.2 Sf+T yn+ 1 n {(q,v): vs:Oorv>Oand n
q~Ln} is in the contractive leaf of Yn . By C.10, on the
other hand, we know that the h.p. in St+Tnyn+1 does not
interact with the light particles which are in the region:
{ (q,v): v~O, q>Ln} . So the h.p. remains 2-n close at all
times to the h.p. in Yn . As a consequence the h.p. in the
configuration St-t*+T Itx0 is at all times 2-(n+1 l+2-n n n
close to the h.p. in Yn From this eq. (3. 14) follows.
From the invariance of xo +
follows that
r2 so + t
4. The isomorphism theorem
In this section we prove the
4. 1 Theorem. The map rl+ on xo +
tablishes an isomorphism between
Proof. We first prove that
We will prove that for x = r2 x 0 +
and that
and eq. ( 3. 14) it easily
on xo +
following
defined in Thm. 3.7 es-
0 0 0 (X ,~ ,St) and (X,~ ,St)
rl+ is invertible on 0 X+ .
the following limit exists:
(4. 1)
(4. 2)
280
Et is defined as follows. Let n be the largest integer
such that t~Tn~(Tn+Tn+ 1 l then Et acts on x by taking
away from x the h.p. and all the light particles which
are not in CL 'rL By Thm. 3.7 and eq. (3.15) we have n n
Given t let k be the largest integer so that
We choose t so large that k~N and we take n > k+1 . We
consider St' yn and we have that the h.p. does not inter
act with the particles of zk for t' < t*-T -t . In fact - n n St*-T -t +T y is in the contractive leaf of yn_ 1
n n n-1 n-1 hence the h.p. has the same order of collisions as in
Yn_ 1 . By iterating the argument the h.p. in
St*-T +t* +T y has the same order of collisions as n n k+1 k+1
yk+1 . By C.8 of Def. 3.6 we know that the h.p. in yk+1 in-
teracts with the particles of zk only after time
t~+ 1 -Tk-t~ By C.3 of Def. 3.6 we can see that the par
ticles of zk are not affected by Et , hence 0
St-t*+T EtS-tyn = zk in CL (Q,oo) • Therefore k k k
0 0 zknC0 (Tk 1 oo) = S_tk+Tkx nc0 (Tk,oo)
because, by C.3 of Def. 3.6,
0 0 { } S_tkx n (q,v): qE[O,Lk), v~O, q+vTk > 0 (a •
We then have
The same holds in the limit
the limit when t diverges
n going to infinity, hence in 0 0
1jJ+fl+x =x
fl+ transforms the measure ~0 in a measure \ on
281
~+~~ , we want to show that ~=~ . We will only sketch the
proof. Let f be a bounded, Lipschitz continuous, cylind
rical function on C ( lR , lR + ) , i.e. the space of paths of
the h.p .. We denote by f(x) the value of f at {q0 (Stxi,
tEJR} . By eq. (3 .14) we have that
lim ~ (fn) = ~(f) (4 .3) n+oo
where ~(f) denotes the ~ expectation of f and
(4 .4)
We condition on the value of t* , say t*=t n n This affects
of zn , cf. C.4 of Def. 3.6; since the distribution
s0 x 0 ~ C for -t' n t'<t . If such condition were absent then
we would be through. In such case the integral in eq. (4.4)
would be
~ (f (St-T x) jxL ) n n
and it would be close to ~(f) , by C.2 of Def. 3.4, for t
large. The problem of the conditioning can be avoided by go
ing to the "next" nice event. We define s* to be the larg-1 n
est time smaller than 'n + 2 ( 1: n+ 1 -1: n) when the n -th nice
event appears and the conditions of Thm. 2.2 apply. By the
proof of Thm. 3.7 we know that
s* > n-1 'n-1
with large probability. After conditioning on
can reduce ourselves to estimate
~ ( f ( s s X) I XL ) • n-1
s* = s n-1 we
Here s > 'n- 1 and xL n-1
is in a rn-1
of all
neighborhood of
XL n-1
We introduce the set z in
{ (q,vJ • q>Ln-1} such that
-(n-1) ~ = {xL U z E ~c (2 ,L 1 )}
c n-1 n-
Then
282
By definition of * (2-n,L ) and for n large enough, c n (s > 1n+1 ) we have
lf(S (xL U z))-f(S (xL U z)) I ~ kf 2-(n- 1 ) s n-1 s n-1
where kf is the Lipschitz coefficient of f . Hence
I iJ. (f (Ssx) X ((zEX )) !xL' ) -I.L (f (S x) c n-1 s
We can now eliminate the condition {zE*c} with an error
~II fD 1J. 0 (X compl.) • vie have c
1~-t(f(S x) lx-L )-~.t(f) I S 211fl2- 112n+2-(n- 1 lllfll+kf 2-(n-1i s n-1
and this shows that \=~.t
Acknowledgements
He acknowledge many useful discussions with Carlo Boldrighini and Sandre Pellegrinotti. One of us (E.P.) acknowledges very kind hospitality at the Institute of Problems of Information and Transmission, ~1oscow, and the State University of Moscow, where this work was started.
References
[1] Aizenmann M., Goldstein S., Lebowitz J. L. Ergodic properties of infinite systems. Springer Lect. Notes in Physics 38, 112 (1975).
[2] Aizenmann M., Goldstein s., Lebowitz J. L. Ergodic Properties of an infinite one dimensional hard rods system. Comm. Math. Phys.
[3] Arnold v. I., Avez A. Problemes ergodiques de la mecanique classique. Paris, Gauthier-Villars, 1967.
[4] Boldrighini c., De nasi A. Ergodic properties of a class of one dimensional systems of statistical mechanics. In preparation.
283
[5] Boldrighini c., oob~ushiA R. L., Sukhov Yu. One dimensional hard rod caricature of hydrodypamics. J. Stat. Phys. 31,577 (1983}.
[6] Boldrighini C., De .t-tasi A., Nogueira A., Presutti E. The dynamics of a particle interacting with a semiinfinite ideal gas is a Bernoulli flow. Preprint, 1984.
[7) Boldrighini c., Pellegrinotti A., Presutti E., Sinai Ya. G., Solovietchic M. R. Ergodic properties of a one dimensional semi-infinite system of statistical mechanics. ?reprint, 1984.
[8] Boldrighini C., Pellegrinotti A., Triolo L. Convergence to stationary states for infinite harmonic systems. J. Stat. Phys. 30 (1983).
[9] Cornfeld I. P., Fomin s. V., Sinai Ya. G. Ergodic theory. Springer-Verlag, 1982.
[10] De Pazzis 0. Ergodic properties of a semi-infinite hard rods system. Commun. Math. Phys. 22, 121 (1971).
[11] Dobrushin R. L., Pellegrinotti A., Sukhov Yu., Triolo L. In preparation.
[12] Dobrushin R. L., Sukhov Yu. The asymptotics for some degenerate models of evolution of systems with an infinite number of particles. J. Soviet Math. 16, 1277 (1981).
[13) Farmer J., Goldstein S., Speer E. R., Invariant states of a thermally conducting barrier. Preprint, 1983.
[14] Goldstein S., Lebowitz J. L., Ravishankar K. Ergodic properties of a system in contact with a heat bath: a one dimensional model. Comm. Math. Phys. 85, 419 (1982).
[15] Goldstein s., Lebowitz J. L., Ravishankar K. Approach to equilibrium in models of a system in contact with a heat bath. Preprint.
[16] Landau L. D., Lifschitz E. M. Statistical physics. Pergamon Press, London-Paris, 1959.
[171 Botnic, Malishev Commun. Nath. Phys. ~(1983-84).
[18] Narnhofer T., Requardt M., Thirring w. Quasi particles at finite temperature. Commun. Math. Phys. 92, 247 ( 1983) .
[19] Nogueira A. Ergodic properties of a one dimensional open system of statistical mechanics. Preprint, 1984.
[20] Ornstein systems. 1974.
[ 21] Reed M., physics:
D. S. Ergodic theory, randomness and dynamical Yale University Press, New Haven and London
Simon B. Methods of modern mathematical III scattering theory. Academic Press, 1979.
[22] Sinai Ya. G. Ergodic properties of a gas of one dimen-sional hard rods with an infinite number of degrees of
284
freedom. Funct. ~al. Appl. 6, 35 (1972).
[23] Sinai Ya. G. Construction of dynamics in one dimensional systems of statistical mechanics. Theor. Math. Phys. 11, 248 (1972T.
[24] Sinai Ya. G. Introduction to ergodic theory. Princeton University Press, 1977.
[25] Sinai Ya. G., Volkovysski K. Ergodic Properties of an ideal gas with an infinite number of degrees of freedom. Funct. Anal. Appl. 5, 19 (1971).
[26] Bowen R. Equilibrium states of the ergodic theory of Anosov diffeomorphisms. Springer, Lect. Notes in Math. 470 (1975).
[27] Ruelle D. Thermodynamics formalism. Addison Wesley, Boston, 1978. Encyclopedia of mathematics and its applications.
285
QUANTUM STOCHASTIC PROCESSES
L. Accardi
Contents
1 .) Quantum Stochastic Processes
2.) The local algebras associated to a stochastic process
3.) Markov processes and dilations
4.) Perturbations of semi-groups: the Feynman-Kac formula
5.) Perturbations of stochastic processes
6.) The Wigner-Weisskopf atom
286
1.) Quantum stochastic processes.
Let M be a *-algebra with identity (usually it wil be a C*- or
a W*-algebra). A quantum stochastic process over M indexed by lR is
defined by a triple {d, (j ) ,., , tp} where t tE"'
-dis a *-algebra with ident:C ty.
- \ : M ~d is an embedding (tElR).
- tp is a state on d.
Example 1.) Classical real valued stochastic processes.
Let ( S1, y;, P) be a probability space and let X t
( t ElR) be a real valued stochastic process. By choosing
(rl, y;,P) ---+ lR
- •= L00 QR) =algebra of all complex valued, Borel-measurable functions
on :R.
- d = L00 (S1, y;,P).
j : /E. c______, j ( I ) = I 0 X t t t
- tp (a) = Jrlad!'. ; a e.w.
/(X ) t
(telR) ( 1.1
The triple { d, ( jt) t E lR , tp} is a quantum stochastic process in the
sense defined above. Conversely, one easily sees that to a given a
quantum stochastic process { d, ( j ) , tp} such that dis an abelian t
C*-algebra, one can associate a classical stochastic process, characte-
rized (up to stochastic equivalence) by the property of having the same
fini t.e dimensional correlation functions as the initial one. Thus,
since the quantum stochastic processes include the classical ones, in
the following we shal only speak of stochastic processes.
Example 2.) (A "small" quantum system interacting with a "larger" one).
Let H0 and F be two Hilbert spaces. One might regard H0 as the
287
quantum state space of a "small system" interacting with an "extended
system" with state space F (a typical situation is : H0 ~ <Cn ; F -
some Fock space); in this case H0 0 F will be the state space of the
"composite system". The evolution of the "composite system" is given
by a 1-parameter group ( t1lf ) of unitary operators on H0 ® F : t
t11ft e ~(H 0 0 F) ~~(H0 ) ®~(F) (1.2
and there is a natural embedding j 0 ~(H 0 ) ~ ~(H0 ) 0 ~(F) of
the algebra of the "small system" into the algebra of the composite
system, given by :
j 0 : b E~(H0 ~ ~ j 0 (b) = b01 E ~(H 0 ) ®~(F)
denoting, for each teJR and ae~(H0 )0~(F):
+ u (a) = t1lf •a• t1lf t t t
one can define, for each t eJR, the embedding :
j : be~(H0 ) ~j (b) =u (j 0 (b))E~(H 0 0F) t t t
(1.3
(1.4
(1.5
Usually a state lfi on ~(H0 ®F) is given (gi?:O and lfi(1H 0 1F) = 1) 0
and, if we are interested only in the time evolution of the "small
system", then all the interesting physical quantities can be expressed
in terms of the correlation functions :
lfJ(j (b1)· ... •j (b)) (1.6 t 1 tn n
where bje~(H 0 ) (j = 1, ... ,n) and t 1 , ... ,tn are real numbers which
need not to be neither time-ordered nor mutually different. Choosing
d= ~(H 0 ®F), and (j) JR as in (1.5), one obtains t t E
a quantum stochastic process in the sense defined above.
Remark 1.) Both in examples ( 1.) and ( 2. ) one could have choosen a
smaller algebra .s;/- for example the norm (in L 00 ( r.l, ?,P) or in
~ (H ®F)) closure of the *-algebra generated by the family
{ \ ( ~ ) : t E JR} . In general, if .s;l is generated, algebraically or
topologically, by the
chastic process { .s;l,
explicitaly stated,
stochastic process".
family {j ( ~) : t eJR} , we say that the stat
(jt) , lfi} is minimal. In the following, unless
by "stochastic process" we will mean "minimal
288
Remark 2.) The occurrence of not necessarily time-ordered correlation
functions in (1. 6) arises naturally, for example in the computation
of moments of observables of the form
L~1 js (bk) ; s1< ... <sn; b1, ... ,bnE .'16(Ho) k
Usually some commutation or anti-commutation relations (arising for
example from Einstein causality) are available, and one is reduced
to time-ordered correlations. Finally, by polarization and eventually
choosing some b. equal to 1, one verifies that the correlations (1.6) J
are uniquely determined by the so called correlation Kernels :
wff (b 1 , ... ,b) = W(jj (b1) ..... j (b 21 22 (1.7 t 1 , ... ,tn n t 1 tn n
(b. E .'16 ; t. E lR ; j = 1, .•. ,n). In [ 3 ] an abstract characterization J . J
of the correlation Kernels is given, and it is shown that any family
of correlation Kernels defines (uniquely up to stochastic equivalence)
a stochastic process.
2.) The local algebras associated to a stochastic process
Given a stochastic process { d, (jt \ E lR , 9' } over a *-algebra
with identity .'16, one can define, for each sub-set I~JR. the algebra
(2.1
where the right-hand side of (2 .1) denotes the algebra generated by
the set { j ( .'16 ) t
t E I} (we leave unspecified the topology under
which this algebra is closed : this will be clear, case by case, from
the context). We will use the notations :
Clearly
v . (.'16) s;;; t Js
~?;t \(.'16)
jt ( ~)
(2.2
(2.3
(2.4
s ;;; t ;::. ds] ;;; d t] ( 2. 5
A family (d ]) lR of sub-algebras of d, satisfying (2.5), is called s sE
a filtration. Given a family ff of sub-sets of lR a family ( d I) of
289
sub-algebras of s( satisfying
(2.6
is called a family of local sub-algebras of s( or simply a localization
on s( based fT.
Example. In the case of a classical stochastic process (X ) m cf. t tE.m.
the Example (1.) in Section (1.), the local algebras s(I (IS:JR) are
sub-algebras of L 00( rl, ~, P), where S01 is the a -algebra generated by
the random variables (X ) I t t E
Given a family ( s(I)ISJR of local algebras (£s() a !-parameter group
of automorphisms (sometimes endomorphisms) of s( is called a shift
(with respect to that localization) if :
uts(I = s(I + t ; 'v'tElR ; ISJR ; (covariance) (2.7
for any ISlR and tElR. If the localization (s(1 ) is defined by a
stochastic process through (2.1), then (2.7) is equivalent to :
V s,tElR
Example. For a classical stochastic process (X ), one has t
j : /EL00{J?..) ~ j (f)= /(X )EL00(rl, SO,P) t t t u (/(X )) = /(X ) ; s,tElR
t s s + t
(2.8
(2.9
(2.10
A stochastic process {s(, (j ) ,IJ!} on 91 is called stationary, if there t
exists a shift (u ) on s( (i.e. a !-parameter automorphisms group of t
s( satisfying (2.8)) such that :
1J! • u t = 1J! ; 'v' tE lR (2.11
Recall that a conditional expectation from s( onto a sub-algebra rc is a linear map E : s( ----+ f(j satisfying :
E(l) = 1 ; E(ca) = cE(a) ; 'v'aEs( ; 'v'cErc (2.12
Sometimes (in classical stochastic processes - always for a natural
choice of the local algebras ( s( 1 )) for any local algebra s(1 ( ISJR)
there exists a conditional expectation E1 : s( ----+ s(I such that
IJ! • E1 = IJ! ( 2 .13
290
i.e. compatible with the state ~· The family (EI) satisfies
I<;;J l> EI·EJ EI (projectivity) (2.14
and if the state ~ is shift-invariant, then
u•E=E •u Vt,VI (2.15 t I I+t t
Any family (EI) of surjective conditional expectations EI :.9/--..9/I
will be called projective if it satisfies (2.14) and covariant if it
satisfies (2.15). In particular, in case of a filtration (a locali
zation based on the "past half-lines" { (- 00 , t] : t E lR} ) conditions
(2.14) and (2.15) become
s :it l> Es]•Et] = Es]
u • E t s]
E • u s+t] t
3.) Markov processes and dilations.
(2.16
(2.17
A markovian stucture on a *-algebra .91 is defined by the assignment
of :
- a "past-filtration"
- a "future-filtration"
- for each t E lR an "algebra at timet", .9/t such that
.9/t c;; .9/tl n .91 [t
- A projective system of conditional expectations Et]
(3.1
.91 -----. .91 t]
i.e. :
s :;: t ==l>E • E = E s] t] s]
enjoying the Markov property
Et]( .91 [t) <;;.s;lt ; V tElR
If the localization {(.9/t]) , (d[t) , (dt)} admits a
u•d =.91 u•d Sit · u•d t s] s+t] t [s [s+t ' t s
(3.2
(3.3
shift ( u ),i.e. t
.91 s+t (3.4
and if the family (Et]) of conditional expectations is covariant,
u t • E s] = E s+t] • u t
i.e.
(3.5
then we speak of a covariant markovian structure .
Example Let \ (II, ?, P) ------+ lR ( t E lR) be a classical Markov process;
291
let ~]' ~t' ~[t be respectively the past, present and the future
o-algebras; denote 00 00
.Sil = L (rl, ~.P) ; .Silt] L (rl, ~t]'P) , ...
and let Et] = l ( • I ~t]) be the ?-conditional expectation on ~t]' Clearly these objects define a markovian structure on .SII. - a covariant
markovian structure if the process (X ) is stationary. t
The connection between markovian structures and semi-groups is
made precise by the following
Theorem (3.1) Let (Es]) be a family of maps (not necessarily condi
tional expectations) satisfying conditions (3.2) - projectivity - and
(3.5) - covariance. Denote ~ the vector space generated by the family: 0
{E0 ]" u t •E0 ]( .s;/0 ) t ~ O}
(;; = .s;1 in the markovian case), and define 0 0
pt = E • I ~ tEJR o] t o
. t -then the fam1ly (P ) is a semi-group of .s;l into itself. 0
Proof. For a E ;; and s, tE JR. one has : 0 0
E ]• u •E ]• u (a ) = E ]E ] u (a ) o s o t o o s s+t o
E ] u (a ) o s+t o
(3.6
(3.7
If the maps (Et]) are completely positive, identity preserving,
(e.g. conditional expectations) then the semi-group (Pt) is completely
positive identity preserving. Any such a semi-group will be called a
markovian semi-group on d . If .J' is a non-commutative algebra, one 0 0
also speaks of a quantum dynamical semi-group.
In the following we shall only consider the markovian case, i.e.
~=.s;l 0 0
.Silo '"---+ ,s;l,
.s;l
joJ
PA
Thus, denoting PA = .s;l and j 0 = the identity embedding 0
one obtains the commutative diagramme :
t .s;l
Eo] .s;l 1 .-1 Jo (3.8
PA pt
292
-1 where j 0 denotes the left inverse of j 0
Definition (3.2) Let ar be a ~*-algebra (with identity} and (Pt) -
a markovian semi-group on ar. A ~*-dilation of the pair { ar, (Pt)}
is a quadruple {d, j , E .1, ( u )} making commutative the diagramme 0 0 t
(3.8) and such that j : Jl ~ d is a ~*-embedding; ( u ) is a 0 t
!-parameter automorphisms group of d; E0 ] : d ~d is a norm-one
projection satisfying :
E ]• ut(j (ar)}!;;;; j (ar) ; V t<: 0 0 0 0
(3.9
If, moreover, denoting d[t - the algebra generated by
{ u • j ( ar) : s ;: t} one has : s 0 -1
E 1·u •E 1·u ld[ = E ]ld[ ; t<:O (3.10 o tot tot
then we speak of a (covariant) markovian dilation of { &I, (Pt)} .
Finally if there exists a state (weight) rp on d satisfying :
f/J•Et] =tp (/J•ut =tp ; t<:O (3.11
then we speak of a stationary markovian dilation of {ar, (Pt)} .
Remark One easily sees that there is a one-to-one correspondence
between covariant markovian dilations of { ar, (Pt)} and covariant
markovian structures (as defined at the beginning of this section} with
d ;;; ar and E • u • j = Pt. o o] t o
A beautiful classification theory of dilations of completely positive . semi-groups has been developed by B. Kummerer and W. Schroder. In the
classical case, i.e. when &I is abelian we know that :
t . i) any markovian semi-group (P ) on ar has a covanant markovian
dilation (obtained through the well known Daniell-Kolmogorov
construction).
ii) (Pt) has a stationary markovian dilation if and only if there exists
a state (weight) rp on fJI such that 0
t rp •P
0 (3.12
In the quantum case the situation is more complicated and only recently
R. Hudson and K.R. Parthasarathy [ 6 ] have shown that the statement
(i) holds; while A. Frigerio [ 5 ] (cf. also the paper by A. Frigerio
and V. Gorini [ 4 ] ) has found the correct quantum analogue of the
293
statement (ii).
In the following sections we will describe the main technical tools
through which the solution of the above mentioned problems has been
achieved.
4 .) Perturbations of semi-group : the Feynman-Kac formula.
Let {d, (dt])' (dt), (d[t)' ( u:). (Et])}
be a given covariant markovian structure, and let be given a covariant
family of local algebras (d[s,t]) (s ;;it ; s,tElR ) such that +
d.[ J c: .r.~[ n.r.~ J (4.1 s,t - s t A markovian cocycle (with respect to the structure defined above) is
a 1-parameter family (M ) of elements of d such that : o,s s ~0
M E d.[ ] ; V t ~ 0 ; (markovianity) o, t o, t
M = M • u 0 (M ) o,t+s o,s s o,t
(4.2
(cocycle property) (4.3
Denoting, for s < t, M = u (M ) , then the two parameter family s,t s o,t-s
(M )s < t is s,t
such that
M Ed ;Vs;;it s,t [s,t]
M •M M ;r<s<t r,s s,t r,t
u (M ) M t r,s r+t,s+t
(4.4
(4.5
(4.6
and the three conditions above are those which, in classical probabili~
theory, define the so-called multiplicative functionals associated to
a given family { 3"[ ]} of a-algebras. Typical examples are given by 00 s,t
d = L (S"l, 3",P) ; (S"l,3",P) -a Wiener probability space; (W) 0 - a t t ~
real valued Wiener process;
M = exp 'l2 {-It V(W )dr + It a(W )dW} s,t s r s r r
(4.7
with V,a : lR ~ lR- sufficiently regular functions.
Theorem ( 4. 1)
t>O
Let (M ) be a markovian cocycle and define, for o,ss>o
Pt(a ) = E (M • u 0 (a )•M+ ; a Ed o o] o,t t 0 o,t o o
t It follows that (P ) is a semi-group d ----. d
(4.8
0 0
294
Proof. For a E .91 and s, t ElR , one has 0 0 +
E](M [u 0 •{E](M u 0 (a)•M+ )}]•M+ o o,t t o o,s s o o,s o,t
E](M ·E 1[u 0 (M )•u 0 (a)•u 0 (M )+]•M+) o o,t t t o,s s+t o t o,s o,t
E•E(M •u 0 (M )•u 0 (a)•u 0 (M )+•M+)= o] t] o,t t o,s s+t o t o,s o,t
E ,(M • u (a )·~1+ ) = Pt+s(a ) oj o,t+s t+s o o,t+s o
Any semi-group (Pt) defined as above, will be called a Feynman-Kac
perturbation of the semi-group Pt = E • 0 (t > 0). - o o] ut '
Formula (4.8) will be referred to
as the Feynman-Kac formula.This formula generalizes several known
constructions :
1.) The classical Feynman-Kac formula. This is obtained by choosing,
in the notations of formula (4.7) :
M o,t
t exp- 1 2 f V(W )ds
0 s (4.9
where Vis a suitably regular function (e.g. measurable bounded below).
2.) The interaction representation . This is obtained by choosing the
markovian structure to be trivial (i.e. all the local algebras are
equal to .91 and E 0 J is the identity map on .91), and the cocycle
M = U to be unitary. In this case, writing instead of Pt the o,t o,t ut Feynman-Kac formula becomes :
u (a) = U • u 0 (a)•U+ t o,t t o,t
aEd (4.10
The cocycle property then assures that ( u ) is a !-parameter autot
morphisms group of .91 ( cf. the proof of Theorem ( 4 .l), with all the
conditional expectations equal to the identity).
The pair { ( u 0 ), (U ) } where ( u 0 ) is a !-parameter automorphisms t 0' t t
group and (U ) is a unitary (markovian) ( u 0 )-cocycle is called an o,t t
interaction representation for the !-parameter automorphisms group
( u ) defined by (4.10). The connection with the notion of interaction t
representation usually met in physics is given by the following formal
considerations : let ( u 0 ) be of the form : t
a Ed (4.ll
295
0 with o/1 = exp itH 0 - a unitary in d, and let H E.s;/ be a self-adjoint
t I operator. Define
HI(t) = u ~(HI) = o/i~·HI• o/1~+ ; tEJR (4.12
and let (U ) be defined by d o, t -U = iU •H (t) ; U (4.13 dt o,t o,t I o,o then (U ) is a unitary ( u 0 )-cocycle (markovian in an appropriate
o,t t localization) and
o/1 = u • o/10 t o,t t
is a 1-parameter unitary d
o/1 = io/1 •[H 0 +HI]
group in .s;l satisfying the formal equation (4.14
dt t t In many concrete examples either H0 + HI or HI(t) are not well defined
as operators so that equation (4.13) or (4.14) has no rigorous meaning.
But we will see that in many cases it is still possible to define, using
quantum stochastic calculus, a markovian cocycle (U ) and a 1-parameter o,t
unitary group ( o/1 t) having all the properties of the formal solutions
of the equations (4.13) and (4.14) (cf. Section (6.) in the following).
3.) Perturbations of the identity semi-group. Consider a markovian
structure as in the beginning of this section, and let .s;l be of the
form :
where H0 and F are complex
shift ( u 0 ) as the form : t
0 vo ut ="ox t
(4.15
separable Hilbert spaces. Assume that the
(4.16 0
where e0 is the identity map on &W(H 0 ) and (Vt) is a 1-parameter auto-t 0
morphisms group of &W(F). In this case the semi-group P 0 = E0 ]• u t is
the identity semi-group on Sll 0 -;;;;_ &W (H 0 ) ®1, and its Feynman-Kac pertur
bation with respect to a unitary markovian cocycle (U ) has the form: o,t
Pt(a ) = E (U •a •U+ ) (4.17 o o] o,t o o,~
A semi-group of this form will be called a Feynman-Kac perturbation
of the identity semi-group.
Theorem (4.2) (cf. R. Hudson - K.R. Parthasarathy [ 6 ] , A. Frigerio,
V.Gorini [ 4] ,A. Frigerio [ 5] ). Let H0 be a complex separable
296
Hilbert space. Any markovian semi-group on 9W(H 0 ) admitting a Lindblad
generator has a covariant markovian dilation which is a Feynman-Kac
perturbation of the identity semi-group.
5.) Perturbation of stochastic process
In the preceeding section we have shown that any markovian cocycle
gives rise to a perturbation of a markovian semi-group. In this section
we show that any unitary markovian cocycle gives rise to a perturbation
of a covariant markovian structure which is still a covariant markovian
structure. This is a purely quantum-probabilistic phenomenon, since
in the abelian case unitary markovian cocycles give rise only to tri-
vial (i.e. identity) perturbations.
Let
be as in Section (5.); let (U ) be a unitary markovian cocycle, and O,t
define
u t(a) U • u 0 (a)•U+ a Ed (5.1 o, t t o, t
Then ( ut) is a 1-parameter antomorphisms group of d and defining
fJit (5.2
one easily verifies that for each a E d
E • u (a) s+t] t
(5.3
thus the family (Et]) is also covariant for the evolution ( ut) defined
by (5.1).
Define now, for t ~ 0
f11[t = ~ ~ t us( 1!4o) ~ ~t us( d 0 ) (5.4
and similarly for 1!4t).Remark that :
1!4 c U • d •U+ (5.5 [t- o,t [t o,t
whence, due to the Markov property of (E ) : t
Et](f!4[t)SUo,t.Et](d[t)·U:,tsuo,t.dt·U:,t = ut(do) = 1!4t (5.6
Thus : (Et)) is markovian also with respect to the localization (fJit))'
(fJit),(fJI[t) or equivalently, defining:
1!4= V JR u (d0 ) (5.7 tE t
+
297
the family {£!J, ( £Wt])' (£tft)' (£W [t)' (ut)' (Et])} is still a covariant
markovian structure. In particular, for any state p0 on £tJ 0 = d 0 ,
defining p = p • E (state on £tJ ) j 0 = the identity embedding o o]
£tJ '---+ £tJ • J. = u • j 0 (t?: 0), the triple { d, (j ) , p} is a o ' t t tt<:o
(markovian) stochastic process. over .s;l , in the sense defined at the 0
beginning of Section (1.). As shown by A. Frigerio and V. Gorini [4],
[5], (in the case of boson dilations) the process will be stationary
if and only if the associated semi-group satisfies a detailed balance
conditions. More generally, in the framework of local algebras, it can
be shown that the stationarity of the pro~ is related to the behaviour
of the semi-group under appropriate "time-reflections" ( cf. [ 1], [ 2]).
6.) The Wigner-Weisskopf atom .
In this section I will outline some results obtained in collabo-
ration with D. Applebaum and which will be published elsewhere. For
the description of the Wigner-Weisskopf model we follow the exposition
given by W. von Waldenfels in (9] and we also refer to this paper for
a more complete discussion of the physical limits of this approximation.
In its simplest version the model describes a 2-levels atom in inter-
action with an electro-magnetic field. In the "rotating wave approxi-
mation" the system is described on the Hilbert space
Jt'= C2®ji'(CIAI):::. C2®[® jl' l AEA A (6.1
where A is a finite set (indexing the frequencies of the EM field),
I AI denotes the cardinality of J\ and, for each A E J\, jl'A:::. r (C) is
the Fock space over the Hilbert space C (with scalar product <u,v> = uv;
u, v E C). On each space jl'A the creation and annihilation operators +
BA, BA are defined in the usual way and they satisfy the commutation
relations :
Introducing 0 0
o+ = (1 ol
the spin matrices
; o_ = (g ~) l (-1 03 = /2 0
(6.2
(6.3
The hamiltonian of the system in the rotating wave approximation is
H tot.
H at.
298
(wa 3 ®1)+['\"' (w +w)l®B+B]+ 0 L>..EA 0 >.. >.. >..
+ [LA.EA(gA.a+®BA. + g>..a_®l<)]
where vJ + w is the frequency of the >.. -th oscillator and g is the 0 >.. >..
coupling constant of the atom with ~:1e >..-th oscillator. Rewriting the
hamiltonian as
H H + H = w [a 3 ® 1 +[ 1 ®B+B ] + tot. o 1 o :;.,. E A >.. >..
['\"' ( 1 B+B B - B+)] + LJ... E A w>.. ® >.. >.. + g>.. a+® >.. + g>.. a-® >..
and remarking that H 0 and H1 commute, we we reduce ourselves to the
consideration of the single term
H =H +H=[' wl®B+B]+['\"' ( ®B +- ~"+)] 1 lo LJ...EA A >.. >.. L>..EA g>..a+ >.. g>..a- l(YL'>..
and H1 is described in interaction representation using H10 as "free
part" and H as "interaction part". This leads to the unitary cocycle
U = U defined by the equation t o, t
d U = -iU H (t) U = 1 ~ t t A o
H ( t) = '\"' ( g a ® B + g a ®B +) A LJ...EA >.. + >.. >..- >..
where
[ -iw t B (t) = g a ® B •_Q, >..
A A.EA >.. + A. The commutator between BA(t) and B~(s) is :
+ 'I I -iw(t-s) [BA(t) , BA(s)] =L g>.. 2 _Q, >.. = KA(t-s)
while all the other commutators vanish.
Introducing on at(S"(~IAI)) the quasi-free state @"characterized by
@" (BA.B1) = @" (B~B:) = 0
+ @" (B>..B11 ) = o A\.1 8 >..
(8>.. a physical constant), one finds
+ + @"(BA(t)•BA(s)) = @"(BA(t)BA(s)) = 0
@"(BA(t)•B;(s)) =[lg>..lz(l + 8A.)_Q,-iw>..(t-s)
The Wigner-Weisskopf approximation is obtained, from the rotating wave
approximation, by replacing
8 = 8 >..
exp(-hw /KT) 0
1 - exp(-hw /KT) 0
299
This means that one substitutes for BA(t) and B;(t) two operators F(t),
F+(t) satisfying :
[F(t)
[F(t) , F+(s)) = x6(t-s)
0 (6.4
(6.5
and on the algebra generated by the family {F(t), F+(t)} one introduces
the quasi-free state characterized by
<B'(F(t)•F(s)) = <B'(F+(t)•F+(s)) = 0
<B'(F(t)·F+(s)) = (l + 8 )• 6(t-s)
(6.6
(6.7
With these approximations the equation for the unitary cocycle becomes
_d- U = -iU •H(t) U (6.8 dt t t 0
H(t) = a ®F(t) +a ®F\t) (6.9 + -
Equation (6.8) is purely formal because, due to (6.5), (6.7) and (6.9),
H(t) is not a well defined operator but an operator valued distribution.
In analogy with the classical procedure von Waldenfels [9) introduced
to methods for the solution of equation (6.8)
I.) The ''Stratonovich method", corresponding to the "singular coupling
limit method" in the physical literature, consisting in three steps :
i) regularize the covariance with the substitution, in (6.5) and cs.n 6(t-s)---+KE(t-s) for some SI:Iooth function KE(•).
ii) solve the corresponding ordinary differential equation, finding
a cocycle UE(O,t).
iii) determine the limit of U E (0, t) - and of the associated process
(Section (5)) as E ~o and K (t-s) ~ 6(t-s). E
II.) The "multiplicative Ito integral method", (corresponding to the
approximation methods in classical probability) in which - instead of
the covariance - you regularize the fields. This can be done in several
ways. In [ 9 ] one considers for each fixed T E F+ a partition
z = {O=t < t 1 < < t =T} of the interval [O,T) and introduces the o n
piecewise constant fields :
t
F (t) =_.:c.__ j k+~(T)dT = F(X( ) : t < t :> t z tk+l-tk tk tk,tk+l) k k+l
One then solves the ordinary differential equation :
300
d --d U (t) = -iU (t)•H (t)
t z z z and studies the limit of Uz(t) (and of the corresponding process) as
I zl maxk(\+1-\)----+ 0
For the Wigner-Weisskopf model the existence of the limiting
cocycle (and of the corresponding process) was established by von
Waldenf~ls [9] in both cases (I.) and (II.). A third possibility, is
to interpret (6.8) as a quantum stochastic differential equation and
use the results of R.Hudson and K.R. Parthasarathy [6] to estab-·
lish the existence, uniqueness and unitarity of the cocycle U(t).
Namely, one considers the Hilbert space
f(L2(R+,dt))®f(L20R+,dt)-) = .Yf
where f(H) denotes the (boson) Fock space of H and H denotes the
conjugate Hilbert space of H. On this Hilbert space one considers the
representation of the CCR with creation and annihilation operators
given by :
F(t) =/Ycoshc!>•a(X[ ])®1 +>'Ysinhcl>•1®a+(X[ ]) o,t o,t
F+(t) =>'Ycoshc!>•a+(X[ ])®1 + .fisinhc!>•1®a()([ ]) + o,t o,t
where a(•) and a (•) are the annihilation and creation operators over
f(L 2 OR+)), and by definition, y = 2 Rex, and :
h2c!> 1 sinh2c!> exp( -W0 /KT) cos = 1 - exp(-w /KT) 1 - exp(-w /KT)
0 0
8
With these notations the unitary (markovian) cocycle Ut is defined as
the solution of the quantum stochastic differential equation
dU t
U { (-io ®dF(t)- iO ®dF+(t))-t + -
- y/2 (cosh2c!>o o ®1 + sintfc!>o o ®1)dt + - - +
Denoting E0 ] the conditional expectation characterized by
E] : x®(Y®Z) E£i(IC 2)®£i(f(L 20R ))®f(L 20R f)) --t 0 + +
--t (X® 1 ® 1) <rl, Yrl><rl, Z~>
(6.10
where S1 ( resp. S'i ) denotes the Fock vacuum in f(L 2 OR)) ( resp.
r(L 2 (R ) -)) and applying the theory outlined in Section ( 4 ) , one +
obtains a semi-group on dW(~ 2 ) = {zxz matrice~ via the prescription:
xe£i(C2) --tE ](U •(x®1®1)U+)E£i(C 2)®1®1::.£i(C 2) 0 t t
whose generator is :
301
-~ cosh2 <l>•y{CJ a ,x} + cosh2 <l>•y•a •x•a + + -- + ~
-Y2 sinh 2 <l>•y{a_CJ+,x} + sinh2 <l>•y•a_•x•a+
( x E §I ( !C 2)). Referring the algebra of 2 x2 complex matrices; §I (!Co" t.o
the standard basis, we find for L the matrix
(
-y8 y8 0 0
y(8+1) -y(8+1)
0 0
0 0
- 1~(28+l)y 0
(6.11
which is exactly the formula found by von Waldenfels via the "multipli-
cative Ito method" [9] (in his notations y ; 2 Rex). To obtain the
formula found by von Waldenfels via the "Stratonovich method" instead
of (6.10) one has to look for the solution of the quantum stochastic
differential equation
+ dUt; Ut·{- ia+®dF(t)- ia ®dF (t)- (y/2 (cosh2 <l>a+a_®l +
- sinh2 <j>a a ®l)]dt- is/2 (28+l)[cosh2 <l>a a® 1 + sinli<!>a a® l]dt) -+ +- -+
where, in von Waldenfels notations: y ; 2Rex, S ; 2Imx. The connection
between the multiplicative Ito (i.e. singular coupling) method and
quantum stochastic differenti.al equations was suggested by Frigerio
and Gorini [4] and the explicit form of the semi-group obtained in the
Wigner-Weisskopf model in the "multiplicative Ito" case (i.e.
corresponding to equation (10)) has been independently obtained by H.
Maassen [ 8].
REFERENCES
1.) L. Accardi. On the quantum Feynman-Kac formula. Rendiconti del Seminario Matematico e Fisico di Milano 48 (1978), 135-180
2.) L. Accardi. A quantum formulation of the Feynman-Kac formula. In: Colloquia Mathematica Societatis Janos Bolyai, 27. Random Fields, Esztergom (Hungary) 1979.
3.) L. Accardi, A. Frigerio, J.T. Lewis. Quantum stochastic processes. Publ. Res. Inst. Math. Sci., Kyoto University~ (1982) 97-133
4.) A. Frigerio, V. Gorini. On stationary Markov dilations of quantum dynamical semi-groups. In: Quantum Probability and applications to the quantum theory of irreversible processes. Ed. by L. Accardi,
302
A. Frigerio, V. Gorini. Springer LNM, N° 1055
5.) A. Frigerio. Covariant Markov dilations of quantum dynamical semigroups. Preprint (1984)
6.) R. Hudson, K.R. Parthasarathy. Construction of quantum diffusions. In: Quantum Probability and applications to the quantum theory of irreversible processes. Ed. by L. Accardi, A. Frigerio, V. Gorini. Springer LNM, N° 1055.
7.) B. Kummerer, W. Schroder. On the structure of unitary dilations. Semesterbericht Funktionalanalysis Tubing en, Wintersemester 1983-84, 177-225
8.) H. Maassen. The construction of continuous dilations by solving quantum stochastic differential equations. Preprint (1984)
9.) W. von Waldenfels. Ito solution of the linear quantum stochastic differential equation describing light emission and absorption. In: Quantum Probability and applications to the quantum theory of irreversible processes. Ed. by L. Accardi, A. Frigerio, V. Gorini. Springer LNM, N° 1055
303
ABSOLUTELY CONTINUOUS INVARIANT MEASURES FOR SOME
MAPS OF THE CIRCLE
P. M. Blecher and M. v. Jakobsen
1. Statement of results. We consider the two-parameter
family of maps on the circle
f q,w x t-+ x+w+ (q/2n) • sin 2nx , x E S 1 = :ffi/ Z
and we find a set M = {(q,w)} of positive Lebesgue meas
ure such that (q,w) EM implies the stochastic behaviour
of f . We present analytical and numerical results which q,w describe the structure of M as follows.
There exists a sequence of points Ak = (qk,wk) I kEN
converging to the limit A = 00 (qoo' wo) I where qoo
1,169701. •. , W00 = q 00 /2'TT I satisfying
Theorem 1. k there exists set 2 of For any a Mk c lR
positive Lebesgue measure, such that Ak is the density
point of Mk , and if (q,w) E ~ then the map f : s 1 ~s 1 --k q 1 W
has an absolutely continuous invariant probability measure
~ . The map f cyclically permutes k t~~~als ,e!il, f'Ew[O,k-11 , ku 1 ,e!il = s 1 .
adjacent in
The support of q,w ._0 q,w
consists of k interval~- s(i) c ,e!i) of equal meas-q,w q,w ~q,w
ure. For any i the map fk q,w
the measure space (s (i) , ~ ) q,w q,w
is a Bernoulli automorphism.
is an exact endomorphism on
, and its natural extension
In order to prove Theorem 1 for a given k it suffices
to verify some conditions of non-degeneracy, see Sect. 3.
For k=1 these conditions are verified analytically. For
2~k~7 they were verified with the help of a computer.
304
If the non-degeneracy conditions hold, then Theorem 1 is
proved by using the argument of [1].
The limit map f is characterized by simple geo-qoowoo
metric features, see Sect. 5.
The full picture is structurally stable in the space
of two-parameter families xt-+x+w+ (q/2TI)h(x) , where
h(x) has period 1 and is c 3 -close to sin 2Tix (Theorem
3).
For any family under consideration the rate of conver
gence of Ak to A00 is the same: lA -A I ~ const I k oo k2
Our work was motivated by a question of Ya. G. Sinai
concerning the appearance of stochasticity in the family
f for q>1 q,w
2. •rransition from q<1 to q>1 . Let us consider the
following map of the cylinder r E (-oo,oo) X E lR/ Z
(rl (1 + ,\(r-1) + (q/2TI)sin 2TIXJ
lxJ t---> lx+w+,\ (r-1) -1 (q/2TI) sin 2TIXJ
If ,\=1 P 1 ,q,w is the so-called standard map, which has
been studied as a model for certain dynamical systems, see
[2], [3]. If ,\<1 , P,\,q,w is dissipative, and for ,\=0
(the case of infinite contraction) it reduces to the map
of the circle:
f : xi->x+w+ (q/2TI)sin 2Tix, xElR/Z = s 1 . q,w
If q~1 then f q,(u is a homeomorphism of the circle. Such
homeomorphisms were studied intensively in connection with
the problem of disappearance of invariant tori in the KAM
theory, see [4], [5]. We shall study f first of all q,w for q>1 , but first we recall some facts about f q,w the critical value of q=1 .
with
For any w the homeomorphism f 1 ,w has a rotation
number p ( w) (for definition see e.g. ref. [ 5] ) depending
continuously on w If p(w) m/k is rational then f 1 , w has a periodic orbit of period k . For all rational num-
305
bers m k there exists an interval
m wEim/k, p(w) k.
such that for
Considering f 1 ,w as a map of the Riemann sphere
z I--+ z + w + ( 1 I 2 rr) sin 2 rr z , z E C , depending on the complex
parameter w , one can prove by the methods of [6] (see
also [7]) the following proposition, which is not directly
used below, but seems to be of some interest.
Pro:eosition. For any I m/k defined above there exists
a unique parameter value wO E Im/k such that f 1, wo
has a
superstable periodic trajectory of period k
The homeomorphism f 1 permutes cyclically k ,wo
tervals: .e (0) 3 .l .e (1 )= f .e (0) .e (k-1) = fk-1 .e (0) 2 ' 1,w0 , •.• , 1,w0
in-
and the graph of f~ j.e(O) looks like figure 1a. ,wo
For q>1 the degenerate critical point c=1/2 bi-
furcates into two simple critical points
c, = (1/2rr)arccos(-1/q) < 1/2
and
c2 1 - (1/2rr)arccos(-1/q) > 1/2
,.Q. 1£.
Figure 1
306
For q close to q 0 = and w close to f still q,w
Permutes k intervals .e_(i) and the graph of q,w '
fk [.e. (0) q,w q,w
q and w looks like figure 1b. A natural idea is to find
characterized by the most chaotic dynamics represented in
figure 1c.
3. Non-degeneracy conditions and proof of Theroem 1'.
Let us denote by F the restriction of fk to .e_(O) q,w q,w q,w
= [z 1 (q,w) ,z 2 (q,w)] , by c 1 (q,w) the point of maximum,
and by c 2 (q,w) the point of minimum of F q,w
Non-degeneracy conditions
1) The equation F (c 1 (q,w))- z 2 (q,w) = G(q,w) = 0 q,w
defines a smooth curve y( 1 ) on the plane
similarly F (c 2 (q,w)) - z 1 (q,w) = H (q,w) q,w
smooth curve y( 2 )
(q, w) , and
defines a
and (2)
y intersect each other at
det f 0 .
Theorem 1'. If the non-degeneracy conditions are satis-
fied then the Lebesgue measure of the set Mk =
= { (q,w) i F : { (O) + { (O) has an absolutely continuous q,w q,w q,w
variant probability measure ~· } is positive, and Ak q,w
in
is
the density point of Mk . For (q,w) E Mk (F ~' ) is q,w ' q,w
an exact endomorphism, and its natural extension is a Ber-
noulli automorphism.
Proof of Theorem 1'. a) The non-degeneracy conditions
imply that y( 1 ) and y( 2 ) intersect each other transver
sally at Ak , and generate the partition of sufficiently
small neighbourhoods of Ak into four quadrants. The in
clusions F (c 1 (q,w)) c .e_(O) , F (c 2 (q,w)) c .e_(O) are q,w q,w q,w q,w
are satisfied for (q,w) belonging to one of these quad-
rants.
with
Let us consider a ray
r(o)=Ak . The maps
r =
F q,w
{r (s)}
with
in this quadrant
(q(s), w(s)) = r(s)
307
Figure 2.
form a one-parameter family of maps such that the graph of
F0 = fq(O),w(O) looks like figure 2. Let y=F 0 (y) be the fixed point different from the end-points of i(O)
-1 -1 and let y 1 , y 2 be the preimages of y . We denote by
T0 the first return map induced by F0 on the interval
-1 -1 y 1 , y 2 , (see fig . 2 . ) .
b) A straightforward calculation shows that for q>1
the mappings f have a negative Schwarzian derivative: q,w
Sf= f"' ,f' - 3/2(f") 2 < 0 .
This implies (see e.g. [8) about the properties of maps
with negative Schwarzian) SF 0 < 0 . From SF 0 < 0 it fol
lows (see [1], [8)) the so-called expanding property:
There exist an mEN and c>1 such that
1 T m I 0 (x) [ > c
308
for any -1 -1 X ( (y 1 1 Y 2 j
Notice that another way to prove (1) is to use the
theory of normal families of analytic functions due to
Mantel.
( 1 )
The non-degeneracy conditions also imply that for
s=O the critical values Fs(c 1 (s)) 1 Fs(c 2 (s)) move with
non-zero velocities to the end-points of l(O) (s) .
c) The whole situation is completely analogous to that
which holds for the one-parameter family of unimodal maps
x 1--->-ax ( 1-x) 1 x E: [ 01 1] 1 with s=O corresponding to a=4
For s close to zero the graph of the induced map Ts
looks like Figure 3.
The methods of [1] (see § 13) are applicable. They
give that the linear Lebesgue measure of the subset M(r)
y-1 1
Figure 3.
309
of the ray r , defined by M(r) = {(q(s), w(s))
= r(s) : Fq(s) ,w(s) has an absolutely continuous invariant
probability measure ll' } is positive, and Ak=r(O) is q,w the density point of M(r)
d) The same arguments are valid for any ray in the
distinguished quadrant, and thus we obtain, by using the
Fubini theorem that mes ~ > 0 , and Ak is the density
point of Mk in this quadrant.
Ergodic properties of endomorphisms (F , ll' ) and q,w q,w of their natural extensions follow from the results of Le-
drappier [9].
4. The choice of the sequence Ak and the proof of
Theorem 1
Consider the values q,w characterized by the follow
ing properties (see Figure 4.):
Then setting z 2 f(z 1 ) , the interval .e_(O) = [z 1 , z 2 ]
triply covers its image .t( 1 ) = [z 2 ,f(z 2 )J under the ac
tion of f . The subsequent iterations of f map .e_( 1 )
homeomorphically on the intervals .e_( 2 ) , ... ,.t(k- 1 ) which
are consecutive, adjacent, and the end of .e_(k- 1 ) = k-1 k [ f (z 1 ) ,f (z 1 )] coincides with z 1 (see fig. .4,
k' l where k=3 ) . Besides the graph of f 1 [z 1 ,z2 looks like
fig. 1c. We take these q,w for qk' wk from Section 1.
If the non-degeneracy conditions are satisfied then accord
ing to Theorem 1' there exists for (q,w) E~ an fkl.t(O)
invariant measure ll~,w « dx . To ll~,w there corresponds
the circle. If
coincides with
a unique f invariant measure l1 on q,w
hence
, w=wk then the support of ll' q,w
, and the support of
then
coincides with s 1 • If
5 [F (c 2 ) ,F (c 1 )] = supp ll' , q,w supp llq,w is the union of k disjoint intervals.
The ergodic properties of l1 follow from those of q,w
ll' q,w which finishes the proof of Theorem 1.
310
1.0
0.9
0.8
0.7
0.6
T 0.5
0{+ I I
c1 0.3
0.2 z,
0.1 --[(2 t< 0) [( 1)
0.0 "-r-T-r-r+-r--.-n-+-r-TT"1-+oTT"r+-r..,.,..o+-rrr-r+T"T"rrt.,...,...rrl..,r-rll-rr..,...r 0.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3
X 0.2 0.1 1.0
Figure 4.
3ll
5. The properties of the map the construction
of the sets Mk
The values (qk 1 wk) satisfy the following conditions:
f2k (c.) J.
The sequence Ak
the map
(i)
(ii)
f may be characterized by: qoo I Woo f has a fixed point u such that
qoo,woo
1 f" w (u) > 0 qoo, oo
f (c 1 ) • qoo,woo
Denote by z 1oo 1 z 2oo
which is triply mapped by
the endpoints of the interval
f on its image qoo,woo
(figure 5.). When k->-oo then the
defined above converges to
Theorem 2. When k->-oo then the sequence of maps f k ,
1 1 (0) . <- converges J.n
qk 1 wk qk 1 wk c 1 -topology to the limit map
f 00 which maps qoo 1 Woo
.t(O) triply on itself. qoo 1 Woo
Condition (i) defines the line q = 2nw in the (q 1 w)
plane.
Condition (ii) defines the curve r which intersects
the line q = 2nw at the point (q001 W00 ) The points Ak
lie on the curve r (see figure 6.). The curves y~) and
y~ 2 ) defined respectively by f 2k(c 1 ) = fk(c 1 ) I and
2k k f (c 2 l = f (c 2 ) 1 have Ak as the points of transversal
intersection. It is natural to suggest that as Ak->- A00 the
curves y~ 1 ) and y~ 2 ) become parallel to the line q=2nw 1
and the angles between y ( 1 ) and y ( 2 ) tend to zero. k k Fig. .6 shows the points Ak = (qklwk) and the curves
y~ 1 ) 1 y~ 2 ) for 3~k~7 • Notice that if k=1 1 then w1=0
and q 1 satisfies the equation: q·cos/q2-1 = -1 . The
curves and (2)
y1 are defined by
312
1.0
0.9
0.8
0.7
06
T c2
0.5
0.4
0.3 z100
0.2
0.1
0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X
Figure 5.
313
q
24
2.2
2.0
1.8
1.6
1.4
1.2
2jfw 1.0 IL,-.-.--r-+-.--r--r--r-f---.-..,...,---,-+-,---,-...,.......,--+-.-----.-...--l---.-,-,---.-+-r-r-,--r-+,.-.l
1.0 1.2 1.4 1.6 1.8 2.0 2.2 24
Figure 6.
314
arc cos(-1/q) + ~ = 2n - (arc sin w/q) - w
and
arc cos(-1/q) + ~ = 2n + (arc sin w/q) + w .
One can check that ( 1 ) y1 and (2 ) y1 have a transversal in-
tersection at the point (q1 10)
6. Structural stabilitv. The full picture described above
is structurally stable. Namely the following is true:
Theorem 3. For any two-parameter family of maps of the
circle
h : xl--+x+w+ (q/2n)h(x) , q,w
where h(x) has period 1 and is close in the c3 norm to
sin 2nx , there exists
and a sequence of sets
of Theorem 1. The maps
a sequence of points Ak = (qk,wk)
Mk3Ak , satisfyi.ng the con(Utions
h converge to the limit map qk,wk
h , which is characterized by the conditions (i), (ii). qco,Woo
The sec-ruence of maps converges on the corres-hk qkwk
hk -invariant intervals to the limit map qkwk
depending or. the far.1ily under consideration. More-
I I 2 over, for any such family we have 1 Ak - A00 1 "' const k
References
[ 1 j Jakobsen !L V. Abolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81,39-88 (1981).
[2] Chirikov B. V. A universal instability of many-dimensional oscillator systems. Phys. Rep. 52, 5, 263 (1979).
[3] Mackay R. S. A renormalization approach to invariant circles in area-preserving maps. Physica 7 D, 283-300 ( 1983) .
[4] Feigenbaum M. J., Kadanoff L. P., Shenker s. J. Quasiperiodicity in dissipative systems: a renormalisation analysis. Physica 5 D, 370 (1982).
[5] Ostlund S., Rand D., Sethna J., Siggia E. Universal properties of the transition from quasiperiodicity to chaos in dissipative systems. Physica 8 D, 303-342 ( 1983) .
315
[6] Douady A., Hubbard J. Iterations des polynomes quadratiques complexes. c. R. Acad. Sc. Paris, 294, 123-126 (1982).
[7] Eremenko A. E., Ljubic M. Yu. Iteration of entire func-· tions. Preprint of Phys.-Techn. Inst. of Low Temp. UkrSSR Acad. of Sci. Kharkov, N 6 (1984).
[8] Guckenheimer J. Sensitive dependence on initial conditions for one-dimensional maps. Cornrnun. Math. Phys. 70, 133-160 (1979).
[9] Ledrappier F. Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. and Dyn. Syst. 1, 77-94 (1981).
317
SOME PROBLEMS IN VORTEX THEORY
C. Marchioro
1. Introduction
In this lecture we want to study the following Hamiltonian
dynamical system, called "vortex system"
<JH a. x.
1 1 ay. 1
0 < i ... N
aH dx.
1
ai e R is called vortex intensity
and where H has the form
1 N
1 H 2 L: . a. a. gD (z., z.) + 2 ~-, J 1 J 1 J i=j
( 1.1)
N 2 I a. yD(zi,zi)
i=l 1
(l. 2)
where gD is the fundamental solution of the Poisson equation
( i . e . /', z gD ( z , z 1 ) = - o ( z -z 1 ) , gD ( z , z 1 ) =0 if z or z 1 D)
and yD is its regular part
1 +-
2'TT lnl z-z' I (1. 3)
The last term in (1.2) takes into account the interaction between
the vortices and the boundary of the domain D (roughly speaking
the interaction with the images). Of course if D=k2
1 2'TT
ln I z - z ~ I (1.4)
318
and the last term in (1.2) vanishes.
We will discuss , shortly, the following problems:
i) the meaning of the dynamical system
ii) its behavior during the time
iii) the statistical mechanics
iv) a mean field limit
v) stochastic vortex system.
2. The meaning of the dinamical system.
The dynamical system (1.1) is strictly related with the
fluid mechanics [1] . Let us consider a non viscous incompressible
fluid moving in a two dimensional domain D. The Euler equation
for the vorticity are
d at w (z,t) + (u• V ) w (z,t) 0
V• u 0
w (z, t) curl u(z,t)
w ( z '0) w 0
(2.1)
u • n 0 ' n normal of (l D
z (; D c R2
u = (ul,u2) (:.; R2 velocity
The relation between the vorticity w (z) and the velocity
u (equation (2.1) 3) can be inverted and it becomes
u(z) ='V ~ JD gD (z,z ') w (z') dz' (2.2)
where
319
We suppose now the vorticity concentrated in N points zi
N w(z) I a. 8 (z-z.)
i=l 1 1 (2. 3)
Using (2. 2) the velocity field is
N
u(z) I a. 'V.J. gD (z' zi) i=l 1 (2. 4)
When z _,_ z. 1
'V gD(z,zi) Qt 1 '(
"21T lz - z. I (2.5)
1
where T is a unit vector orthogonal to z-zi . That is u
becomes singular when z approaches zi. We evolve w (z) via
the Euler equation (in the weak form), we neglect the singular
term and we obtain in a formal way
N
w(z, t) I (2.6) i=l
where zi(t) satisfy (1.1). It means that the vorticity remains
concentrated and the center moves in the velocity field produced
by other vortices but not by itself.
This is a formal proof only.To make it rigorous we consider
N disjoint blobs of vorticity
N
wt:(z,O) I i (z,O) (2. 7) w i=l €
where
i (z ,0) -2
w a. € XII. € 1 1,£
Here
A. 1,E
where
time
sign
open.
320
is the characteristic function of the set
= E2 , diam. A . =)E ,A . 31 z. 1,£ 1,£ 1
We define center of vorticity of the blob i
Mi ( t) -2 J z E w
i (z, t) is the w E We must prove that
l im Mi (t) E-"o
Equation (2. 9)
z. ( t) 1
has
i (z, t) dz E
Euler evolution of
been proved for every
[2] and for every time for N=l [3]
i (z ,o). w E
N only for
and N = 2,
(2.8)
(2.9)
short
al = sign a [4] The general global problem is yet 2 As a remarke we note that the difficulty of a rigorous
proof is related with the singularity of the velocity field
in the limit f-.' 0 ,limit in which the paths of the particles
do not converge. A cancellation of the singular term allows us
to obtain (2.9).
3. Behaviour during the time.
We want to study the dynamical system (1,1). It is easily
to recognize the existence of some first integral of motion.
In fact the energy H itself is a constant of motion. Moreover,
if H is translationally invariant, then
N
M I a. z. const i=l
1 1 (3.1)
If H is rotationally invariant, then
N
I I 2 a. z. const i=l 1 1
( 3. 2)
321
When we deal with two vortices in R2 the above first '
integral are sufficient to determine the motion. If a1 ; a 2 they rotate with uniform speed around the center of vorticity.
If a1 ; - a2 they move on parallel straight lines with
uniform velocity.
For more vortices (or complicated domain) the problem is
more hard. Not even we are sure of the existence and the uniqueness
of the solutions since the logarithmic divergence of the Green
function may generate singularities in finite time not prevented
by the first integral of motion. It is possible to show that
special vorticities and initial conditions lead to a catastrophe
in a finiterime (5] . These collapse exclude the possibility
to give a general theorem of existence and uniqueness of the
solution of (1.1). However we can hope that these collapse are
exceptional in the sense that the set of the bad initial
conditions have null Lebesgue measure in DN. Equations (1.1) so
define a dynamical system, i.e. an evolution of every measurable
set. This property has been proved adding some technical
hypothesis on the vorticity and/or on the domaid D [ 6 7]
Let us discuss now the qualitative behaVPur of the vortex
motion. If D ; R2 for N ~ 3 the system is integrable, while
for N > 4 there is a numerical and a theoretic evidence that
a caotic motion may happen for many initial conditions [R J
Neverthless it is possible to exhibit a positive measure set
of initial conditions for which a quasiperiodic motion takes
place
D ; R2 '
[9] We skecth the main idea. we· suppose N ; 4,
H
a 3 o.-a4, a1 + a2 ~ -(a3 + a4l. The Hamiltonian
H + V where H is the interaction of the can be written as 0 0
two pairs of vortices and the interaction of their centers of
vorticity and V ; H - H0 that is a small perturbation if the
two pairs are far enough. The thesis follows by an application
of the KAM theorem [ 10] since the imperturbed motion given
by H0 is quasiperiodic. It is then easy to generalize to
arbitrary N by induction.
The previous idea can be extended to a bounded domain D.
322
In ttiis case of course we cannot put the pairs very far.
However ~7] we can constract weakly interacting clu~rs using the singularity of the interaction and scalcng the
problem
z + a z t .... large
4. The statistical mechanics.
The statistical mechanics of system (1.1) has been introduced
by Onsager with the aim to discuss the turbolence [11]
Formally the problem is very simple: system (1.1) is Hamiltonian,
the Liouville theorem holds and111e can introduce a "Gibbs
measure"
exp( -B H) I normalisation.
Here B plays the role of a factor which can be called
"temperature" in analogy with the usual statistical mechanics.
Of course a priori I; can be positive or negative. If we consider
a gas of vortices in a box when B is positive the particles
spread down in all the region, while when B is negative two
cluster of different 'vorticity will be formed. In fact vortices
interact as a coulomb system, i.e. via a logarithmic interaction
and different B correspond in some sense to attraction or
repulsion of vortices of the same sign. Of course in the first
case only there is a thermodynamical limit as it has been proved
rigorously in [12] .
For the Euler flow H is not the only first integral of the
motion. In fact every integral on D of a funtion of the vorticity
is a constant of motion. In particular
s
called enstrophy plays an important role. We do not enter in
this theory and we send the reader to the review paper [13]
323
5. Mean field limit.
In this section we want to discuss an other relation
between system (1.1) and the Euler equations. Suppose that
w " Ll /l Leo is an initial profile of vorticity and 0
it is approximated at time zero by N 1 LN 0 w N 0 x.
i=l 1
where { x1 , .... ,~} are suitable chosen points.
We call xi(t) the evolution of these points according to the
vortex dynamics (1.1) and w(t) the evolution of W 0 via
the Euler equation.
The question is: N 1 w t N
is an approximation of
More precisely if
N N 1 L w N 0 i=l
w ~ Ll ·1 Leo 0
then
1 N
L 0 N i=l xi (t)
N
L i=l
w ( t)
0 X.
1
weak N-+eo
0 xi (t)
weak N-+eo w
0
w (t) ?
The answer is positive if we add an essential technical divice:
we smooth the kernel and we remove the cutoff simultaneously
of the limit N-+oo [14, 15, 16, 17]
This result, interesting per se, gives a numerical algorithm
for the study of the Euler flow in two dimension. The rapidity
of the convergence depends in a deep way on a care choice of
the cutoff and on the regularity of the initial condition w 0 •
324
We note that this mean field theorem is analogous to the
relation in kinetic theory between the molecular dynamics
and the Vlasov equation, the so called Vlasov limit [ 18,19,201
6. Stochastc vortex model.
We want to approximate by a similar procedure using
a vortex model the Navier-Stokes equation for a viscous
incompressible two dimensional fluid . The equation are
Clw at + ( u• V ) w = V 11 w
'V • u 0
v > 0
We try to take into account the visco~i~ V of the
problem adding to equations (1.1) a Brownian motion.
In this case the stochastic vortex model is described
in a region without boundary (R2 or T2) by the stochastic
equations
where {b.} 1
N
l k=l k=i
+ 0 db.(t) 1
N 1
are N independent Brownian motions.
(6.1)
First we must prove that equations (6.1) define a dynamical
system, that is we must show that the probability of a collapse
is negligible. This has been done in T2 in [6]
325
Then, we want approximate (as in the previous section)
a solution of the Navies-Stokes equations by a measure
N
w N(dx) l 'i 6 (x - x~(t)) dx t N i=l
1
where x~(t) 1
are the solutions of (6.1) with a. = l/N 1
We can perform this program [16] smoothing the kernel,
removing the cutoff when N-><x> and choosing v = 0 2/2
We note that this problem is more complicate that the
deterministic case. In fact in this case the vortex model is
in some sense (neglecting an infinite term !) a weak solution
of the Euler equation and the mean field limit is a continuity
theorem with respect to the initial data. Viceversa (6 .. 1) is
not a weak solution of the Navier-Stokes equations. When N -·J '"
the large number law allow us to obtain the result.
When the boundaries are present the problem is more
complicate. In fact the boundaries can produce vorticity and
this important effect must be take into account. Chorin '~21'
introduced a numerical algorithm based on a stochastic vortex
model useful to appoximate numerically the Navier-Stokes
equations for high Reynolds number. Its rigorous justification
is yet missing. For a partial result see [22]
For a more detailed review of the argument of Section 2,3,5,6
see [7]
326
References.
[1] H.Helmholtz, Phil.Mag.,~, p.485 (1867)
G.Kirchhoff, Vorlesungen ueber Math.Phys. Teubener
Leipzig (1883)
H.Poincare, Theories des Turbillons, George Carre (1983)
Lord Kelvin, Mathematical and Physical Papers, Vol IV
(No.lO,l2) Cambridge Univ.Press p.563
C.C.Lin, On the motion of Vortices,in Two Dimension,
Toronto University Press p.39
[2] C.Marchioro ,M.Pulvirenti, Commun.Math.Phys.,2l,563 (1983)
[3] B.Turkington, On the evolution of a concentrated vortex in
a ideal fluid (preprint 1984)
[4] C.Marchioro,E.Pagani, Euler evolution of two concentrated
vortices moving in a bidimensional bounded region. (in
preparation).
[5] H.Aref, Physics of Fluid, ~ ,393 (1979)
[6] D.Durr,M.Pulvirenti,Commun.Math.Phys. 85, 265 (1982)
[7] C.Marchioro,M.Pulvirenti, Lecture Notes in Phys. 203,
Springer-Verlag (1984)
[8] H.Aref, Ann.Rev.Fluid Mech.,l5 (1983)
[9] K.M.Khanin,Physica D ~. 261 (1982)
[10] A.N.Kolmogorov,Dokl.Akad.Nauk. SSSR 98,527 (1954)
V.I.Arnold, Usp.Mat.Nauk, ~. 13 (1963)
J.Moser, Math.Ann.,l69, 136 (1967)
[11] L.Onsager, Suppl. Nuovo Cimento, ~. "79 (1949)
[12] J.Frohlich, D.Ruelle, Commun.Math.Phys.,~,l (1982)
327
[13] R.H.Kraichnan, D.Montgomery, Rep.Prog.Phys.,43,547 (1980)
[14] O.Hald, V.Mauceri Del Prete, Math.Comp.,~,791 (1978)
[15] O.Hald, SIAM J.Numer.Anal.,~,726 (1979)
[16] C.Marchioro,M.Pulvirenti, Commun.Math.Phys.,84,483 (1982)
[17] J.T.Beale,A.Majda, Math.Comp.,~,l (1982),~,29 (1982)
[18] W.Braun,K.Hepp, Commun.Math.Phys.,56 ,101 (1977)
[19] R.L.Dobrushin, Sov.J.Funct.Anal. !l,ll5 (1979)
[20] H.Neunzert, Lect.Notes in Math. 1048 (1984)
[21] A.J.Chorin, J.Fluid Mech.,2I,785 (1973)
[22] G.Benfatto,M.Pulvirenti, Generation of vorticity near the
boundary in two dimensional flows (Comm.Math.Phys 1984)
329
THE MAXWELL RULE AND PHASE SEPARATION IN THE HIER
ARCHICAL VECTOR-VALUED ¢4 -MODEL
P. M. Blecher
Summary. We investigate Dyson's hierarchical vector
valued ¢4 -model at S>Sc where Sc is the critical in-3 verse temperature. The "Gaussian" case, 2 > a > 1 , is con-
sidered. We prove that the thermodynamical potential is con
stant in the coexistence region of phases (the Maxwell rule) ,
and find the principal non-constant term of the asymptotics
of the small partition function when V+oo in this region.
We also study the phase separation at S>Sc .
1. Introduction
As it is well-known, in classical ferromagnet spin mod
els, such as the d -dimensional Ising model for d~2 , and
the d -dimensional Heisenberg model for d~3 , various spa
tially homogeneous pure phases exist at low temperatures.
The phenomenon of phase separation for Ising type models
consists in the fact that under fixed density p of par
ticles any typical configuration of spins represents a
"drop" of particles of a pure phase which is submerged into
another phase if p is such that p 1<p<p 2 , where P 1 , p 2 are densities of particles in the pure phases. The phase
separation with finite number of pure phases was investi
gated in Minlos, Sinai [1], Gallavotti, Miracle-Sole [2],
Kuroda [3] and others. For models with continuous symmetry
group such as the classical Heisenberg model,where the num
ber of pure phases is infinite, as it was proved in the pa
per [4] of Frohlich, Simon, Spencer, there is no descrip
tion of phase separation at present time as well as no suf
ficiently constructive description of the pure phases them-
330
selves.
In such a situation it is natural to try to study phase
separation in simpler models with continuous symmetry group.
Such a study will be done in the present paper where we con
sider phase separation in the hierarchical vector-valued
~ 4 -model. Moreover we shall establish rigorously the Max
well rule for this model according to which the thermody
namical potential is constant in the phase coexistence re
gion (see, e.g. [5]), and find the principal term of the
asymptotics for the small partition function when V+oo •
We use essentially the results of the paper [6] of Major
and the author, where pure phases of the hierarchical vec-4
tor-valued ~ -model were constructed,furthermore we use
some ideas of the paper [7] of the autho~where the Maxwell
rule and phase separation in the hierarchical scalar ~ 4 -
model was studied. It is noteworthy that phase separation
in the scalar and vector-valued models is similar but not
the same.
2. Description of the model and formulation of results
The hierarchical model was introduced by Dyson (see
[8]). We use a slightly modified definition of the hier
archical model. We follow the paper [6].
Let V = {1,2, ••. } , Vkn = {jEVi (k-1)2n<j~k·2n}
k,n 1 ,2, •••• Put v 1n=Vn • Define for i,jEV
n(i,j) = min{n I 3k such that i,j E Vkn}
The hierarchical distance is defined as
d(i,j) = 2n(i,j)- 1 if i#j 1 d(i,i) = 0 •
In a vector-valued hierarchical model the spin variables
a(i), iEV , take m
space lR , m~2 •
values in the m -dimensional Euclidean
The energy of a configuration a = = { a ( i) , i E V kn } is defined by the Hamiltonian
L U(i,j) (a(i);a(j)) , (i,j) :i#j i, j E V 1n
( 2. 1)
331
where U(i,j) = -d-a(i,j) and (·;·) denotes the scalar
product in lRm . In particular,
I u ( i , j J· (a ( i l ; a ( j l l (i,j) :i#j i,j EVkn
The value a , 2>a>1 , is the parameter of the model.
Given a probability measure v on lRm such that
J m exp(A(x;x)) v(dx)<oo for any A>O, we define the lR
Gibbs distribution in the volume Vkn at inverse tempera-
ture S = T- 1 by the formula
1kn(do) = 1kn(do;S,v) n v(do(i)) iEVkn
~kn :Okn(S,v) =fexp[-SHk (o)] IT v(do(i))
n . c:v ~- kn
In this paper we consider the vector-valued
the measure v is defined as
v(dx) L- 1exp[-%(x;x) 2 - ~~x)]dx
L f ( u( )2 _ (x 2;x_))dx . lRm exp l-4 x; x
(2.2)
(2.3)
4 ¢ -model where
(2.4)
Here u>O
spin model
is a parameter of the model. In the classical
v(dx) = L- 16(1xl-1)dx , L = f 6(lxl-1)dx lRm
For the classical hierarchical model Dyson proved in
[ 8 ] that if S is large enough then there exists sane b>O such that
< (a ( i) ; a ( j) ) > n = f (a ( i) ; a ( j) ) lln (do; S, v) > b for all n=
=1,2, ... and i,j EVn . Dyson's estimate means the exis
tence of long7range order. In paper [6] a detailed
study of the structure of the pure phases in the hierarch
ical vector-valued ¢4 -model was carried out for 1<a<~ Let us recall some results obtained there, which we shall
use in the following.
space
We say that a function ¢ ( s) , sElRm , belongs to the
ck+E (lRm) , 1 >E>O , if ¢ (s) E Ck (lRm) and for any
R>O ,
sup isi~R; O<it[~R
332
where a= (a 1 , ... ,a) is any multiindex such m a I al a1 , am
that [ai =
We say that a = a 1+ ... +am = k and D =a ;as1 •.. ~sm
function ¢ (s), sEJRm belongs to the class S , if the fol
lowing conditions are fulfilled:
1 ) ¢ ( s) = ¢ ( I s I ) (isotropy)
2) ¢ (s) E c 2+E (JRm) , E>O ;
3)3M>O suchthat ¢'(M)=O and cp"(r)>D>O,if 4
r > SM Note that here and later we use the convention to write
any function ¢(s) which depends on lsi both as ¢(s)
and as ¢ (Is I )
Let
Z (s) =Z (s;B,u) =!0([2-n I o(i)-sl1exp[-BH (o)] x n n iEV n
n
x II { L -l exp [ -~4 (a ( i) ; a ( i) ) 2--21 (a ( i) ; a ( i) ) ] do ( i) } iEV n
be the small partition function of the hierarchical vector
valued ¢ 4 -model. Note that the function pn(s) = Zn(s)/~n is the density of the probability distribution of the ran
dom variable 2-nLiEV o(i) with respect to the Gibbs measure n
Jln(do) .
In [6] a theorem was proved on the asymptotics of the
small partition function Zn(s) when n~oo. We slightly
strengthen now this theorem. Note that it implies a local
limit theorem with large deviation asymptotics for the ran
dom variable 2-nLiEVno(i) with respect to )Jn(da) , when
n~oo Let x(x) E C00 (JR1) be an arbitrary function such
that
0. 1 when x 5: 0. 1 ,
x(x)
x when x ~ 0. 2 ,
333
and 0.1 < x!x) < 0.2 when 0.1 < x < 0.2 . Write the func
tion Zn(s) in the form
[ 2n(2-a) Sao 112
Zn(s) = exp- --2- 1s, - 2n <Pn (s)) , (2.5)
where ao = (1-2 1-al-1 •
3 Theorem A. Let 1<a<2 3 such that E < (a-1) 12 -a) •
and E>O be arbitrary numbers
Then 3u0 such that Vu ,
0 < u < u 0 , 3Sc>O such that
= <j>(s;S,u) E S, <P'(M) =0,
that the asymptotic formula
V S > Sc 3 function <P (s) =
M"- (S-Sc)/u , <P" (M) "'S-Se such
(2.6)
holds when n~oo and
tion of the equation
lsi ~ T~_ 1 . Here is a solu-
<P' (TE ) + 2(n-1 ) (1-a)Sa (1-E)TE = 0 n n-1 1 n-1
such that lP~ (r) > a 0 > 0
depend on n and
when where
ln 2 m 1 ljin(s) =-n - 2 - [1+(m-1) (2-all-tzlnrr- 2 ln<P"(s) -
When
- m~ 1 L 2-j ln[x(Sa 1+2(n+j) (a-1)<1>' (lsl)/lsll j=O
E Is I ~ Tn_ 1 , the estimate
holds.
does not
The notation f = O(g) used here and in the following
means that f ~ Cg when C depends only on a . The nota
tion f"-g means that c1f ~ g ~ c2f •
Theorem A gives the asymptotics of the small partition E function Zn(s) in th~ domain lsi ~ Tn_ 1 and its estimate
in the domain lsi ~ Tn_1 It slightly strengthens the re
sult of the paper [6] where the asymptotics of Zn(s) was
334
proved in a somewhat smaller domain lsi :::: M(1-0,01·2n(1-al) .
We can verify that a local property ensuring the validity
of the asymptotic formula (2.6) holds in the domain
lsi <: T~_ 1 and it enables us to extend formula (2.6)
to the above domain.
In the present work we obtain the asymptotic behavior
of Zn(s) in the domain lsi<M.
Theorem 1. Let 1 <a< 3/2 , 0 < u < u 0 , S>Sc • Then for
lsi<M
¢n(s)-¢n(M) 2 2 lim 2n ( 1 a) = a 1 (M -I s I ) , n->-oo
where
In particular this theorem implies the phenomenoillgical
Maxwell rule for the investigated model. According to this
rule the thermodynamical potential ¢(s) =
=-lim [v i-1ln Z (s) is a constant function in the do-n->-oo n n main lsi~M . Indeed,
[ (2 ) Sao 2 n J <jJ(s) =lim 2-n 2n -a - 2-Jsl +2 <Pn(s) =lim <Pn(s) n~oo n+oo
In Theorem 2 we investigate phase separation in the
hierarchical vector-valued ¢4 -model. Denote
-n \ sk (o) = 2 L
n 'EV J kn
() ( j) 1
Prn {A [ sn (o) =s} = ~ lln (do sn (o) =s) ,
where
-1 11 (do[s (o)=s)=Z (s) o(s (0)-s)exp[-SH (0)] X n n n n n
x II { L - 1 exp [ -~ ( 0 ( j ) ; 0 ( j ) ) 2 - ~ ( 0 ( j ) i 0 ( j ) ] } jEVn
is the small Gibbs ensemble. The value Pr {A[s (o)=s} is n n the probability of the event A in the small Gibbs ensemble.
335
Theorem 2. Let 1 <a< 3/2 , O<u<u 0 ,
lsi<M , £>0 , and
n+oo lim Pr {A [s (o)=s} = 1 . n £ n Then
Roughly speaking Theorem 2 means that when n+oo , then
the small Gibbs ensemble 11 (do[s (o)=s) is the mixture n n
of pure ~es in the subvolumes v1,n-1 and v2,n-1 ·
3. Proofs of Theorems 1, 2
For the sake of simplicity we consider the case m=2 .
The extension of our considerations to the case of arbitrary
m~2 is straightforward. In the proof we use constants C ,
c 0 , c 1 etc., which will be positive and they depend only
on the parameter a of the hierarchical model. They may
differ in different estimates. The notation f = 0 (g) has
the usual sense:
co£ ~ g ~ c1g .
f ~ Cg , the notation f<vg means that
The functions ¢n(s) satisfy the recurrent equations
¢n+ 1 (s) = -2-(n+1)ln{4 JJR 2 exp[-2n( 2-alsa1 (v;v) -
- 2n ¢n(s+v)-2n ¢n(s-v)]dv}
(3. 1)
(see [6]). The main step of our proof will be the demonstra
tion that the integral at the RHS have these equations is essen
tially concentrated in the neighbourhood of some maximum
points which we shall find. Denote
;p ( s) n
and Fn(v;s)
easily that
when £
lsi ~Tn_ 1 ,
n(1-a) - -= 2 Sa 1 (v;v)+¢ (s+v)+¢ (s-v) . One can see _ 1 n n ¢ (s) E C (JR2 ) as T£ 1 is a solution of the n n-
336
equation
s= (r,O)
¢' (TE )+2(n- 1 ) ( 1-a)i3a (1-s)TE n n-1 1 n-1
, r = Is I , and
min F (v;s) 2 n
vCJR
Lemma 3.
v*"' J 0 ,
l ( 0 I
Fn(v*;s)
when
when
where Tn is a solution of the equation
¢~(Tn) + 2n(1-a)i3a1 Tn = 0 .
Proof. Denote
( 1 ) i3a1 2 ;:; ( I s I l = ¢ ( I s I l + 2n -a - 2-1 s I n n
0 . Let
Then F (v;s)=L; (ls+vi)+L; (ls-vl)-2n( 1-a)Sa 1 1sl 2 n n n . As the
value 2n( 1-a)sa1 lsl 2 does not depend on v , we have to
find only the minima of the function
+L;n(ls-vl) . First we show that the
of the function ;:;n(r) . Indeed,
-o Fn(v;s) = ;:;n(ls+vl)+
Tn is a minimum point
;:;~(Tn) = ¢~(Tn) + 2n(1-a)i3a1Tn = 0 ;
;:;~(r) = ¢~(r)+2n( 1 -a)Sa 1 ~a 0 +2n( 1 -a)Sa 1 >o, if
Sa ;:; (r) = ¢ (Ts )+2 (n-1) (1-a) _1 (1-s) ((TE )2-r2)+2n(1-a) n n n-1 2 n-1
By the above relations we have
if (3. 2)
£ E 2 2 L:n (r) ~ L:n (Tn-1) +a1 ( (Tn-'-1) -r ) ' if
E 0 ~ r ~ Tn_ 1 , (3.3)
where 13 ~ 1 [2a- 1 (1-s)-1] >0 1 SO is the minimum
point.
337
Consider the circumferences c1 = {vlis+vi=Tn} I c2 =
{ v I I s-v I =Tn} . If Is I $Tn 1 they intersect each other in the points
v 112 = (o 1 ±/T~-Is1 2 ) so
-o min Fn(v;s) ~min ~n(ls+vl)+min ~n(ls-vl) ~ v v v
as we claimed.
If Is I > Tn the circumferences c1 I c2 do IlOt inter
sect each other. Now we show that the minimum point is v=O • As -o the function Fn(v;s) is even in
v = (v 1 ~v 2 l we may assume that
=(lsi-TniO) If ls+vl > ls+v0 1
and in v 2 1 where
1 v2~o • Let v0
so the minimum point does not lie in the domain ls+vl >
> ls+v0 1
If ls+vl ~ ls+v0 1 1 v= (v11v 2) 1 v 1 ~o 1 then lsl+v1 ~
~ ls+vl ~ lsl+lv0 1 so v 1 ~1v0 1 and F~(v;s) ~ F~(v;s) 1
where v= (v110) I as ls±vl ~ ls±vl ~ Tn and so
~n(ls±vl) ~ ~n(ls±vl) . Thus the minimum point lies on the
segment [0 1v0 ] • On this segment ls±vl ~ Tn > T~_ 1 so
a2 -o ( 1 l --2 Fn(v;s) = ~~(ls+vl)+~~(ls-vl) ~ ao+2n -a Ba1 I
av1
hence
i.e. v=O is the minimum point. Lemma 3 is proved. * Let v denote the set of the minimum points of the func-
tion F (v;s) in v . By n Lemma 3 V*={O} when Is I ~ Tn
and V* = {v11v2} where v1 12 = (o~±h~-lsl 2 ) when O<lsl~
~Tn Denote
p(v;V*) inf lv-v 1 I . V 1 EV*
Lemma 4. For any E 1 > 0 there exists a constant C 1 > 0 such £
338
that for lsi~£' and v*CV* ,
r c p4 (v;V*) if p(v;V*) s: ~~
£ F ! v · s > - F ! v* • s > n ' n ' 2
l c£ , p (v;V*) , if p(v;V*) ~ 1
Proof. By the estimates (3.2), (3.3),
(3 .4)
If £' S: lsi ~ Tn , then simple geometrical considerations
show that
rc(£')p2(v;V*)
max{ I Js+vi-Tnl, lls-vJ-Tnl} ~ ~ [co p (v;V*) ,
Therefore
if p (v;V*) ~ 1
if p(v;V*) ~ 1 •
F (v;s)-F (v*;s) =<; (Js+vl)+<; (Is-vi)-<; (ls+v*l)-1; (ls-v*l) n n n n n n
fc1 (£') p 4 (v;"", if p (v;V*) s; 1
=l;n ( ls+vl )-l;n (Tn)+~ (Is-vi )-<;n (Tn) ~ i [c1p2(v;V*), if p(v;V*) ~ 1 •
The case J s I ~Tn is considered similarly. Lemma 4 is proved.
Denote vs = {v ! lvi-T 1 t n < 21 s I }
Lemma 5. If 0. 1 > £I > 0 then for lsi<£' we have
Fn(v;s) - Fn(v*;s) ~ c p2(v;Vs)
Proof. The desired inequality is a consequence of (3.4)
and the simple geometrical inequality
1 s 2 p(v;V)
Indeed,
339
Lemma 5 is proved.
Proof of Theorem 1. We estimate the function ¢n(s)
from above and below.
Estimate ¢n(s) from below. We shall prove by induc
tion the inequality
Sa "' ( ) >A: (s) ="' (TE: ) +2 (n- 1 ) (1-a) - 1 (1-E:) ( (TE: l 2-isi 2l (3 5) '~'n s - '~'n '~'n n-1 2 n-1 ' ·
when lsi ~ T~_ 1 For n=O this inequality is verified
easily with the help of the explicit formula ¢0 (s) =
u 4 1-Sao 2 = 4 1s1 +-2-- lsi • Indeed, the function ¢ 0 (rl-¢ 0 (r) is
a polynomial of fourth degree and by construction it has
second order zeroes at the points r = ±T~ 1 , and it has no
other zeroes, hence ¢ 0 (rl-¢0 (r) ~ 0 .
Assume that inequality (3.5) holds. for the function
¢n(s) . Prove it for the function ¢n+ 1 (s) . By the recur
rent formula (3.1),
n - ] } - (n+1) { n - } - 2 ¢n (s-v) dv = -2 ln 4 JlR2 exp[ -2 Fn (VJS) ]dv
By Lemma 4 when lsi~E' and by Lemma 5 for lsi~E' ,
If I sl ~Tn ,
Fn(v*Js) = 2 ¢n(Tn) + 2n(1-a)Sa1 (T~-Is12)
so
¢ (s) ~ -2-(n+1 )ln{4C exp[-2n+1¢ (T )-2n( 2-a)Sa x n+1 n n 1
Sa x(T2-Isi 2 lJ} = ¢ (T )+2n( 1-a)_1 (T 2 -isi 2 )-2-(n+1)ln(4C)~ (3.6)
n n n 2 n
~ ¢ (T )+2n( 1-a) sa, (1-E) (T 2-Isi 2 l-2-(n+ 1 )ln(4C) n n 2 n
340
Let us estimate now the RHS of this inequality. In the
proof of Theorem A the following estimates were established:
(3. 7)
We use now these estimates. Introduce the function
( 1 ) Ba1 E 2 2 r;, (r) =¢ (r)-¢ (TE)-2n -a - 2-(1-E) ((Tn) -r ). n 1E n n n
It has the following properties:
r (TE) - 0 "n 1 E n - 1
!;;" (r) n 1 E
¢" (r)+2n( 1-a) Ba (1-E) n 1
Hence by the Taylor formula
where E e E [Tn1Tnl 1 or in a more detailed form
E -n x 0 (Tn u 2 ) +
Substituting this formula in the RHS of (3.6) we get
( 1 ) Ba 1 E 2 2 ¢n+ 1 (isl) ~ ¢n(T~)+2n -a - 2-(1-E) ((Tn) -lsi ) -
(TE-T ) 2 n n (¢"(e)+
2 n
+ 2n ( 1 -a) Ba 1 ( 1 - E ) ) 1
341
or taking into account (3.7),
( 1 ) Sa 1 E 2 2 ¢n+ 1 (1sl) ~¢n+ 1 !T~)+2n -a - 2-(1-E)((Tn) -lsi)+
It remains to estimate
¢"(8 )(TE-T )=¢'(TE)-¢'(T )=¢' (TE)-¢'(T) + n 0 n n n n n n n+1 n n n
E -n + 0 (Tn u 2 ) ,
where Hence
¢ 11 ( e ) + 2n ( 1 -a) 13 n 0 a1
Thus
(TE-T ) 2 n n (¢"(8)+2n(1-a)l3a (1-E))
2 n 1
2n ( 3- 2a) S 2 M2 E 2 cv 2-n x
¢" (M)
As M2 cv (13-13c)/u, ¢"(M) cv 13-Sc we get finally that
(TE-T ) 2 (3 2 ) 2 2 n n (¢"(8)+2n(1-a)l3a (1-E)) cv 2-n x 2n -a 13 E
2 n 1 u
(3 .8)
(3. 9)
As 2n( 3- 2a)S 2E2/u »n then by the inequality (3.8) we have
now
342
( 1 ) Sa1 E 2 2, ¢n+ 1 (lsi) ~ ¢n+1 (T~)+2n -a - 2-(1-E) ((Tn) -lsi J
We proved the estimate (3.5) for Jsi~Tn . It remains
to prove it for Tn < lsi ~ T~ • Here we shall consider two
cases: 1) T < lsi ~ (Tn+TnE)/2; 2) (T +TE)/2 <lsi ~TE . n n n n In the first case we use the equality Fn (v* ;s) = 2¢n (Is I)
and similarly (3.6) we get the estimate
Further estimations may be proven in the same way as in the
case lsi~T and we omit them. n
In the second case, when (T +TE)/2 < lsi ~ TE , the n n n function ¢n+ 1 (lsi) satisfies the estimates of Theorem A
and according to these estimates
¢~+ 1 (r) > 0 > ¢~+ 1 (r) , when
Since by definition ¢n+1 (T~) = ¢n+1 (T~) , ¢~+ 1 (T~) =
= ~~+ 1 (T~) hence the last inequalities imply the estimate
¢n+1 (r) ~ ¢n+ 1 (r) . The proof of estimate (3.5) is canpleted.
Estimate of ¢n(s) from above. Let us prove the esti-
mate
(3. 10)
when lsi~Tn . For this purpose we shall use the estimate
AI *'2 r;n (I s±v I) -r;n (I s±v* I l ~ 2 v-v 1
when '1v-v* 11 ~ T -TE , where n n-1 A = sup r;~ ( I s I l =
n ( 1-a) = sup(¢~(1sll+2 Sa 1 l and sup is taken with respect
t TE 0 n-1
~ lsi ~ 2 T -TE 1 . Substituting this estimate inn n-to the recurrent equation (3.1) we get:
n(2-a) exp[2 x
343
Note that (2nA) 112 (T -T* ) » 1 n n-1
relation implies:
The estimate (3.10) is proved.
Let us show now that
(see (3.9)), so the last
As ·~(M) = 0(M·2-nu) , ·~(Tn) = -2n( 1-a)Ba1Tn then
M-T <v 2n 11 -a) Ba 1M I ... (M) • So • (T ) -• (M) <v ... (M) (T -M) 2 <v n n n n n
'V 22n (1-a) 82M2 I ... (M) 'V 22n (1-a) B2 I u • Hence
when n~oo what was stated. Similarly
•n(T~-1>-•n(M) lim ( 1 ) 0 • n~oo 2n -a
We get now by the estimate (3.5),
• (Is I)-· (M) Ba 1 2 2 lim n 2 (n- 1 ) (~-a) ~ - 2-(M -lsi ) (1-c) n~oo
During the proof we used an estimate of £ from below only
once when we assumed that 2n (3 - 2a) s2c 2 I u » n • Hence it is
clear that we may consider a sequence £=en~ 0 when n~oo •
Then the last relation
lim <Pn ( Is I l -<Pn (M)
2 (n-1) (2-a) n-+-oo
Similarly with the help
<Pn ( Is I l -<Pn (M) lim ~-=-. .. ~7Tn-+-oo 2 (n-1) (2-a)
hence
<Pn ( Is I ) -<Pn (M) lim ~~----~~n-+-oo 2 (n-1) (2-a)
Theorem 1 is proved.
344
gives
sa, (M2-Isl 2 l -2- .
of (3. 10) we get
Proof of Theorem 2. We use the formula
Pr {A Is (cr)=s} n E n
n II ls±vi-MI<Eexp[-2 Fn(VJS)1dv
J 2 exp[-2nF (v;s) 1dv lR n
(3 .11)
where Fn(v;s) =2n( 1-a 1sa1 ivi 2+<Pn(s+vl+<Pn(s-v) (see [71). Using Lemmata 4, 5 we get:
J exp[-2n Fn(VJS)1dv ~ lR2 '{ lls±vi-MI<d
~
~
On the
Hence
J exp[-2n Fn(v;s)1dv ~ :JR2 '{ II s±v 1-M I <E}
C exp [-2n - n 4 Fn(v*;s)-2 c0E 1
C exp [-2n n 4 Fn(V*Js)-2 c 0E 1
other hand using the estimate (3 .1 0) we get:
J exp [-2nFn(v;s)]dv::!: c 1 exp [-2nFn(v*;s)-c2nJ lR2
345
n n 4 C exp[-2 Fn(v*;s)-2 C0E]
n c 1exp[-2 Fn(v*;s)-C 2nJ
when n~oo • Theorem 2 is proved.
[ 1]
I2l
13]
[ 4]
[ 5]
[6]
[ 7]
[8]
References
Minlos R. A., Sinai Ya. G. Phenomenon of "phase separation" at low temperatures in some lattice models of gase. I. Matern. Shorn. 73, No 3, 375-448 (1967), II. Trans. Moscow Mathern. Soc. 19, 113-178 (1968).
Gallavotti G., Miracle-SoleS. Equilibrium states of the Ising model in the two phase region. Phys. Rev. B5, 2555 (1972).
Kuroda K. The probabilistic treatment of phase separations in lattice models composed of more than two types of particles. Publ. RIMS Kyoto Univ. 18, No 1, 275-305 (1982).
Frohlich J., Simon B., Spencer T. Infrared bounds, phase transitions and continuous symmetry breaking. Cornrnun. Math. Phys. 50, 79-95 (1976).
Stanley H. E. Introduction to phase transitions and critical phenomena. Clarendon Press, Oxford, 1971.
Blecher P.M., Major P. Renorrnalization of Dyson's hierarchical vector-valued ¢4 -model at low temperatures, preprint Math. Inst. Hungar. Ac. Sci. 1984, Cornrnun. Math. Phys. (in press).
Blecher P. M. Phenomenon of phase separation in ¢ 4 -hierarchical model. Theoret. Math. Phys. (in press).
Dyson F. J. Existence of a phase transition in a onedimensional Ising ferromagnet. Cornrnun. Math. Phys. 12, No 2, 91 (1969).
347
CONSTRUCTIVE CRITERION FOR THE UNIQUENESS OF GIBBS
FIELD
R. L. Dobrushin and S. B. Shlosrran
1. Introduction
In this report we consider v -dimensional lattice
systems with a given translation-invariant potential U
Our main result is the construction of a set of conditions
CV on the potential U , where vczv is any finite vol
ume, such that if for some V the condition CV holds for
the interaction U , then the Gibbs state with this inter
action is unique. The complexity of the conditions CV in
creases, of course, with the volume V . The one-point cri
terion, C{t} , was introduced earlier by one of us [1]. It
was intensively used later (see [2]-[7]).
Under the conditions CV , some property of exponen
tial decay of correlations can be proven. This property is
very Strong (see (3.10)), though not the Strongest one
(see (3.11)). (In this report only finite range potentials
are treaten - otherwise the exponential decay statement has
to be replaced by a slower one. The infinite-range poten
tials will be treated in a separate publication.)
To discuss the effectiveness of our conditions CV ,
we restrict ourselves to the models with finite single-spin
spaces. In this case the set A0 of the interactions, for
which for some V the condition CV holds, includes the
set of those interactions, for which the unicity is accom
panied by the Strongest exponential decay of correlations
mentioned above. In a subsequent paper [8] we show that this
last property follows from a natural property of analyticity
of the partition functions. This and other speculations make
348
plausible the following
Conjecture: For a wide class of potentials the bound
ary of the set A0 is the surface of phase transitions.
But from our definitions it follows that A0 = u A n=1 n
where A1cA 2c ... and the boundary of the set An can be
specified by a finite number N(n) of arithmetical and
logical operations. This is why we call our criterion "ef
fective". Of course, N(n) + oo with n+oo • Nevertheless,
our criterion provides one of the first possibilities to
locate the phase-transition threshold for a general situa
tion, in principle with any given accuracy (compare, how
ever, with [9]).
This has to be compared with many wellknown numerical
methods for exploring the phase-transition region, utilized
by mathematical physicists (for example, MC-simulation).
The main difference lies in the following. While both the
traditional methods and ours are finite-volume ones, the
results given by the former are necessarily approximate
when attributed to infinite-volume systems; so their im
plications about phase transitions cannot be justified with
mathematical rigour. On the other hand, the criterion de
scribed in this paper can be used as a basis for computer
assisted rigorous proof of the absence of phase transition.
This is because our result has the form of the following
Theorem: If Something (=Cvl happens in some finite
volume (=V) , then surely there is no phase transition (in
infinite volume)!
we are going to finish computer-assisted proofs of
unicity for certain models in the near future.
The content of this report is the following: in Section
2 we state our conditions CV and prove that Cv implies
unicity. The next Section 3 deals with various definitions
of correlation decay, and we prove the exponential decay,
provided the condition Cv holds. The setting of both of
the Sections 2, 3 is abstract - we study random fields with
a specification given. We turn ourselves to the Gibbs case
in Section 4, in which in addition to the general results
349
of Sections 2, 3, some specific results can be proven -
using ground state structures. The final Section 5 deals
with the effectiveness of the criterion presented. We show
that the set A0 is effectively enumerable and present an
algorithm, which enables one to check uniqueness for any
given potential. We finish this section by fixing some nota
tions used throughout this paper:
z;v - v -dimensional lattice, v~1 ,
s - single-spin space ,
B - o -algebra of subsets of S ,
o, o, T,... : z: v + S - maps, or "configurations on :l!:v ",
~ - the set of all configurations,
V, W, /1., ••• c Z v - subsets,
lVI - the cardinality of V , zv,v ,
v i { t E :l!: : -n~t s:n i=1, ... ,v} -the n -cube,
centered in the origin,
~V- the set of all configurations on V:{ov:V+S} ,
ov olv- the restriction of oE~ or or:~W on vcw
ov u ov E ~v uv is defined by (ov1 u ov2 l ivl. ov. 1 2 1 2v l
i=1,2 for '1,v2 cz: , v 1nv2 = 0 , ov. E ~v. , i=1,2 , l l
BV - o -algebra of the subsets of ~v , generated by
the cylinders {ov E stv'otEBt , BtEB , tEV}
B = Bzv ,
av = a v r
{tEVc: dist(t,V)S:r} , where r~O •
2. Unicity
Let (S,B) be a measurable Polish space (or finite
space, if preferred), and p(·,·) be a metric on S , such
that the function p(o,T) on sxs is measurable with re-
spect to the o -algebra BxB of probability measures on S
~ 1 , ~ 2 we call any measure ~
~(BxS) ~ 1 (B)
Consider a pair ~ 1 , ~ 2 By a joint distribution of
on sxs such that
( 2. 1)
350
for all BEB • The set of all these ~ will be denoted by
~0
on
For any measure
S let
on sxs and any measures
R (~l p
f p(a,T)~(da,dT) sxs
(2. 2)
RP(~ 1 ,~ 2 J can be shown to
of the measures ~ with
be a metric on the space
RP(~,~ 0 J < oo for some
is called Kantorovich metric, (see [1]). The metric R p
or Kantorovich-Rubinstein, or Kantorovich-Rubinstein-Orn
stein-Vasserstein and so on (see a recent historical-orien
ted review [10], which contains about 150 references begin
ning with Monge's paper published in 1781).
For
p (a 1 T)
( 1 1 afT 1
i (2. 3)
(0, a=T I
RP (·,·) coincides with the variation distance Var(·,·)
For any V c Z v , any probability measure P on
(~v,Bvl and any WcV we denote Pw the projection of P
on (~W' Bwl , given by
(2 .4)
By a random field we mean any probability measure P on
(~I B) •
By an r -specification (r~O) we mean a system of
functions Q = {Qv:vczv,IVI<oo}, Qv = QV(AioJ, where
AEBV I oE~ I such that
1) for any A the function QV(AI ·) is Ba V -measurr
able,
2) for any a the function Qv(·lo) is a probability
measure on (~v,Bvl
on
Sometime we shall think of
~a V , or even on r
~a vnw r
Qv(AI ·) as of a function
for some W c X v . This is
351
done for notational convenience and is justified by 1). We
denote by QV,A(· Ia) the projection of Qv(· Ia) on a set
Ac.v
A random field P is V -consistent with a specifica
tion Q for a set V c. Z v , IV I <oo , if the conditional dis
tribution of P with respect to the a -algebra B c , v
Pv(·l ·) , coincides with Qv(·l ·) almost everywhere.
A specification Q is called self-consiste~t, if for
any finite Vc.Zv and aES"l the random field Q~, defined
by the following conditions -
1 ) (Q~lv coincides with Qv(·la) ,
0 -( Qv l c (a l = 1 ,
v 2)
is W -consistent with Q for any Wc.V •
A random field P is called consistent with a self-
consistent specification Q , (or Q -consistent) if P is
V -consistent with Q for all finite Vc.Z:v .
The main example of self-consistent specifications are
Gibbs specifications (see Sect. 4).
In what follows, we shall consider only translation
invariant specifications. This is done solely in order to
simplify notations.
\'le say the random field P to be of exponential growth,
if for some c 0Es , g, G < oo
(2.5)
We denote the set of all Q -consistent fields P , satis
fying (2.5) for some T 0ES and G<oo by P0 (g)
Theorem 2.1. Suppose that for a given r -specification
Q there exists a volume V , such that the following con
dition holds:
Condition cv : there exists a function kt~o , tEdV
with the following properties:
1 ) for tEdV and any -1 a2 E r2
-1 -2 for any a , , a a s s
s;lt
352
(2.6)
where
1 2 o ,o E 'Jv (2.7)
for lVI <oo
2)
y < 1 • (2.8)
Then there exists a value g 0 = g 0 (v,r,y) > 0 , such
that the set PQ(g 0 ) contains no more than one element.
Remarks
2.1. In the case IVI=1 Theorem 2.1 was proven by one
of us in [1]. This particular case is called "mean-field"
bound (see [3]), so our theorem is an improvement of the
latter bound.
2.2. In the case when S has finite diameter, the
theorem implies the unicity in the class of all Q -consis
tent fields. On the other hand, for S unbounded, the re
striction (2.5) is natural (see [11] for Gaussian case),
and the theorem fails without it.
2.3. Suppose that
d = diam S = sup p(o,T) ~ 1 o,TES
(2.9)
(as in the case (2.3) of the variation distance).
Then the following more simple condition implies uni-
city:
For some finite V c Z v and any
exactly in one point t E Z v
with some E>O .
-1 -2 o , o E 'J different
( 2. 10)
As for the proof of the above statement, one has only
to put kt = IVI/13VI for tE3V and to apply Theorem 2.1.
353
The proof of Theorem 2.1 is based on the following
lemmas.
Lemma 2.1. Suppose that condition CV holds for the
specification Q • Let W be any given volume, and
T(W) {tEZv: (vuav)+tcw}. (2.11)
Let P1 , P2 be two random fields, (V+t) -consistent with
Q for all t E T(W)
Then, for any o >0 there exists an element \l E: 1 2
E K (PW,P\.Y) (which is a measure on rlwxr.\v ) , such that for
1 2 1 2 J p(ot,ot)\.l(do ,do ) , tEW (2. 12) rlwxrlw
and for all s E T(W)
I f s tEV t+s
(2.13)
Proof of Lemma 2.1. It is enough to take for the meas-1 2
\l any element of the family K(PW,PW) with the prop-ure
erty
(2.14)
(the definitions (2.2) and (2.7) implies the existence of
such a \l } • To see (2.13) holds for this measure, let us
fix some and check (2.13) for it. To simplify
the notations, suppose -1 -2 that for any a ,a E rl
E K ( QV ( • I 0 1 ) , QV ( • I 0 2 ) )
\) s 0 = 0 E:rt: From (2.6) it follows
there exists a measure (l ( ·,. 101 ,02) €
on such that
(2.15)
Now we are going to apply to the measure \l on rlvxrlv the
following "surgery" construction (see [1], [12]). For any
W 1 c W denote by Bv.; the a -subalgebra of Bwx BW gener
ated by the pairs B 1 xB" c rlwxrlw , where B 1 ,B" E Bw . Now 1
354
the V -surgery, applied to ~ , results in a new measure
~ on nwxnw , which is defined by the following two prop
erties:
1) ~ equals ~ on BW'V
2) its conditional distribution with respect to BW'V
is equal ~-everywhere to 0(·,· 1·,·) • From the Q -consistency of P1 , P2 it follows that
- 1 2) ~ E K (Pw,Pw , hence
- 1 2 L ft?: R !Pw, P\vl (2.16) tEW Pw
where ft is defined by (2.12) with ~ instead of ~ •
By definition,
From (2.15) it follows that
t - - -1 -2 1 2 1 21-1 -2 1.. ft= J ~(do ,dol J Pv(o ,a lP(do ,do a ,a)~
tEV r1\'1xnW Vriy
From (2.14), (2.16), (2.17) it follows that
I L f -tEV t
hence from (2.18)
L f < L k £ + o tEV t - tEav t t
which together with (2.17) implies (2.13).
(2. 17)
(2. 18)
(2.19)
(2.20)
Lemma 2.2. Let condition 2) of Theorem 2.1 hold (see
(2.8)). Let a volume W<=Zv, IWI<co, and a positive func
tion ft , tEW be given, such that for all sET (W) the
bounds (2.13) hold with some o>O • Let AcW be given and
for some g>O let us define the function
C(t) =C -(t) =exp{-gdist(t,A)}, tEZv. A,g
(2.21)
where
355
-Then for some value g
I tEW
f(tlc(tl ~c I tdvw
{tEW : dist(t,Wc) ~ diam(VU3V)}
(2.22)
(2.23)
Proof of Lemma 2.2: After multiplying each of the
bounds (2.13) by the corresponding value c(s) and summing
them over all sET (W) one arrives to the following esti
mate:
(2.24)
S: L ft[ L c(s)- L kt-s c(s) l + cS IWI • tEW sE(T(W))c:t-sEV sE(T(W))c:t-sE3V J
Let
c(s) m1 = max c(t)
t,s:t-sEVU3V
c (s) , m = min ----2 t,s:t-sEvuav c(t)
(2.25)
Then the left-hand factor in (2.24) can be estimated using
(2.8) as follows:
L c (s) -
s E :I: v :t-sEV
L kt-s c (s) ~ c(t) lVI (m2-ym1) (2.26) sEZV:t-sE3V
which in turn is larger than c ( t) IV I K provided 0 < K < 1-y and g = g ( K) is small enough. The analogous factor in the
right-hand side of (2.24) is non-zero only for t: aVW • It
can be estimated from above by 21VI , provided g is so
small that m1 ~ 2 • From this and (2.26) follows (2.22).
1 2 Proof of Theorem 2. 1 : Let P , P E P Q (g) . To prove the
theorem it is enough to show that for any finite A cZv
P ~ = P~ Let AC:W for some finite W c Z v . From Lemmas
2.1, 2.2 it follows the existence of a measure ]JE:K(P~,P~) such that
356
(2.27)
Using definitions (2.21), (2.12), (2.7) one has
(2.28)
On the other hand from (2.5) and (2.12) it follows that
(2.29)
Now take W to be the n -cube Vn (defined in Sect. 1).
It is clear that for g~g 0 <g and for a suitably chosen
o=o(n) the right-hand side of (2.27) goes to zero as n4oo 1 2
From (2.27) and (2.28) it follows then that R (PJ\,PJ\) = 0 pi\
But is a metric, hence
3. The decay of correlations
In this section we shall show that for any Q -consis
tent random field P the correlations decay exponentially,
provided condition CV of Theorem 2.1 holds. In what fol
lows we suppose for simplicity that condition (2.9) on the
diameter of s holds.
Theorem 3.1. Let Q be a self-consistent r -specifi
cation and the conditions of Theorem 2.1 hold. Let P be
the (unique) random field consistent with Q Then there
exist constants G1 <oo , g 1 >0 depending only on y , r , V
such that for any 1\cW c zV and any oErl
R (QW 1\(· lo),PJ\) ~ G1 L exp{-g 1dist(t,J\)} Pi\ ' tEdvW
(3. 1)
An analogous bound holds for the specification Q itself: -1 -2
for o ,o Erl
RPA <cw,A<·Io1J,CW,A<·Io2JJ ~ G1 I exp{-g1dist<t,AJ}
tEaJ"
(3. 2)
Proof: The proof is very similar to the proof of Theo-
357
rem 2.1. To prove (3.2) one has to consider the measures . i l 0 · 1 2 ( S t 2) h ( t) P = QW , l= , see ec . . T ese measures are V+ -
consistent with Q for t E: T (I'V) , hence r~emmas 2. 1 , 2. 2
are valid for them. The condition Pi € P (g) follows from
(2.9). The inequality (3.1) is proved in the same way.
Now we state the decay property in a more familiar
form.
Theorem 3.2. Under the hypothesis of Theorem 3.1 let
1\i c:z:'J be finite and ljli be B/\. -measurable for i=1,2 l
Suppose that l)l 1 is bounded while l)l 2 is Lipschitz: for
c 1 , c2 < oo
(3. 3)
Then
IH1 (o)lji 2 (o)P(do)-H 1 (o)P(do)!l)! 2 (o)P(do) I ~ s:J s:J s:J
(3. 4)
where G1 , g 1 -are the same as in Theorem 3.1.
Proof: we begin by the following remark due already to
Kantorovich: for any function lji (a) , a E S , which is Lip
schitz with a constant C>O and for any measures ~ 1 , ~ 2
if lji(ol~ 1 (do) - f lji(o)~ 2 (do) I ~ CRP(~1'~2) s s
(3. 5)
Indeed if ~EK(~ 1 ,~ 2 ) then
II lji (a) ~ 1 (do) - f l)l(o)~ 2 (do) ~
s s
~ f llji(o)-lji(T) l~(do,dT) ~ C f p(o,T)~(do,dT) sxs sxs
so (3.5) follows. Now without loss of generality one may
assume that
358
f 1J! 2 (a) P (da) = 0 . (]
( 3. 6)
Let be the conditional distribution of P with
respect to the a -algebra 8(~ 1 ) • Then
p~ [ (/ 1 (·I;:;)) A_,P~] :5G1 L exp{-g1 dist z --:.:! z tEa (~ l c
v 1
which follows from (3.1) if one takes ~=~ 2 , W=Vn\~ 1 and
then puts n~ro . From (3.5), (3.6), (3.7) it follows that
~, IH2 (a)P (dal;:;li:5G1c2 L exp{-g 1 dist(t,~ 2 J} (3.8)
rl tEav(~ 1 Jc
But
- - ~, -f lJ! 1 (a) lJ! 2 (a) P (da) = f P (da) lJ! 1 (a) Hz (a) P (da I a) (] (] Q
Hence (3.8) implies (3.4).
Remarks.
3.1. In the case when p is given by (2.3) any
bounded function 1jJ on rl~ is Lipschitz with respect to
p~ with Lipschitz constants C = sup lj!(a) . So in this aErl ~
case the bound (3.4) is applicable to all bounded functions.
3. 2. The bound ( 3. 1 ) is more powerful than ( 3. 4) (be
cause the 6 -function is not Lipschitz!).
3.3. One would like to improve the bound (3.4) sub
stituting exp{-g dist (~ 1 .~ 2 )} instead of the right-hand
sum. However this "improved" bound does not hold in general.
Really note that in the general case the measure P~ u~ 1 2
is singular with respect to the product measure P~ xp~
{t=(t1 ,t2 ) E:ll: 2 :t1=a.} 1 2
if for example ~. = , i=l, 2 for l_ l_
any a 1ta2 It is clear because in the general case the
expectations of random variables of the type
n ( 1 L ~ X ,x I k=-n (a 1 ,k) (a2 ,k) J
are different for both measures and the law of large num-
359
bers proves that these variables are almost constant for
large n .
3.4. From (3.1) the unicity of a Q -consistent field
follows.
3.5. Consider now the case of p given by (2.3). Then
for the corresponding metric pA ("Hamming distance") the
following estimate is evident
where p is the metric on QA , satisfying (2.3). Hence
for any measures ~ 1 , ~ 2 on QA
(3. 9)
In our case the estimate (3.2) implies the "Very Strong
Decay Property": for any ;;: 1 , ;;: 2 E Q
-1 -2 I Var(QW,A(·Io ),QW,A(·Io )) :s; G1 L exp{-g 1dist(t,A)} (3.10)
tEavw It is important to compare (3.10) with the following
"The Strongest Decay Property": for some G2 ,g 2 > 0 and any
o1 , ;;:2 E Q such that a! ;;:~ unless S~tO , t 0 E 3W
( 3. 11 )
The main difference between (3.10) and (3.11) is the fol
lowing. In the case of (3.11) the conditional distributions -1 -2 in W corresponding to boundary conditions a ,a dif-
ferent only in a point t 0 differ essentially only in a
vicinity of t 0 while in the case of (3.10) these distribu
tions may differ essentially along the whole boundary of
W. In [8] we show that (3.11) is equivalent to a natural
analyticity condition on the partition function.
Theorem 3.3. Condition (3.11) implies condition CV
of Theorem 2.1 for some V •
Proof: Let
Vn\Ad • Let
V = Vn and Ad= {tEVn : jt-t0 j~d} ,
il E K ( QV A ( • I ;;: 1 ) , QV A ( • I o 2 ) ) be n' d n' d
such
that
360
RpA (~) ~ 2 G2 exp{-g2d} d
(3 .12)
Consider the measure ~ on nV xnV given by the follown n
ing two properties
1) its projection ori AdxAd is equal to ~
2) its conditional distribution on AdxAd with respect to the o -algebra BA (see the proof of Lemma 2.1) is
1 2 d for any 1: ,< E r.A given by the product
d
QA [· [< 1ua1_ ] d (Ad)c
r , 2 -2 ) x Qj\ t • I' Uo _ I
d (A )CJ d
Clearly ~EK(QV (·lcr 1 ),QV (•lcr 2 )) and n n
It is clear that for some choice of d=d(n) the right-hand
side is o(n) and so (2.10) is true for d large enough. We shall discuss below (see Remark 4.2 and [13] for
more details) some examples where (3.10) holds while (3.11) does not. In some of these cases condition Cv holds true for some v and in others they are violated for all V • Note that condition CV for IV I= 1 does imply ( 3 • 11 ) (see [ 1] ) •
4. Gibbs fields with finite single-spin space
In this and the following sections the space S is
assumed to be finite endowed with the metric (2.3).
Let Ar be the set of all translation-invariant inter
sections U with radius r, U {UA(o) == UA(oA), AcZv,
IAI<oo} , UA (o) = 0 if diam A>r For U EAr let
[jUjf max ; UA (o)[ oEr2,AcZ"
(4. 1)
To each interaction U EAr certain r -specification Q0 =
=={Q~, vczv} corresponds, which is given by
where
361
Z(Vjcr) = 2 exp{-Rv(ojcr)t oEs:Jv
is called the partition function and
Rv(ojcr) = 2 UA(ovnA U a A:Anvt~ venA
( 4. 2)
(4.3)
14.4)
Any Q0 -consistent field is called a Gibbs field with
the inter U (or a U -Gibbs field) . The results of
Sections 2 1 3 are certainly valid for them. By Av cAr we
denote the set of interactions for which condition ~ of
Theorem 2.1 holds. Let Er u A • It is easy to see VC'Ev V
that E is r an open subset of Ar Let UEA and u 13 = 8UEAr The factor r
the inverse temperature. Let V C'Ev IVI<oo
c~ be the set of "conditional ground state
B is called
and GV(cr}C
configurations" 1
i.e. of those configurations oEs:Jv for which
Rv(oicr) =min HV(o' lcr) . o'Es:Jv
( 4. 5)
The ground specification 6° is defined by the formula:
-u - liGv(crll-1 if QV(oio) =
0 otherwise
(4.6)
This is a self-consistent r -specification and
0 s - -u -lim QV (olo) = QV(olo) S+oo
for all v I oEs:Jv (see [ 14] for more details) .
Let v czv be a finite volume and t 0Eav Let
M0 (V) c V be the set of all points tEV such that if to 1 -1
o2 EGV(cr 2 ) and -1 -2
unless then oEGV(o) 1 0 = 0 s=t0 s s
necessarily 1 2 In other words the conditional ground 0 = ot . t
configuration does not depend on the condition at t 0 out-
side M~ (V) 0
362
Theorem 4.1. Suppose that for some finite V c :£v
I I V\M~ (V) I < I vI ( 4. 7) t 0Eav o
Us Then condition CV holds for the specification Q 1 pro-
vided S is large enough. In particular the (BU) -Gibbs
field is unique and the decay condition (3.10) holds for
it.
for
u 8 -1 Proof. Let JJ S E K (QV ( · I a ) 1
s=t0 Denoting M~ (V) by 0
us -2 Qv (·Ia ))
M one has
1 2 1 2 f pv(av 1 avlJJ 8 (dav 1 dav) ~ IV\MI +
11vx "v
where -1 -2 a =a s s
(4. 8)
where (JJS)M is the projection of JJB on QMxQM . But
(JJ ) is concentrated on the pairs of identical configura-S H
tions as s~oo • Hence the right-hand side integral in (4.8)
goes to zero as s~oo and (4.7) implies condition~.
Remarks
4.1. Consider the following condition: for some
f . 't VC"'\! d - 1 - 2 E - 1 - 2 <f -+t any ~n~ e a. an any a 1 a ll 1 as = as ~ sr
aiEGV(-;;i) 1 i=1 1 2
1 a s
a; if dist(s 1 t) ~ d
d>O I
and
( 4. 9)
Clearly (4.9) implies (4.7). Under condition (4.9) the uni
city is proven in [ 12] 1 though the condition itself was not
clearly formulated there. This condition implies also (see
again [ 12]) the decay property (3.11).
4.2. In [ 12] it was introduced the condition: for some
d>O 1 all finite VC:£v and -;; 1 1 -;;
2 E ll 1 a 1 EGV(-;; 1 ) 1 a 2 E
EGV (-;;2)
2 a s
if dist(s 1 Vc) ~ d . ( 4. 1 0)
363
This condition is also implied by (4.9). The difference be-·
tween (4.9) and (4.10) is the same as between (3.11) and
(3.10). The change of the boundary condition in one point
t 0 results in a change of the conditional ground configura
tions only in a vicinity of the point t 0 in the former
case while it may lead to a change along all the boundary
av in the latter case.
The first example of an interaction satisfying (4.10),
but not (4.9) was constructed by Navratil (Prague), in the
context of criticizing the paper [12], where these condi
tions were confused. The corresponding class of models is
studied in details in [13], where they are called "Czech
models". In spite of the fact that (3.11) does not hold for
them for large S (which erroneously was stated in [12])
the unicity statement for them holds true which is shown in
[13] together with the weaker decay property (3.10). Condition (4.10) follows from the statement of unicity of the
periodic ground configuration with Peierls (or GPS) condi
tion (see [12]).
These Czech models provide examples mentioned in Remark 3.5. They have many other unusual features. For example, the
uniqueness in the whole space Zv for them can be accompa
nied by the non-uniqueness in the half-space Z~ = {tEZv:
t 1 ~0} • Another example of unusual behaviour is that the
zeroes of the partition functions Z(Vicr) tend to the real axis as V+Zv violating thus the natural analyticity as
sumptions (see [8]). Although, the limit free energies of
the Czech models are complex analytic functions! As a result
one is driven to the conclusion that the phase transitions
of a special type take place in Czech models. In this sense
they do not provide a counterexample to the hypothesis
stated in the Introduction of this report.
5. Effectiveness
In this section we discuss the effectiveness property
of the conditions of Theorem 2.1. We begin with some elemen
tary facts of algorithm theory. It is generally agreed that
the effectively calculable functions are the recursive ones
364
(we are speaking about functions from (Z+)k to z+ where
Z+ {1,2, .•• } and kEZ+ ). A set ReX+ is called enurrer
able, if it is the image of some recursive function. This
means the existence of an algorithm which allows one to con
struct certain finite subsets of the set R , Rn in the
n 'th step, in such a way that RnfR when n+oo . At the
same time the complement Z+'R is not necessarily enumer
able, so, in general, there is no algorithm which allows
one to check that k ~ R
Now let us consider the set
tials: U E A' iff all the values r
A~cAr of rational poten
UA(aA) are rational.
The set A~ is countable so it is possible to identify it
with z+ • Let E~ = Ern A~ .
Theorem 5.1. The set E' is enumerable. r
we give here only a sketch of the proof. Note that the
exponent can be approximated by its Taylor series which has
rational coefficients. Hence, it is easy to see that one
can define the conditional probability distributions
Q~' n ( • I . ) , n E Z\ such that
jQ~'n(Bio)-Q~(Bio) [ .S: (s(U,n))- 1 , nE:Z+' UEJ\~ (5.1)
where
1) for V c z; v , BEBV , aESi () fixed the function
U n -Qv' (8Ja) takes on only rational values, and is recursive
(the set of rationals was identified with Z, ), T
2) the function s(U,n) +oo in a monotone way as n+oo
s(U,n) EZ+ and is a recursive function of (U,n) EJ\~xz+.
(Warning! The functions Q0 'n(·l·) need not to be consisV
tent!)
Consider now the family of pairs (V,~_ 1 _ 2 ) where a , a
vczv is finite, -1 -2 a , a E Q av is a pair of boundary condi-
tions which differ only at a point t E av and, finally,
~- 1 _ 2 is a probability measure on a , a
with rational
values depending on -1 -2 a , a . Now let us define the function
<P = <P (U,n, (V,~_ 1 _ 2 )) to be equal to a , a
u if
1) for all
2)
-1 -2 0 , 0
365
the measure · U n -1
\.1_1 -2 E:K(Qv' (•lo) 0 ,o
- lVI lVI RP (\.1_1 -2) ~ l<lVI- s(U,n) 1 <5 • 2 ) v 0 0
otherwise .p is equal to some fixed u0 ( E~ • The function
.p is defined on a countable set and after a natural iden-
tifying of its domain with Z, it is recursive. It is easy T
to see also that its image is exactly E~ •
The same idea can be stated otherwise. For each pair
(U,n) , U€A~ , n E Z:+ consider a ball O(U,n) in Ar of
of radius n- 1 centered at U The set of all these pairs
can be identified with Z+ • We call the subset A cAr ef
fectively enumerable if for some enumerable set W of pairs
(U,n)
U O(U,n) (5.3) (U,n)EW
Theorem 5.1. The set Er (of Sect. 4) is effectively
enumerable.
The proof is almost the same as above.
Of course, the algorithm described in Theorem 5.1 is
not suitable for application: it was constructed exclusively
to simplify the proof of the theorem. In real situations the
key part of the problem is to look for a measure \l minimiz
ing - or almost minimizing - the integral (2.2) for p=pv •
This is an usual linear programming problem. But usual algo
rithms are of no help here because the size of the problem
increases exponentially fast with lVI , and also they do
not use specific features of our problem. These are the fol
lowing: it is intuitively clear that the main contribution
to the optimal value of (2.2) comes from those points sEV
which are in a vicinity of the point tEClV where the bound--1 -2
ary conditions o , o differ. Hence to construct an al-
most optimal measure one has to begin somehow with the con
struction of a \l which is good in this vicinity.
We finish our report by presenting such an algorithm
366
which results in the joint distribution sought, i.e. a
measure ~ on nvxnv which is close enough to optimal.
We believe algorithms of such type to be of importance in
attempts to prove the unicity by using computers.
The algorithm. Let a volume VcZv as well as a pair -1 -2 -1 -2 a , a of boundary conditions be given with as = as for
u -1 u -2 s;etov . Ne construct a measure ~ E K {QV ( · I a ) , QV ( · I a ) )
We shall use the following notations:
F. J
H. J
{sEV:r(j-1) < dist(s,t) ~ rj}
U F k~j k
( 5. 4)
The
are the
input data for the i -th step of our algorithm
following:
1 ) a measure Pi-1 on 1 2
qi-1 I qi-1 2) measures on nH , j -1
with the following property: for any 2
t; E supp qi-· 1 and any j=1, .•• , i-1
r:: 1 E, supp- qi_ 1 ,
(5.5)
(Warning: In general these measures are not probabilistic!)
The parameter i runs from to [diam V/r]+1 . As
for the initial conditions
Po - 0
1 2 qo qo
(the set Ha=~ and
We describe now
n~ consists of one element) .
the i -th step.
(E,,t;) I E,E Let us consider the set of all pairs 1 E supp qi_ 1 , and let us number these pairs by an index
k=1, .•. ,k(i) such that if k 1<k 2 then
k k k k p ( E, 1 I 1; 1 ) ~ PH ( E, 2 I 1; 2) Hi-1 i-1
The i -th step is again
an inductive procedure on k . The input data for the k -th
substep are the following:
367
1 ) a measure p i-1 ,k-1 on DVxDV ' 2) 1 2
nH. measures qi-1,k-1 ' qi-1 ,k-1 on l.
For k=1 the measure Pi-1,o=Pi-1 and
i i u -i q i -1 , 0 ( t; U 1 ) = q . -1 ( t; ) QV'-H F ( 1 I 0 U t; l
l. i-1' i (5. 6)
i= 1 , 2 , t; E: DH , 1 E: DF. i-1 l.
For k = k (i) the measure
i i qi-1,k(i) = qi ' £.== 1 • 2 • ( 5. 7)
To describe the transformation of the measure let us
introduce for t;, n E: DH , a E: DV,H. the quantity i-1 l.-1
:l\-1,k-1(i;,I:,G) =
Now we define for a 1 ,o 2 E Dv , Cl: E DH· . l.-1
p 1 2 i-1(o ,a)
1 qi-1,k(t;)=
Pi-1,k-1 (o1 ' 02 )~~i-1,k-1 (~,~,a) if
1 qi-1 ,k-1 (!;) otherwise
( 5. 9)
( 5. 10)
r I
2 ( \- J qi-1,k r,, - I
l
368
( 5. 11 )
2 qi- 1 ,k_ 1 (r,) otherwise
From the definition (5.9)-(5.11) it follmvs that the
inductive condition (5.5) is reproduced.
The algorithm presented allows one to construct a seU -1 U -2
quence Jli E K(Qv(·lo ) ,Qv(·lo ) ) namely
12 121 12 2 ]1 1. (o ,o) = P. (o ,o )+q. 0 (o )q. 0 (o)
1. 1., 1.,
(see (5.6)). The quantity R (Jl.) is estimated as follows: Pv 1.
R (P.)+Ivd1- ' P(o 1 ,o 2 )l . p 1. , L V o 1 o2 E rl )
v
R ( l1. l Pv 1.
Hence if for some i pi<kt where kt' tEav satisfy (2.8)
one may terminate at the step
do not claim the convergence of
general it does not hold!
i Note, however, that we
to an optimal J1 - in
Finally the following a priori bound holds:
pi~ ). (VarF.l jHjj+(VarF. , 1 ) lVI J~l_ J l.T
( 5. 12)
where for AcV
u -1 u -2 Var A = var (Qv, A ( • I o ) ,Qv, A ( · I o ) ) ( 5. 13)
Indeed if Pk ( o 1 , o 2 ) f- 0 then by ( 5. 9)
On the other hand for l=1,2
1 2 Pv ( o , o l ~ I Hk I •
At last
Pi(o 1 ,o 2 )=1- L qf(o)
oErlH. 1.
VarF i+1
1-VarV'H. 1.
because the r -Markov property of the measures Q~(·lol)
369
implies that the conditional distributions, induced by
these measures, for the restriction a of av to V'Hi+1
under the condition that aH. is fixed, coincide ~
for l=1 and l=2 . If condition (3 .11) is true then pi< lVI I I avl for
large enough i and V and so the algorithm proves the
property CV
References
[1] Dobrushin R. L. Prescribing a system of random variables by the help of conditional distributions. Theory Prob.•. and its Appl. 15, N 3, 469-497 (1970).
[2] Gross L. Decay of correlations in classical lattice models at high temperature. Comm. Math. Phys. 68, N 1, 9-27 (1979).
[3] Simon B. A remark on Dobrushin's uniqueness theorem. Comm. Math. Phys. 68, N 2, 183-185 (1979).
[4] Levin S. L. Application of Dobrushin's uniqueness theorem toN-vector model. Comm. Math. Phys. 78, N 1, 65-74 (1980).
[5] Gross L. Absence of second-order phase transitions in the Dobrushin uniqueness region. J. Stat. Phys. 25, N 1, 57-72 (1981).
[6] Kunsch H. Decay of correlations under Dobrushin's uniqueness condition and its applications. Comm. Math. Phys. 84, N 2, 207-222 (1982).
[7] Klein P. Dobrushin's uniqueness theorem and the decay of correlations in continuum statistical mechanics. Comm. Math. Phys. 86, N 2, 227-246 (1982).
[8] Shlosman S. B., Dobrushin R. L. Complete analytical Gibbsian fields. This volume.
[9] Lieb, E. H. A refinement of Simon's correlation inequality. Comm. Math. Phys. 77, N 1, 127-136 (1980).
[10] Rachev S. T. Monge-Kantorovich problem of a mass shift and its application in the stochastic. Theory Prob. and its Appl. (in print).
[111 Dobrushin R. L. Gaussian random fields - Gibbsian point of view. In "Multicomponent random systems", Marcell Dekker, N. Y. -Basel, 119-151 (1980).
[12] Dobrushin R. L., Pecherski E. A. Uniqueness conditions for finitely dependent random fields. In "Random Fields", vol. 1, North-Holland, Amsterdam-Oxford-N.Y., 223-262 (1981).
370
[13] Dobrushin R. L., Shlosman S. B. The problem of translational invariance in statistical physics. Soviet Scientific Reviews, Ser. c. Vol. 5 (1984).
[14] Shlosman s. B. Meditation on Czech models: uniqueness, half-space non-uniqueness and analyticity properties (in preparation).
371
COMPLETELY ANALYTICAL GIBBS FIELDS
R. L. Dobrushin and s. B. Shlosman
1. Introduction
In this report we describe the class of interactions
A for which the corresponding Gibbs field is unique and
possess every possible virtue one can imagine. vle mean by
this the well-known regularity properties of Gibbs fields
in the high-temperature region. This class of interactions
is defined axiomatically by ten (!) very natural properties.
But the reader should not be confused by their great amount
because all of them turn out to be equivalent! This fact
alone shows that the class considered is natural. We call
the potentials of the class completely analitical. The
boundary of A corresponds to phase transition surface.
Furthermore, the region A can be defined constructively
by a certain algorithm. Here we use the word "constructive"
in the same sense, as we used it in report [1], which is
conceptually close to this one. For the sake of simplicity,
we restrict ourselves to random fields on ~v , with finite
single-spin space S and finite-range, translation-inva
riant interactions with finite values. The ideas of the
present report and those of [1] can be applied in more com
plex cases; in particular, we study perturbations of
Gaussian fields in [2]. Throughout this report we shall
use the notations of [1], §1.
The Ten Conditions, which define our class of inter
actions A and which are equivalent, can be divided into
three types. Conditions of the first type are those of ana
lyticity. Namely, in one of them we ask for the following
estimate to hold:
372
for UEA , the partition function in any finite volume
V , with any boundary condition a satisfies the following
bound:
( 1. 1)
uniformly in V, a and u I where the latter is any com
plex (not necessarily translation-invariant) interaction
small enough.
Condition (1.1) is inspired by the well-known Lee-Yang
method to locate the points of phase transition, which are
interpreted as the limit points in the complex plane of the
zeros of the partition functions, when the underlying vol
ume V goes to infinity. From (1.1) the analyticity of the
free energy in U follows. But condition (1.1) is even
stronger than the usual requirement of analyticity of the
free energy, because it excludes the cases when the latter
is accompanied by accumulation of zeroes of the partition
functions near the real axis (see an example in [3]).
Conditions of the second type are stated in terms of
the variation distance, Var(·,·) , between the projections
Q fl(·la) of the conditional Gibbs distributions Q (·Ia) v, v in the volume V into the subvolume !lev for different
boundary conditions
for any finite
a ; one of them is the following: v -1 -2
\t:Z: a , a E r2 , which differ only vc
in one point t E Vc
-1 -2 Var(Q [l(·la ), Q [l(·la )) S: K exp{-adist(t,A)}, v, v, ( 1 • 2)
where K=K(U)<oo,a=a(U)>O
Finally, conditions of the third type are stated in
terms of decay properties of semi-invariants which are
similar to the properties studied in the papers [4]-[8].
For completely analytical potentials the central limit
theorem holds with various improvements, as vlell as many
other properties of "very good" fields.
Two basic ideas are used in the proof of the equiva
lence of the Ten Conditions. To show that (1.1) follows
373
from (1.2), we use induction on lVI The same idea was
used earlier by one of us ([9, 10]) in the one-dimensional
case, but the general case can be treated as well. Since
(1.2) for U implies (1.1) for small perturbations of U,
we obtain that (1.1) is valid in high-temperature region
(taking for U the non-interacting potential). The same is
valid for the low-temperature region when the ground-state
is unique, as well as for the small activity region at any
temperature.
To derive (1.2) from (1.1) we use a modified version
of the conform transformation method, which was developed
earlier in [4]-[8]. The same modification is used for estima
tion of the semiinvariants and other aims.
These two methods enable us to obtain all natural prop
erties of the Gibbs fields under consideration without in
voking the cumbersome machinery of correlation equations or
cluster expansion (cf. [11], [12]). The same techniques can
be applied in more general cases (see [2] ), where the usual
approach requires cumbrous calculations.
2. The main result
v Let A={Aic:Z, i=1, ..• ,k} beafinitecollectionof
finite subsets of Zv and !J. = !J. (A) ={A c: Z : A= A. +t for l.
i=i(A)=1, •.• ,k, t=t(A)EZv}. We denote by A8 ,
the real and the complex Banach spaces of translation-
invariant interactions with support in !J. • An interaction
UE A8 (A~) is thus a family U={UA(a):.:UA(crA)' Ac:Zv,
IAI <oo, crEst} , such that
crs+t ' ( 2. 1 )
u11• - 0 unless AE !J.(A) (2.2)
The norm of U is given by
:; u~ sup [uA (a) l A,a
(2. 3)
The radius of interaction U is the number
374
r r(U) =max diam (A) A:UAtO
If 1:!. = {A c Z: v: diam A~r} , then the corresponding spaces
A8 , A~ will be also denoted by Ar' A~ Throughout the
most part of this section the families A and 1:!. will be
fixed and so the corresponding index will be omitted, and
we shall speak about the spaces A , AC , with r = r (A)
being the maximum radius of interactions UEA .
For any set c1 cA we define its main component M (cl)
to be the maximal open connected subset of a, which con
tains the zero interaction u0 = {u 0 = 0} A-We shall denote by cla the set of interactions satis-
fying condition a - which is one of the Ten Conditions to
be formulated below. ( a runs through Ia, Ib, IIa, IIb,
IIc, IIIa, IIIb, IIIc, IIId, IIIe.)
We present now the main results of our report
Theorem 1.1. The main components M(cla) coincide.
(a= Ia to IIIe.)
This common main component is called the set of com
pletely analytical potentials.
Theorem 2. 2. The classes of interactions A , B , C , V , E
to be defined below, consist of completely analytical inter
actions.
We now start to formulate our Ten Conditions and five
classes of interactions.
Let Ac be the set of all complex interactions which
satisfy (2.2), but not necessarily (2.1), with the norm c ~ ~ v
( 2 • 3) . Of course, A c A • For any U A , V c Z: finite
and boundary condition cr€0 let
( 2. 4)
where
(2. 5)
Condition Ia. U Eclia if there exist C<oo and E>O ,
375
such that for any v, o the function Zv(Uiol is holo
morphic and nonvanishing in the region
(2.6)
and moreover,
(2.7)
Remark 1. The function ZV(Uio) depends only on those
UA(oA) , for which Anv~~ . So by being holomorphic we mean
the usual property of functions of several complex variables.
Remark 2. ln Zv(Uio) is the uniquely defined holo
morphic function, which coincides with the usual (real)
logarithm for U real. Its analytic continuation to OE(U)
is possible and unique because the latter set is contract
ible and z1 0 (Ul ~ o • E
Condition Ib. U Eaib iff there exist two constants
C<oo , £>0 , such that for any V , o the function
Zv(Uio) is holomorphic and nonvanishing in the region
OE(U) and moreover
lln[Zv(Uicrl/Zv(UiolJ I ~ clvn supp(U-Ul 1 ,
where for any <I> E if
supp <I>
(2.8)
(2.9)
We recall now the definition of semiinvariants (see,
e.g. [11], [12]). Let ~; 1 , ... ,/;mES be random variables
with the joint probability distribution q = q (x1 , •.. ,xm) ,
xi E S • The semiinvariant of order (k 1, ... ,km) , where
ki>O is the number k k k k1+ ... -+k '1 '2 'm a m 1
<1;1 ,1;2 , ... ,~ > = k k ln<j>(z1, .•. ,zm) 1 (2.10) q a 1z1 .•. a ~m 'Z =z2= .•• =zm=O
where
(2.11)
376
is the generating function, and
Now let
z. E C • l
be the conditional Gibbs distribution in v , and
l/J 1 (oA1 ), • .. , l/Jm (a Am) be real functions, where Ai c v
(not necessarily distinct) subsets, then
where
UA + L z.lj! .• i•A =A 1 1
. i
are
(2.13}
( 2. 14)
Condition IIa. UEaiia iff for some constants C>O ,
E>O and for all V, o, l)J 1 , .•. ,lj!m, k 1 , ... ,km with !l/Jil~1
,k1 ,k -and Ai E !:J. , the function <l)J 1 , •.. ,lj!m m:u,v,o> , defined
by (2.13) for real 6 , can be extended to a holomorphic
function on OE(U) , with the following bound to hold:
6 E 0 E (U} •
Condition IIb. U Eaiib iff there exists a constant
C>O , such that for all V, o, l)J 1 , ... ,lj! , k 1 , ... ,k with m m and A.E!:J.
l
( 2. 16)
377
where G(A1 , ••• ,Am) is the set of all trees r with m
vertices identified with the sets
set of all bounds y = (A. , A. ) ~y Jy
A1 , .•• ,Am , E (r) is the
of r , lyl =dist(A. ,A. l ~y Jy
and finally cp (d) > 0 is a decreasing function on Z+ with
L cJ> (It I ) It I v-1 < co
t E Zv
(2.17)
Condition IIc. U Eel IIc iff for some constants C>O ,
a>O and for all V, a, w1 , ••• ,wm, k 1 , ••. ,km with
!wii~1 and AiEA
,k1 ,k k1+ ••• -+k I m1 - 1 m { < w1 , ••• rWm ;U ,V,o > 1 ~ (k1 ! ... km!)C exp -Q.d(A1,_.,Am)} , ( 2. 18)
where d (A1 , ••• ,A ) = min I B I m B: BU(A1u ••• uAm) is connected
and connectedness is meant in the sense of the graph zv
with edges joining nearest neighbours.
For AcV we define
u -QV,A (Bio) with BcS1A • ( 2. 19)
Condition IIIa. U Eciiiia iff for some constants o<1 and p>O and for all finite vczv , t E av , o1 ,o2 En with
o1 = o2 for s#t s s
U -1 U -2 I 1-1 var <OV,B(t, p,V) (·lo ) , OV,B(t,p,V) (·lo ) ) < o1B(t,p,V) 1 , (2.20)
where
B(t,p,V) = {sEV: p<Js-tl~ p+r}, r=r(U) (2.21)
Condition IIIb. U E <l IIIb iff for some decreasing function cj>(d) with
lim cj>(d)dv-1 = 0 , d->-oo
for the same -1 -2 v, t, 0 , 0 as in Ilia, and for any
u -1 u -2 Var(Qv,A<·Io ),QV,A(·Io )) ~ L cJ>(Is-tl)
sEA
(2.22)
Acv
(2.23)
378
and
Condition IIIc. U Eaiiic iff for some constants K<oo -1 -2 y>O and for the same V, A, t, a , a as in IIIb
Condition IIId. U Eaiiid iff for some K<oo , y>O ,
for the same
a A E nA
-1 -2 V, A, t, a , a as in IIIb and for all
~ K exp{-y dist(t,A)} . (2.25)
Condition IIIe (CONSTRUCTIVE condition). U E G IIIe iff
U satisfies the same conditions as in IIIc but only for
volumes vcV n (see [1] for the definition of
n = n(K, y, v, r)
vn ) , where
(2.26)
is some constant which can be EXPLICITELY estimated.
We define now the five classes of interactions, which
were mentioned in Theorem 2.2.
Class A (High-temperature class). u E A iff for some
u0 E A0 , liu-u0 1! < E (~) , for u E A= A~ and E (~) small
enough.
Class B (Large magnetic field - or chemical potential -class). U E B iff U is of the form q,J..I , where for some 4>EA and s 0 E S
(2.27)
provided J..1 ~ ~(114>1, r(A), v), where the latter is a func-
tion large enough.
To define class C we recall the B (d) -property, in
troduced in [ 13] • For UEAr , V c :1: v , crErl denote by u -Gv(a) the set of all aEnv such that
(2.28)
379
The potential U is called B(d) -potential, with d>O \) -1 -2 -1 -2
if for any vcz , tE:lV , a ,a E S1 with os =as for s#t
and any oi E G~(aiJ , i=1 ,2
(2.29)
for all uEV such dist(u,t) ~ d . Let
US = SU , S>O . (2. 30)
Class C (Low-temperature B(d) -case of a unique ground
state). Suppose that a potential UEA as well as all the
potentials u~ defined by (2.27) have B(dl -property for
some d>O, s 0Es and all ~, O<~<~(IUII,r(A),v) .
UEC iff U =US , defined by (2.30), provided
S ~ S(IUI ,r(A) ,v) where the latter is a function large
enough.
Class V (Ferromagnet in large external field). U E V
iff the space S = {-1 ,+1} , and
r osot for A={s,t} s-t
UA (oA) = -h ot for A= {t}
0 for other A
( 2. 31)
where u s-t ~ 0 and
I hi > I us (2.32)
sEzv,o
Class E (One-dimensional and almost one-dimensional in
teractions). U E E if IIU-¢11 s E ((II <PI) ,r,v) , for some
longitudinal potential <PEA , and for some E
The interaction ¢ is longitudinal if ¢A r 0
small enough.
implies that
for any two points (t 1 , ... ,t),(s1 , .•• ,s) EA, si=ti for
i = 2 , ••. , v (see [ 1 3] ) .
Comments
2.1. We believe that all the families aa are connec
ted, thus implying M(aa)=aa. However we have no proof of
this hypothesis. We need the condition of being in the main
component only when proving the implications IIa ~ IIc,
380
Ia~IIId.
2.2. If t:,.ct:,.' then the inclusion Ar:,.cAr:,., induces
the inclusion of the corresponding sets of completely ana
lytical potentials: M(aa<Ar:,.ll cM(aa (Ar:,.,ll • This follows
from the fact that the definition of the class aiiib does
not use the value r(A) . Our statement is, however, not so
innocuous: it means in particular that if the interaction
UEAr:,. is completely analytic in the sense of Ia, i.e. all
partition functions ZV(U+Uio) are nonzero and (2.7) holds - -£ for the complex perturbations UEAr:,. , then the same is true
-c -for perturbations from At:,. , for all V and a with any
6.' ! This is by no means a priori evident, and in this
place the fact that the interaction U lies in the same
component with the zero interaction is again important. Sim-·
ilarly conditions IIa, IIb, IIc can be extended to semiin
variants of functions w. (aA ) with arbitrary finite Ai J. i
but with constants C, £ depending on Ai
2.3. Recall that by Gibbs fields with interaction U
in a volume V c Z v which is not necessarily finite) , with
boundary condition aED we mean any probability measure P
on DV , such that its conditional distribution in any fi-
nite /\cv subject to the condition a ED equals /\cnv /\cnv
a.e. the Gibbs specification Q/\(·ia Uo ) Applying /\cnv vc
now condition IIIc with vnVn instead of V , and taking
the limit n+oo , we find immediately that the Gibbs state
for completely analytical potentials is unique in any volume
v . 2.4. The novel features of the analyticity condition Ia
are two-fold: we consider all complex perturbations of the
interaction U - and not only the translation-invariant
ones, and we consider all boundary conditions o • These
features are essential. Of course, using Van Hove's theorem
and the compactness criterion for the analytic functions,
one can prove that condition Ia implies the existence of
the limit (in the sense of Van Hove)
-flU) =lim IVI- 1 ln Zvlfila) V+oo
(2.33)
381
for all u E 0' (U) = 0 (U) n Ac , which is holomorphic on E E
O~(U) . The main point here is that the converse implica-
tion does not hold - see [1], §3, and [3] for details. The
potential constructed in [3] is not completely analytic,
and the unicity of the Gibbs field in Zv is accompanied
with a non-unicity in the half-space Z~={tE:Z:v: t( 1 )~0} for a special boundary condition. So it is natural to think
that the corresponding potential exhibits some sort of
phase transition.
2.5. Condition Ib improves essentially condition Ia.
For example, it implies the following bound. Let Ac(U) =
= {'u E Ac: Jsupp(U-6) I< oo} • For any vcz:v , oE(t , UEA and
u E Ac(U) let
(2.34)
where is a Gibbs field in -boundary condition o • Then
OE(u)nAc(U) and
V with interaction u and
ln Zv(Uio) is holomorphic in
Jln zv(Uio) l s c1 Jvn supp(U-6) I , (2.35)
where c, does not depend on u, v and 0 Indeed, for
finite v (2.35) is the same as ( 2. 8) • In general case one
has to take v n vn instead of v in (2.8) and then tend
with n to (X) The bound (2.35) can be used to derive the
properties of completely analytical interactions in infinite
volumes.
2.6. The bounds on semiinvariants of the type IIa-IIc
have occured earlier for specific classes of potentials in
many papers (see [6]-[10]). For example, a statement of [8]
is close to our II a.,. IIc.
2.7. In the same way as (2.33) for the free energy,
follows from IIa the existence of the limits
,k 1 ,k 2 ,km _ ,k 1 ,km 1
_ _
<~ .~ 2 , .•• ,~m Ju> = lim<~ 1 , ... ,~m ,U,V,o) V->oo
(2.36)
for 6 E 0 E (U) . They are semi invariants of the Gibbs field
in Zv for real 6, and are holornorphic on OE(U) . From
382
our equivalence statement follows that the bound (2.18)
holds for the limiting semiinvariants (2.36). We show below
(Prop. 4.6) that the convergence in (2.36) is exponential.
But the example already discussed in Comment 2.4 shows that
the bounds on infinite-volume semiinvariants do not imply
the complete analyticity.
2.8. Equivalences IIb..,IIc and IIIb..,IIIc illustrate
the general principle that in the finite range case between
the exponential- and power-law no intermediate correlation
decay can occur (see [13]-[15]). Equivalence IIIb..,IIIc
with v 2+E instead of v in (2.22), was obtained in [13].
Nevertheless, our result does not cover that of [13], be
cause we are restricted to the main component. Condition
IIIc was formulated already in [13], where it was derived
from the unicity condition of [13] (see also [1], §3).
2.9. The main peculiarity of condition IIIe is its con
structiveness. Using it, one can prove the complete analyt
icity of a given potential after a finite amount of calcula
tions. Putting it more rigorously, the set of completely
analytic interactions is effectively enumerable (see the
discussion in [1], §4). However, the proof of equivalence
of IIIe and all the other conditions is very involved, and
we do not present it in this report. Even our estimate on
the number n(k,y,v,r) is so awkward at present that we do
not demonstrate it here. We hope to improve it in a forth
coming publication.
Let us compare condition IIIe with the unicity crite
rion of [1]. We showed in [1] that IIIb implies this unicity
criterion. On the other hand, the example of [3] shows that
the latter can hold together with a non-uniqueness in a
half-space. Hence it is valid not only for completely ana
lytical potentials - though these extra interactions are in
a sense "exotic". It seems that now only criterion of [1]
can be recommended for computer-assisted proofs.
2.10. Consider the random variable
L ¢ ({ot+s' tEB}) , B:B+scl\
(2.37)
where ¢ is some function on ~B , 1\cv , and the distribu-
383
ov is the conditional Gibbs distribution pY in 0 tion of
vcz:v with boundary condition a . Then the characteristic
function of SA is
zSA <e >= zv<u+z¢Jol
where
r¢(oA-s) I if A=B+scA
l 0 otherwise.
(2.38)
From (2.35) follows that its logarithm is holomorphic and
bounded for lzi~EO , where Eo is independent of A
Hence the general result of Statulevi~us [16] together with
positive lower bounds on the variance of SA in the limit
IAI~oo (see [17]) imply the central limit theorem for SA
with various improvements. The local limit theorem in our
case follows from the global one (see [18]).
2. 11 . The interactions of class A correspond to high
temperature, and those of class B to low density. These
classes are highly explored by now, and the properties dis
cussed above can be proven for them using correlation equa
tions and cluster expansion. We think, however, that our ap
proach is more straightforward and easier.
2. 12. Class C corresponds to the case of unique ground
state. Condition B(d) is an additional essential restric
tion. Without it one can construct examples of not com
pletely analytic interactions with unique ground state
(see [1] and [3]). We require condition B(d) not only for
U itself, but also for U This is done only to ensure \.1
the inclusion of the low-temperature interaction SU in
the main component. Conditions of the type III follow for
SU even if only U itself is B(d) , but we do not know
whether SU is contained always in the main component.
2.13. For the case of the ferromagnetic class V, it
is well known that the unicity holds for h#O . But the
known techniques for locating the zeroes of the partition
function (beginning with Lee-Yang method), enables one to
384
prove condition Ia only under the additional restriction
(2. 3 2) .
It is worth mentioning that under this restriction the
set G~(cr) (see (2.28)) always consisits of exactly one
configuration: ov = sign h •
The scheme of the proof of the theorems. We shall prove
the following implications between the Ten Conditions:
'"L ""i j't'" I rt~L_ rfb IIIb (2.39) t I
I rJrc +
I lie~ IIId
I l t IIIe
The arrows X+ Y mean that condition Y is satisfied for
the main component of the set of interactions with property
X . The theorem follows from the fact that in the oriented
graph (2.39) there is a circuit passing at least once
through each vertex. The implications Ib + Ia, Ib + IIa,
IIc +lib+ IIa, IIId + IIIc + IIIb +Ilia are almost evident. +
IIIe
Therefore we make only minor remarks concerning them. To
prove Ib + Ia it is useful to note that for U real
lln zv (U I a) I ~ c I vI with c independent of v, a • Proving
Ib+IIa one notes first that isupp(U(z 1 , ••• ,zm)-U) I = m
= I U A. i ~ c 1 m, where i=1 J.
c 1 = c 1 (r, v l , and then applies the
Cauchy formula for the derivatives of
To prove IIId + IIIc one needs to note
ability distributions Q1 (x), Q2 (x)
holomorphic functions.
that for any two prob
en a finite set X
Var(Q 1 ,Q2 l = ~ L IQ 1 (xl-Q 2 (x) I ~ xEX
I Q1 (x) I L Q (x) 1-Q(XT ~ xEX 2
1 I Q1 (x) I ~ 2 max 1 -~) •
xEX I ld2 \XI I
The implication IIIa+Ib is the content of Proposition 3.1,
which follows from Theorem 3. 1. The proof of IIIe + IIIc fol
lows from Theorem 3.2, the proof of which is, however, not
385
easy, and will be published separately. Propositions 4.5
and 4.4 prove the implications IIa +lie and Ia + IIId; they
follow from Theorem 4.2. In Section 4 we also explain the
implication lib+ Ia (Prop. 4. 6) •
One can prove Theorem 2.2 in many ways, because it is
enough to check any one of the Ten Conditions.
The easiest of the conditions to check is condition
IIIe. Indeed, one can easily show that it is true for the
classes A, B , C (see the end of §3). Vle present also other
ways to prove the theorem, especially because we do not
present in our publication the proof of IIIe "* IIIc.
Note that for any interaction u0 E A0 condition Ilia
is evidently fulfilled, because the corresponding field is
a field of independent random variables. But Ilia+ Ia, and
Ia is valid in an open subset of A , thus the complete
analyticity for the class A is proven.
The specification for the classes A, B are close to
those for independent random variables (for class B those
are the variables, taking the constant value s 0 ) . Thus
again condition Ilia holds for A , B • It follows for class
C from the main result of [ 13] • For E it was also checked
in [ 13] •
On the other hand, one can check Ia straightforwardly.
For class V it can be done using the techniques of the
Asano-Ruelle-Slawny method (see [19], Prop. 4.2, 4.1). For
class E it follows from known formulas for partition func
tions through transfer-matrices ([20] ).
To see that all the five classes of ours are in the
main component, one has to note that each of them is con
nected and u 0 E A n E , B n A f ~ , C n B f ~ , V n B f ~ .
3. Estimates on partition functions
In this section we present a new method for estimating
partition functions. We shall consider an arbitrary self
consistent r -specification Q={Qv<·lo),VCZv,IVI<oo,crEQ}
For any function ¢ on Q let
<¢> -v ,o ( 3 • 1 )
386
For Acv we denote by QV,A (·lo) the projection of O~(·io)
onto A • I
(3. 2)
Finally, for any u E i\~ I finite vczv and oErl let (see
( 2. 4) l
zv(Q,Uio) =<exp{-I\<·Io)}> _ v ,a
(3. 3)
Theorem 3.1. Let p>O , o<1 and E>O be specified.
Suppose that
1) For any finite vcz:v , tEVc and 01 ,02 E rl with
01 = 02 for sit s s
-1 -2 Var{QV,B(t,p,V) (•IO ),QV,B(t,p,V) (·Ia )) .$
where B(t,p,V) is defined by (2.18);
2) U E 0 E , where
{uEA~: sup IUv<ov>l <E}. v
v c z: , ovErlv
(3.4)
(3. 5)
There exists a function E0 (o) , depending only on v, r ,
such that conditions 1), 2) with E ~E 0(o) imply that the
functions ln ZV(Q,Uio) are holomorphic in the variables
{UA(oA); oAErlA' AnVi~} . Moreover, there exists a function
C ( E, o, p) , depending only on v, r , C ( E, o, p) -+ 0 as E-+0 ,
such that (cf. (2.9))
(3. 6)
The proof is based on the following lemma.
Lemma 3.1. Let conditions 1), 2) of Theorem 3.1 hold,
vcz:v be finite, tEav and
V = {sEV: is-t: > p+r}
Then for all oErl and E S E0 (o,p)
the partition functions Zv{Q,Uio)
(3. 7)
, with some E0 (o,p) ,
and Zv{Q,Uio) do not
387
vanish. Furthermore:
o =a for s s 1) for any 0 I a E n I such that
s E av n av (here and in the sequel Z~=1 by definition)
zv<Q,Uiol/Zv(Q,ulcrl =1+9,C1 exp{-Kdist (t, vnsupp u)}, (3.8)
2) for any 01 ,0 2 E n , such that_ cr 1 =cr 2 for stt s s
- -1 - -2 -Zv(Q,Uio l/Zv(Q,Uio )=1+9{2 exp{-K dist (t,Vnsupp U)} (3.9)
~ere K = K ( o ) , C • = C • ( £ l -+- 0 J J
with £-+-0 are functions, de-
pending only on v, r, p, and - -1 -2 9 1 =9 1 (o,o), 92 =9 2 (o ,o l
are such that \9. \~1 , j=1,2 . J
The proof of the lemma goes by an induction on lVI
Supposing (3. 8) , (3. 9) to be valid for all W c :& v
with IWI~IVI-1 , and some K, C. , which will be specified J
later, we first
prove (3.9) for
prove (3.8) for the volume V • Next, we
V, assuming both (3.8), (3.9) are valid
for smaller volumes, while (3.8) is valid for V itself.
The initial step of induction is valid automatically, be
cause Z~=1 • The variables Q, u, t, p will be fixed
throughout the proof and omitted in all notations. We also
denote by
d = dist (t, V n supp U) d dist (V n av I v n supp U) (3 .1 0)
We begin with (3.8). Let
(3. 11)
By induction, ZfO . By the definition (3.1)
(3. 12)
where
z- [x - u a ) v avnv (avnv)c
(3 .13)
and
¢ (x -) = exp{- L V'V A:AnVfO
AnV=O
(3 .14)
388
But the boundary conditions 0 and X - U a differ av-...v ( avnv) c
only inside avnv~B I hence, applying [avnv[ times (3.9)
to the volume V which is valid by the induction hypothesis,
we have
(3. 15)
where 8 = e (xavnvl I lSI ~1 . Now we have to consider two
cases:
I) d > p+r
II) d ~ p+r
(3. 16)
( 3. 17)
Inthefirstcase ¢(xv-...vl=O, hence from (3.12), (3.15)
follows that
(3.18)
(Here and in the sequel quantities §j satisfy [ej [~1 .)
But d ~ d+(p+r) and IBI ~ [B(t,p,Xv) [ , hence for some
c 1 = c 1 (c 2 ,K,p,r,v)
While K 1 p 1 r t V
which goes to zero together with c2 remain fixed, from (3.18) follows that
(3. 19)
As for the second case, from (3.5) and (3.14) follows that
(3. 20)
where K=K(p,r,v). Hence from (3.12), (3.15) we deduce
that
(3. 21)
where c1 = c1 (C2,E,K,p,r,v) and goes to 0 if C2+0 I E+O I
while K, p, r, v are fixed. Defining
- - eK(p+r)) max (C 1 ,c 1 (3. 22)
we arrive to (3.8). The condition that c 1+0 together with
E is satisfied, provided the same holds for c 2 Now, let us prove (3.9). Evidently, one has
389
(Zv(a 1 )-Zv(a2 )z-1
zv(cr2)Z 1 (3. 23)
Denote by ¢j , ~=1,2 the functions defined by (3.14) with
aj instead of a • Using now (3.12), one can rewrite the
numerator of (3.23):
(Zv(a 1 )-Zv(a 2 ))Z- 1 =
- --1 - --1 = < Z(xavnv>z -1> -1-<z(xavnv>z -1> -2 +
v,a v,a (3. 24)
Using the inclusion oVnVcB and the estimate (3.15) (which
follows from the induction estimate (3.9)), we have
( 3. 25)
Now let us take functions K(6) , s 0 (6) to be so small
that for c 2 (s) =IE the following estimate would be true:
- IB I (1+C 2 e-Kd) -1 ~ IBIC 2 e-Kd 6- 1/ 2 ,
provided d .S: d+ (p+r) and E .S: s 0 (6) • Applying then condi
tion (3.4), one has
I - --1 - --1 I -Kd 1/2 <z(xoVW)Z -1) _1 -< Z(x0Vw)Z -1) _2 5 c2 e 6 . (3.26) v,a v,a
Again, one has to consider cases I, II. In case I, ¢1 =¢ 2=o , so the only nontrivial term in (3.24) is already estimated
in (3.26). Using the already proven estimate (3.8) and tak
ing s 0 (o) to be small enough, we can estimate the denomi
nator of (3.23) by
(3. 27)
hence (3.26) implies for case I that
390
-1 -2 --1
I
(Zv(a )-Zv(a ) )Z 1
~
zv!o2>z-1 (3. 28)
Thus in case I the estimate (3.9) is inductively reproduced.
In case II, from (3.15) and (3.20) follows that for j=1,2
iz-\z!xavnv>tPj!xv-...vl> -jl :> KE:(1+c2liBI . V ,a
(3. 29)
From (3.24), (3.26), (3.29) and (3.27) follows that
-1 -2 --1
I (Zv (a l -zv (a ) ) z I ~ I I I c2e-Kdo1 4+2KE:(HC2) IB o-1 4 (3.30)
zv!o2)z-1
As o is fixed, d :> p+r , and K ( o) can be taken smaller
than 1, it follows that for E: 0 (o) small enough the right--Kd hand side of (3.30) for E: < E: 0 (o) is less than c 2 e
(because of our choice of c 2 to be equal to IS ) . Thus
condition (3.9) reproduces itself also for case II. The proof
of the lemma is thus finished.
The proof of the theorem. Let v 0=v, v 1 ,v2 , ... ,Vk=.0
be a sequence of volumes, such that V o +1 = V o for some 1 1 0
t =tiE av i . Consider the boundary conditions o 1 E r2 I which
satisfy the relation
for t E av 0 n av 1
for t E av i n v ,
where TES is some fixed point. Then
k-1 ln zv(Q,Uiol = l:
i=O
From (3.8) follows that
ln
k-1 11nz.__(Q,Uioli52C1 l: exp{-Kdist(to,vonsuppu)} ~
-v i=O 1 1
::;2c 1 L exp{-Kdist(t,vnsuppli)}S tEV
5 2c 1c1v n supp 61 ,
(3. 31)
(3 .32)
391
provided £ , and in turn c1 , are small enough. Here
e = C(K,V) ( 00 • From (3.32) follows (3.6).
Proposition 3.1. uEaiiia implies uEaib
Proof. Note that for the Gibbs specification Q0 ,
given by (2.12), one has
(3. 33)
where zv<· I·) is defined by (2.4). Now, condition I of the
theorem follows from (2.20). Condition II is valid for the
potential U-U , provided U E a£ (U) (cf. ( 2. 6) with £
small enough). Hence (2.8) follows from (3.6).
Definition 3.1. Let Mc2l:v and ¢(p) > 0
zero as
M , if ;1 = ;2
s s
p+oo • A specification Q is called -1 -2 for any finite VcM , tEoV , a ,a E S"l
for sft , and any p ,
and goes to
¢ -mixing on
such that
-1 -2 Var{QV,B(t,p,V)(·Io ), QV,B(t,p,V)(·Io )) ~¢(p) (3 .34)
If M = Zv the specification Q is called ¢ -mixing.
Note 3.1. If Q is ¢-mixing and
lim ¢(p)pv-1 = 0 , (3. 35) p+oo
then condition I of Theorem 3.1 is valid with some p •
Theorem 3.2. For
¢(p) = C exp{-ap} , (3. 36)
where C>O , a>O , there exists a number m(C,a,r,v) , such
that if the r -specification Q is ¢ -mixing on any cube
(3. 37)
for some m~m(C,a,r,v) , then Q is ¢-mixing.
A proof of Theorem 3.2 together with an estimate on
m(C,a,r,v) will be published later. Note, however, that an
important ingredient of the proof is the algorithm of con
struction of a certain joint distribution on S"Jvxs-JV , pre
sented in §4 of [1].
Proposition 3.2. Let the translation-invariant specifi
cation Q be C exp{-ap} -mixing, and the family QY of
392
translation-invarian specifications converge to Q in the v - y -sense that for all vc:.: , oEn Bcnv lim Qv(Bio)
y-+0 = QV (B I o). Then the specification Q y satisfies condition I of
Theorem 3.1, provided y is small enough.
Proof. For y small enough the specification QY is
2C exp{-ad} -mixing on any V(s,m) , with m=m(2C,a,r,v) •
Hence our statement follows from Theorem 3.2 and Note 3.1.
From the last proposition condition IIIa follows for
the classes A, B, C. Indeed, the corresponding specifica
tions are close to C exp{-ad} -mixing ones. For classes A,
B these are the specifications of independent random vari
ables, \'lhile for class C it is the ground specification
(IG~(o) i-1
l 0
for
otherwise.
In this case the mixing condition follows from B(d)
4. Estimates on the functionals of partition functions
The variant of the conform mapping method we use in
this section is based on the following simple property of
analytic functions:
Proposition 4.1. Let G cC 1 be a bounded, connected and simply-connected open region and z 0 E G • For any point
z 1 E G , there exist positive constants C = C (z 1 ,z 0 ,G) ,
a = a ( z 1 , z 0 , G) , such that for any function f , holomorphic
in G , conditions
1 ) ! f ( z ) I ~ M , z EG
and
2) 1 f (z 0 ) = f (z 0 ) f (n-1) <zol 0
with some n>O
imply that
!f<z 1 ) i ~ MCexp{-an}.
Proof. We begin with the case when
ball centered in z 0=0 . Then
G=G 1
(4. 1)
(4 .2)
(4 .3)
is the unit
393
f (z) = zn <P (z) (4 .4)
where <P is again analytic in G1 . From the maximum prin
ciple it follows that
i <P (z) I .S: M (4 .5)
(because lzlaG 1 [=1 ) • Hence from (4.4)follows that
jf(z) I .S: Mjzjn, (4. 6)
which implies (4.3). For the general case, one uses first
the well-known Riemann theorem to construct a conform map
lJ!: G1 ->- G with lj! (0) = z 0 and then applies the statement al
ready proven to f(lj!(z)) •
We shall use consequently the following theorem:
Theorem 4. 1 . Let W cAr be open and connected. Let
also u0 E W where u0 is the zero interaction. Then there
exist functions a(U) =a(U,E) >0, C(U) =C(U,E) <oo, UEW,
such that for any analytic function g (U) , 6 E 0 E (W) c Ac conditions
1 ) (4. 7)
and
2) for n>O all the derivatives
- 0 , (4. 8)
imply the bound
jg(U)j ~ MC(U) exp{-na(U)}, UEW (4 .9)
to hold.
Remark. By the partial derivative (4.8) we mean the
family
394
1 (4.10)
t . . > 0, t. 1+ ... +!. =!., ai,j E 0 , j=1, ..• ,u1., u1. >0, l.,J J., J.,Ui J. -l3i
'-1 1 J.- I ••• ,q f • Proof. As W is connected, for any UEW there exists
0 -a path f(s) , 0:S:s:S:1_, f(O) =U , f(1) =U . Using the Weier-
strass theorem, we can assume without loss of generality s that all the functions UA(aA) are polynomials in s ,
where f(s)=UsEA . Let r
f (s) = g (f (s)) • ( 4. 11 )
This function can be continued analytically into the region
{ z E c: -o < Re z < 1 +o, I Im z I < o} , (4. 12)
where
0 = 0 (U IE) (4. 13)
is small enough. Now, the function f satisfies the condi
tions of Proposition 4.1 with z 0 =0, z 1 =1. Thus (4.9)
follows from (4.3).
We shall apply Theorem 4.1 to functions of 6 , which
depend on partition functions and semiinvariants. In order
to simplify the notations, we identify the logarithm of the
partition function with the semiinvariant:
,k1 ,km - -ln ZV (U I a) = < 1j! 1 , ••• , 1j! m I U, V, a > , (4. 14)
where m=O and the families {Ai}, {lj!i} are empty.
To check condition (4.8) we shall use the following
simple fact. 4 { V I I • } Definition .1. A family Cic Z : 1Ci1 <oo, J.=1, ... ,s
is called connected, if for any proper subset I c { 1 , ... , s}
[ u c.J n [ u c.J f 0 (4.15) iEI 1 i~I 1
Proposition 4.2.
I.
395
t 1+ .. . +iq ,k 1 ,km _ _ 3 < \j! 1 , ••• , \j! m [ U, V, <J)
£_1 i a OB ••. a q uB 0
1 q U=U
0 (4. 16)
unless the family {A1 ,.:.,Am' B1 , .•. ,Bq} is connected.
II. Let V.cz J
and oj E Sl . I j=1 I 2 be such that avJ
for V = [( ~ A.} U [ ~ B. J 1 2 1 2 , v n v =Vnv , v n av = v n av , i=1 l. i=1 l.
and for s E V n av1 01 = 02 s s
• Then for any choice of the other par-arreters,
f1+ .. . +iq ,k1 ,km - -1 a <lj! 1 , ... ,lj!m [u,v1 ,a>
a 16B ••• a q uB 1 q
(4. 17)
t 1 + .. • +iq ,k1 ,km,- _2 a ( \j! 1 I ••• I \j!m I u I v 2 I (J )
a 1 uB • • • a q uB 1 q
Proof. The Gibbs field in V with interaction u0 and -any boundary condition a is nothing else than a family
{~t' tEV} of independent random variables, uniformly dis
tributed in S . The components of the derivative (4.10) are
the semiinvariants of the form
i ,k1 ,k ,f1 1 ,f1 ,2 ' s,us 0 -
<lj! 1 , .•. ,lj!m m,x 1 , 1 ' ,x 1 , 2 , ••. ,x [u ,v,a >,
where
x. . (avl l.,J
provided
for
otherwise,
s,us
i,j 0 vnB.
l.
(4. 18)
(4. 19)
396
i,j a avnB.
~
for all j = 1 , ••• , l i , i=1 , ••• , s ; ( 4 • 20)
otherwise the corresponding component is zero, because the
partition function ZV(Uio) does not depend on u (ai,j) B. ~
Now statement II follows immediately from (4.18)-(4.20~
To prove I, note that if (4.15) is violated with (c1, ••• ,Cs)=
=(A1, ••• ,Am' B1 , ••• ,Bq) , the family (~ 1 , ••• ,~m' x1 , 1 , •••
... ,x ) can be divided into two groups, depending on in-s,us
dependent sets of variables. Hence the corresponding semi
invariant vanishes. (As in this case the generating function
(2.11) is a product of functions depending on disjoint sub
sets of variables zi .)
Now we shall apply Theorem 4.1.
Proposition 4.3. U E M(c!Ia) implies that u E cl!rrd .
Proof. Consider the function (see (2.25)):
u ,-1 Ov,A ({a A} a )
ln o~,A ({aA} lcr2)
g(U) (4. 21)
By definitions ( 2 • 1 2) , ( 2 • 19)
(4. 22)
From condition Ia follows that the function g(U) can be
continued analytically, and then Theorem 4.1 can be applied.
Namely, we define W = W to be the main component of the c,e:
potentials, which satisfy (2.7) with constants C, 2£
Surely, the sets W exhaust all c!Ia • From (2.7) fol-c,e: lows that
(4 .23)
Now, the derivative of the type (4.8) can be nonvanishing
only if the family {B 1, ••• ,Bq} is connected, with t E Bi
for some i and with An Bj I~ for some j • Indeed, if
connectedness does not hold, then the derivative vanishes
397
according to Proposition 4.2, I. If, on the other hand, the
family is connected, but t rf_ Bj for all j , or all the in
tersections B. n /1. are empty, then, according to statement J
II of the same proposition, we have the complete cancella-
tion of equal terms with different signs. So, if the deriv--1 ative is nonzero, then q ~ q 0 = [r dist (t,/1.)] . Hence con-
dition II of Theorem 4.1 holds with n=q 0-1. From (4.9),
(4.23) follows that
lg (U) I ~ c (U) lVI exp {-y (U) dist (t,A)}. (4. 24)
This is not, however, what we need, because of the extra
factor lVI . But
property of Gibbs
lows that for any
we can eliminate it by using r -Markov
fields. Note that from (4.21), (4.24) fol-1 -2 o , o E ~
u 1-1 QV,A ({oA} o ) I u 1-2 QV , A ( { 0 A } 0 )
(4. 25)
where ll(o 1 ,0 2 ) = {sE<lV: 01 -F 02 } • Indeed, one has to cons s sider any sequence 01 ,0 2 , ••• , o _ 1 _ 2 , where the first
ill (0 ,o ll -1 . -2 I - - I term is o , the last one ~s o , and ll(oi,oi+1 l = 1 ,
and then apply (4.21), (4.24) to each pair oi, oi+1 . Now,
considering again the initial problem, define
~ = {sEV: dist(s,A) ~ dist(t,A)} (4.26)
From r -Markov property it follows that
j=1,2
Let us apply (4.25) with V instead of V . For any
o', o" E ~v ..... ~
(4. 27)
398
(4.28)
x exp {-y(U)dist(t,A)} •
But lVI ~c 1 1AI (dist(t,A))v, lavl ~c2 1AI (dist(t,A))v-1 ,
where c 1 =c 1 (v), c2 =c 2 (v,r), hence (2.25) follows from
(4.27), (4.28) for any y <y(U) and some K, thus the
proof is finished.
Let us turn now to semiinvariants.
Proposition 4.4. UEM(C!IIa) implies that UEC!I!c
Proof. Let us apply Theorem 4.1 to the function
,k1 ,k - m1- -g(U) =( 1/1 1 , ••• ,1/Jm 1 U,V,a> (4. 29)
An a priori bound is (2.15):
M k1+ .•• +k
C mk 1 ! ••• km!
Statement II of Proposition 4.2 implies that the derivative
(4.8) can be nonzero only when the family {A1 , ••• ,Am'
B1 , ••• ,Bq} is connected. But the bound diam Bj ~ r , and
the definition of the quantity d(A1 , ••• ,Am) (see IIc)
result in the estimate
for some c = c (r, v) , which proves lie.
We mentioned in Comment 2.7 that the convergence of the
semiinvariants is exponential when v~oo • This follows from
the following estimate (which is, in fact, quite exact):
Proposition 4.5. Let U be a completely analytical in
teraction. Then for some C (U) < oo , a= a (U) > 0 , for all fi-v -1 -2 I I nite v1 ,v2 c:z , a ,a EO, 1/1 1 , ••• ,1/Jm with 1/Ji !1 and
Ai EV1 nv2 , k 1 , ••• ,km the bound
399
,k1 ,km -1 ,k1 ,km -2 I< 1)!1 , ••. ,lj!m lu,v1a > -< 1)!1 , ... ,lJ!m lu,v2,o >I ~
k1+ .•. -tk ~ C m(k1! .•. km!)exp{-a(dist(A1U ... UAm,ll)+d(A1, ... ,Am))}
(4. 30)
holds with
( 4. 31 )
Proof. We apply Theorem 4.1 to the function
Condition IIa implies the a priori bound:
k1 + ..• +k lg (U) I ~ 2 C m(k 1 !. .. km!) (4. 33)
From statements I, II of Proposition 4.3 follows that the
non-vanishing of t.he derivative implies that the family
{A1 , .•• ,Am' B1 , ... ,Bq} is connected with Bint.#fl' for
some i. Hence for some c=c(v,r)
which proves (4.30).
The only remaining thing to prove is the following
statement.
Proposition 4.6. UEaiib implies that UEOIIa
Proof. From (2.13), (2.14) the following formal expan
sion follows:
+I .e.
[ ) -1 ,.e.1 1 ,.e.1 2 I S,U -.e.1 1! .e.1 2! • • • .e. ! ( x1 1 I tX1 21 l•••tX SIU,V,O)X
I I S IUS I I S 1 US
(4.34)
Here the functions X· . are defined by (4.19), and the l.,J
400
summation goes over all s>O , all sequences (B 1 , ••• ,Bs)
with different elements, such that Bin V f ~ , all integers
t 1 1 ,t1 2 , ••• ,! , such that .e.. 1 +£.. 2 + ••• +£.. > 0, I ~ • s,us l., l., l.,Ui
with ol.,J running over DB. , j = 1, .•. ,u. = IDB I . ]_ ]_ i
We are going to show that the series (4.34) is abso-
lutely convergent and then to estimate its sum. For this
purpose we shall use the bound (2.16) and we suppose that
U E 0£ (U) . Then the sum is estimated by
.e., ,+.e., 2+ •• • +i
L (C£) s L I I S,U [
fEG(B 1 , ••• ,Bs) II ¢(1yi)J I (4.35)
yEf
where the range of summation is the same. But u. = IDB I ~ ]_ i
~ C(r,v) , and performing the summation over all .e. .. , we l.,J
bound the sum (4.35) by
(4. 36)
if £ is small enough. But the sum (4.36) is finite, be
cause the sets Bi have to be different. So the series
(4.34) defines a holomorphic function on 0£(U) , provided
£ is small. The bound (2.7) follows now from (4.36), (2.17)
and the following combinatorics:
Lemma 4.1. Let G be a finite set. Suppose that the
function ¢(g 1 ,g2 ) > 0 is such that for any g 1EG
(4 .37)
Let G(g0 ) be the set of all trees r with vertices from
C , such that the point g 0EC is one of them. Let lfl be
the number of bounds in r Then there exists a value
t: 0 (K) such that if £ < t: 0 , for any g 0EG
L £1fl II ¢(g',g") s: C(t:,K) < co • (4 .38) fEG(gO) (g',g")Ef
Proof of the lemma. Let R (f) be the maximal number go
of the links of a path on rEG (g 0 ) beginning in g 0 . Let
GR (gO) ={rEG (g 0 ): R (f) S: R} . Let go
401
(4. 39)
The tree r E GR (g 0 ) can be specified by the set g 1 , ••.
• • . ,g.t E l>'-{g0 } of vertices, which are nearest neighbours
g 0 in the tree f , together with the trees f j E GR-l (gj) ,
j=1, •.• ,.t (which can be empty). Now,
II ¢(g',g") ~ (g' ,g")Ef
n (1+E(1+CR_ 1 l<Pig0 ,gll-1 ~ gEG'-{go}
~ exp { E(1+CR_ 1 l l: ¢(g0 ,gl}-1 ~ exp{EK(HCR_1l}-1. gEe>'-{ go}
As is arbitrary, we have
(4. 40)
(4. 41)
c0 being equal to zero. Let ya be the unique positive
solution of the equation
y = ea(y+1) - 1 , (4.42)
which exists for
hence CR ~ YEK
(4.38) holds with
References
-1 O<a<e Supposing CR_ 1 < yEK one has
for all K . This means that the bound -1
EO (K) = (Ke) , C (E ,K) = y £K .
[1] Dobrushin R. L., Shlosman S. B. Constructive criterion for the uniqueness of Gibbs field, this volume.
[2] Dobrushin R. L. A new method of the study of the Gibbs perturbations of Gaussian fields (in preparation).
[3] Shlosman S. B. Mediation on Czech models: uniqueness, half-space non-uniqueness and analyticity properties (in preparation).
402
[4] Lebowitz J. L., Penrose 0. Analytic and clustering properties of thermondinamic functions and distribution functions for classical lattice and continuum systems. Comm. Math. Phys. 11, 99-124 ( 1968) .
[5] Gallavotti G., Miracle-SoleS. On the cluster property above the critical temperature in lattice gase. Comm. Math. Phys. 12, 269-274 (1969).
[6] Duneau M., Jagolnitzer D., Souillard B. Decrease properties of truncated correlation functions and analyticity properties for classical lattice and continuous systems. Comm. Math. Phys. 31, 191-208 (1973).
[7] Duneau M., Souillard B., Jagolnitzer D. Analyticity and strong cluster properties for classical gases with finite range interactions. Comm. Math. Phys. 35, 307-320 (1974).
[8] Duneau M., Souillard B. Cluster properties of lattice and continuum systems. Comm. Math. Phys. 47, 155-166 ( 1976).
[9] Dobrushin R. L. Analyticity of correlation functions in one-dimensional classical systems with slowly decreasing potentials. Comm. Math. Phys. 23, 269-289 (1973).
[10] Dobrushin R. L. Analyticity of correlation functions in one-dimensional classical systems with power decay of the potential. Matern. Sbornik 94, No. 1, 16-48 ( 1974) •
[11] Malyshev V. A. Cluster expansions in lattice models of statistical physics and quantum field theory. Uspehi Mat. Nauk 35, No. 2, 3-53 (1980).
[12] Malyshev v. A., Minlos K. A. Gibbsian random fields (the method of cluster expansion). Hoscow, 1985 (in print).
[13] Dobrushin K. L., Pechersky E. A. Uniqueness conditions for finitely dependent random fields. In "Random Fields" Vol. 1, North-Holland, Amsterdam-Oxford-N-Y, 232-262 ( 1981) .
[14] Simon B. Correlation inequalities and the decay of correlation in ferromagnets. Comm. Math. Phys. 77, 111-126 (1980).
[15] Lieb E. M. A refinement of Simon's correlation inequality. Comm. Math. Phys. 77, 127-136 (1980).
[16] Statulevicus v. A. On large deviations. Z. Wahrsch. verw. Geb. 6, .123-144 (1966).
[17] Dobrushin R. L., Nahapetian B. S. Strong convexity of the pressure for lattice systems of classical statistical physics. Teor. Mat. Fiz. 20, No. 2, 223-234 ( 1974) .
[18] Dobrushin R. L., Tirozzi B. The central limit theorem and the problem of equivalence of ensembles. Comm. Math. Phys. 54, 173-192 (1977).
403
[19] Gruber S., Hintermann A., Merlini D. Analyticity and uniqueness of the invariant equilibrium states for general spin classical systems. Comm. Math. Phys. 40, 83-99 (1975).
[20] Van HoveL., Sur l'integrale de configuration pour les systems de perticules a une dimension. Physica 16, 137-143 (1950).
405
DIFFUSION, MOBILITY AND THE EINSTEIN RELATION
Pablo A. Ferrari , Sheldon Goldstein
and Joel L. Lebowitz
Abstract. We investigate the Einstein relation o= BD between the
dif~usion constant D and the
interacting with its environment
"mobility" , of a "test particle"
a-1 is the temperature of the system
where D is measured and oE is the drift in a constant external field
E • The relation is found to be satisfied for all model systems in which
we can find a unique stationary non-equilibrium state of the environment,
as seen from the test particle 1n the presence of the field. For some
systems, e.g. infinite systems of hard rods in one dimension, we find
non unique stationary states which do not satisfy the Einstein relation.
For some models in a periodic box the Einstein relation is the most
direct way of obtaining D . A precise macroscopic formulation of the
Einstein relationwhich makes it mathematically very plausible is given.
I. Introduction.
We investigate stationary states of various classical model- sys
tems in which a charged "test particle" (tp) is subject to a constant
external field E in the x-direction . The suitably defined drift or
mean velocity of this tp , u(E) , is generally expected, for small
fields to be proportional to the diffusion constant D of the tp at
E = 0 , when the system is 1n equilibrium. More precisely,
o - lim u~E) E-->0
BD (1.1)
where B the reciprocal of the temperature characterizing the equili
brium state of the system at E = 0 • (We set Boltzmann's constant and the
charge of the tp equal to unity. We use the usual physicist's normali
zation for D : for standard Brownian motion, (dlJ) 2 = dt , D = 1/2 ,
406
and for the general one dimensional Brownian motion <Wo(t) 2 > = 2Dt) •
Equation (1.1) is the first example of a class of general rela
tions between linear transport coefficients and equilibrium fluctuations
e.g. o and D . It was derived by Einstein for Brownian particles in a
fluid using physically intuitive quasi-equilibrium arguments [ 1 ]. Their
most general formulation, as Einstein-Green-Kubo' (EGK) relations, is
usually derived via formal perburbation arguments around the equilibrium
state, see below and refs.[2).
The validity of the EGK relations, or at least of some of their
experimental consequences appear well established in many cases. A con
vincing mathematical derivation, and in some cases even a precise formu
lation, is however lacking at present [3] . The purpose of this pre
sentation is to discuss the meaning and status of equation (1.1) for
various model systems
Formulation of Problem.
We shall call a system for wbich the tp has differentiable
spatial trajectories, i.e. in which the microscopic velocity d ~, ~ 'E R
of the tp is well defined a mechanical system. For such a system
u (E) M < v 2 > X 0
( 1. 2)
where M is the mass and vx the x-component of ~ • The subscripts
E and 0 refer to expectation values in the appropriate stationary mea
sures with and without the field. The microscopic action of the field in
such a mechanical system is an acceleration M-lE of the test particle
in the x-direction - while leaving all other interactions unchanged. We
shall later discuss different models of such mechanical systems. Note
that according to our terminology a particle whose velocity undergoes
an Ornstein-Uhlenbeck process is a mechanical system. A system which
evolves in a deterministic manner according to Newton's equations of
motion will be called Newtonian.
In addition to mechanical models we shall also consider systems
where the microscopic dynamics is modeled by Brownian motion, a continuous
spatial process whose velocity is not well defined. In these models the
407
electric field acts by adding a drift term proportional to Edt to the
displacement of the tp • The definition of a appearing in (1.1) is now
related to the behavior of the tp in equilibrium with an external
"confining potential" . A similar situation occurs for models in which
the position of the tp takes values on a lattice, X E ~d • Here the
electric field acts to "suitably" bias jumps in the x-direction •
In general, for mechanical and non-mechanical systems in which
the tp 0 0 d d 0
1s free t:_o·movemall of lR or ~., there w11l be no stationary
probability measure completely describing the entire systems, environment
plus tp , since the position of the latte< will not be localized in the
"steady state". However, if we ignore the position of the tp and consi
der only the remaining coordinates, describing the environment relative
to the tp (including the velocity
of the tp in the case of mechanical systems), this problem disappears.
It is to stationary probability measures for this relative description -
the environment seen from the test partic:e- tn which< > 0 and< >E refer.
For mechanical systems it i3 ~atural to define u(E) as
< v >E ' but this makes no sense for non-mechanical systems. However,
X
1n all cases X(t) (the x-coordinate of) the position of the tp
at time t (X(O)=O) can be naturally defined as a random variable on the
path space for the evolution of the envi~o~ment seen from the tp ,
equipped with the inva~iant path measure arising from < We
then have that < X(t) >E/t is independent of t and we define u(E)
by (we abuse notation and write < >E for the expectation with respect to
PE) ,
( l. 3)
This defini:ic~ ag~ees with the previous on~ in the case of mechanical t
systems, since in this case X(t) = J v(s)ds • We also note th2t if the
proc~ss describing :he evolcticn of0:he environment seen by the tp star
ting from the state < >E is ~rgodic, then
. X(t) l1m -- = u(E) t-- t
(1.4)
a.s. with respect to PE .
408
Observe also that while for mechanical system;the diffusion cons-
tant D is given by J < v (0) v (t) > dt , an expression which makes O X X 0
no sense for non-mechanical systems, in all cases (under consideration)
we have
D = lim (2t)-l < X(t) 2 > 0
(1. 5) t-..o
Note that equation (1.1) implies in particular that a is zero
whenever D vanishes. This is trivial in the case where the tp is
confined to a compact spatial region by an external potential, e.g. by
walls. It is more interesting in the one dimensional system of Brownian
particles with hard cores or lattice gases with jumps restricted to
nearest neighbor empty sites. In these cases it is known [ 4 ] that
< x2(t) > ~ t 112 so that D is zero. We show indeed that in these cases
u(E) = 0 : In fact the infinite volume stationary measures < >E for
E f 0 turn out to be limits of finite volume E f 0 , stationary states
containing N particles with either periodic or rigid wall boundary con
ditions. In the latter case the finite volume states are generalized Gib~
sian for which of course u(E) =0 , while in the periodic case we verify
(1.1) for finite N , with uN(E) , DN ~ N-l
The situation is less clear for interacting Newtonian particles
with hard cores. The diffusion constant for a tp , having the same mass
~1 as the other "fluid" particles, is non-zero [ 5] . However, just as
for Brownian particles, we explicitly find stationary measures as limits
of finite volume (generalized)Gibbs states. These states, which are spa
tially identical to those found for Brownian particles, have Maxwellian
velocities, so that u(E) = 0 . (They are however no longer limits of
stationary states with periodic boundary conditions). Moreover, by
Galileaninvariance, these states are imbedded in a famil~members of
which can be found assigning u(E) any value whatsoever. Presumably,
none ,.f these states have anything to do with the Einstein relation, not
even the one(s) for which it is satisfied. We expect, but do not prove,
that there are also other stationary stat•·s, arising in the limit t -+ ""
(under the evolution with E # 0) from the initial state < > 0 , for
which we expect that the Einstein relation is satisfied. Indeed, in cases
where there is more than one stationary state < >E (for given 8 and
density), it is only for those arising in the above way that we expect
the Einstein relation to be satisfied and hence we believe that u(E)
409
should be defined in these states. In fact this is essenti~lly what the
usual derivation of the Kubo formulas (EGK relations), via perturbation
of the equilibrium state in effect does (2]. We present it here as heuris
tics.
Perturbation Argument.
Let the time evolution of the appropriate probability measure
be given by the forward generator of the process 1 0 + EL1
where L 0
t > 0 when
Then by the
and Ll are independent of E We consider
ll(O) = ll . the equilibrium state, when E 0
standard Dyson formula
IJ(t) t
ll 0 + E f exp[ (t-t 1 )L0 )L11J(t 1 )dt 1
0
(1.6)
now the state at
= 0 ; L o ~'o = 0
(1. 7)
time
Let now U U + U E 0 1
be the function whose expectation value in the
correct stationary state is u(E) (For a mechanical system U0 = vx ,
U1 = 0 ; !:or a diffusion system U1 is a constant). We then write formally,
assuming that the average of U converges as t -+- oo to the correct u(E)
and that the limit (1.1) exists and is given by letting t -+-oo after expanding
< U > in E ,
0 = < ul > + f < u<o) (t 1 )A > dt I 0 0 0
(1. 8) 0
where
A -1
Ll Ll log ll ll ~'o 0 0 (1. 9)
The subsript < > means that th~ average is with respect to p 0 and the
superscript u<ol(~ 1 ) indicates that the time evolution is taken with the 0
gene~~to:.- L 0
For a mechanical system (1.8) is simply
0 = a f < v (t 1 )v O X X
> dt' 0
SD (1.10)
A similar formula [6)is obtained for a diffusive system with the appropriate
choice of U 0
410
We have avoided making precise statement about the limits
t + "', E-> 0 leading to (1.8). This is deliberate; we have no rigorous
(or even very convincing) arguments about the validity of (1.8) for general
systems [3). It is precisely this lack of knowledge which led us to the pre
sent work. We believe that in addition to the specific models considered ex
plicitly here the formulation in the last section has some promise of lea
ding to rigorous results for general systems. In particular that formulation
makes Einstein's original argument precise and very convincing.
We proceed to give some examples of models with "static" environ
ments for which the Einstein relation holds. In later sections we will
investigate models with ''dynamic" environments in which the tp ~s one
of the particles in an interacting system.and give a new formulation of
the Einstein relation.
II. Static Environments.
a) Markovian Mechanical Models.
In this widely used class of models the velocity of the tp
undergoes either i) a Markov jump process specified by a transition rate
K(v,v')dv or ii) an Ornstein-Uhlenbeck process. (We take here v E lR
since extra dimensions do not introduce any essential new elements.)Set-t
ting x(O) = 0 the position at time t is given by x(t) = J v(t' )dt'
The position and the velocity distribution of the tp satisfies a
"linear Boltzman equation"
af(x,v,t) + v of + l [- au(x) + E) af at ax M ax av
(Kf)(x,v,t) (2.1)
411
We have included here also a force term coming from an external
potential U(x) (which we generally take to be zero) in order to clarify
the role of E • The operator K on the right side of (2.1) represents
the effect of the environment : it is the forward generator of the
Markov velocity process and is independent of U , E and x • K is
assumed to have a non-degenerate eigenvalue zero, corresponding to the
Maxwellian velocity distribution ha(v)
Kh 6 = o (2.2)
-1/2 2 (2n/6M) exp[-(1/~a Mv ] (2.3)
The stationary solution of (2.1) for E = 0 is the equilibrium distri
bution
(2.4)
C is a normalization constant whose meaning is clear when exp [-BU(x)]
is integrable - or the particle is 1n a rigid box. When U(x) = 0 or
is bounded periodic with period L it is simplest to interpret
(2.4) as holding for x 1n ~/L, i.e. we can think of the tp defined
in the periodic box of length L . Alternatively f 0
is the stationary
Poisson density of independent particles in the x,v plane.
The diffusion constant D for the tp undergoing the process
defined by (2.1) with U zero or bounded periodic can be shown to exist
under mild assumptions on K , and to be given,as usual, by the integral
of the velocity autocorellation function as in Equation (1.10) with
< > 0
referring to the normalized equilibrium distribution (2.4). It
might appear that, in this case at least, very few additional assumptions
are required to jusfity (1.8) and thus prove (1.1) (see however eq. (2.9)
and the subsequent discussion). We do not attempt to investigate this here
instead we limit ourselves to two exa~ples where h(v;E) can be computed
explicitly and (1.1) can thus be shown to hold.
412
For the OU process [9]
(2.5)
and D = (yll)-l • It is easy to check explicitely (or by performing a
Galilean trans format ion) that for U = 0 the stationary velocity distri
bution is given by
h(v;E) E hll(v-E/y) (2.6)
and so o = y-l = I!D •
A problem may arise, however, when K is an integral operator
(case i).
(Kh) (v) =I K(v, v' )h(v') dv' - J K(v', v)h(v) dv' (2. 7)
so that £L1 becomes a singular perturbation : E multiplying the
highest derivative. To see this let us consider the simplest jump pro
cess, one without memory,
(2.8)
with T a constant.
The stationary solution of (2.1) can now be easily obtained for the
case U = 0 ;
v h(v;E) = I aexp -a(v-v') hll(v')dv', a= M/TE (2.9)
Clearly h(v;E) is not differentiable at E = 0 . The Einstein relation
is nevertheless satisfied,
< v >E = TE/M = I!DE • (2.10)
The first part of (2.10) is obtained most easily by multiplying (2.1) by
v and integrating over v , while the second equality follows readily
from (1.10) (In fact the same method can be applied to the case
K(v,v') • W(v-Av') , W(n) even and integrable, A < 1 • It gives
a • t/(1-A)M .- llD , t-l "'J W(v)dv) .• Thus the existence of an expansion of
413
h(v;E) about E"O, is certainly not necessary for (1.1) tb hold, what
is then? Our formulation in section 5 suggests that (1.1) is valid in
essentially all cases where the motion of the tp , for E "0 , converges
un the limit of macroscopic length and time scales) to Brownian motion.
The problem however remains open.
b. Diffusion in a Random Environment.
We continue the investigation of static environments by conside
ring the motion of a test particle diffusing in a potential U with a
"conductivity" a(x). The distribution function p (x, t) for this model
(there is no velocity variable) sat'sfies the equation
ap(x,t) + 2._ ( -1[ au l <lt <lx Y - a;z+E p) (2.11)
with -1 ay : B , independent of x , required to make the stationary
measure, for E = 0 , assume its equilibrium value, c.f. (2.4). For
U = 0 and a constant, the Einstein relation is then satisfied by
assumption. The question about (1.1) now occurs when U and/or a is perio
dic or more generally themselves form a spatially stationary random pro
cess. We shall consider this case 1n one dimension where one can obtain
explicit expressions for a and D and show that they satisfy (1.1).
Consider ergodic, translation invariant random fields U(x) and/
or a(x) , x E lR, defined on some probability space < n,A,P >.(We
assume that w may be identified with Uw,aw , or (Uw,aw) .)We write
< • > for the expectation w. r. t. P • For each w E: Q the forward gene
rator of the corresponding process X(w,t) is
Lwp =-_i__ [ -1(- <lUw + E) p ]+ J-k <l J d X y 'W OX dX W ;J;Z p
d Lr Ef;z< flap) (2 .12)
and the stationary solutions to (2.11) satisfy
According to (2.12)
tor).
0
div Jw(p) , where J w
(2.13)
is the current (opera-
414
Thus the first integration constant of (2.13) is the current
J cJ(pw) , which can be interpreted as
N (t) 0
t
where N ( t) is the (expected) signed number of crossings of the origin 0
(total flux) up to time t of independent particles, each performing
X(w,t) and distributed initially according to a non homogeneousPoisson
process of density P (x) w
If P is ergodic one can prove (using ergodicity of the process . . ) . X(w,t) . 1nduced 1n the space of the env1ronments that the llm ---t--- ex1sts
t._ and is independent of w . Thus the effective velocity of the system is
given by
u(E) lim X(w,t) t
t--><» , P a.s.
and we have also that < pw(O) > ~(E) J(p) • so
u(E) J(p)
Now we compute u(E) in two cases
(2.14)
(2.15)
a) random potential (a is constant [= 1) in eq. (2.11)) and
b) random conductivities (U is zero in eq. 2.11).
For the case of a random potential the stationary density pw is
the solution of the equation
Putting the first integration constant J(pw)
P (x) = f e-eE(x'-x) ee[U(x')-U(x)] dx' w
X
Taking x • 0 and using eq, (2.15) we get
(2. 16)
1 we obtain
(2 .17)
Thus
1 u(E) • <p (o)>
w
415
a _ lim u(E) • B(< e-BU(O) > < eBU(O) >)-1 E-+0 E
(2.19)
The rightmost factor is the diffusion constant D of the tp
moving in the random potential (provided e.g. e-aU(O) E t 2(p) and P is
mixing under translations) , so we have that a • BD •
For the case of random conductivities,stationary densities pw
satisfy the equation
Taking again the first integration constant "' -BEx
x = 0 • pw(O) = f ~ dx Q alXJ
J(p ) 1o1
we obtain, for
Thus 1 a : u(E) • <illO)>
-1 BE < 1/a > which implies that
w
. u(E) -1 hm -- a /!l< 1/a > E-+O E
It is known [7 ,6] that < 1/a >-l equals the diffusion coefficient
in this case, so relation (1.1) has been verified for the cases of
random potentials and random conductivities: see also Appendix B.
III. Pynamic Environments.
1. Non-crossing Particles in one dimension.
a. Ornstein-Uhlenbeck Dynamics.
We consider first the case of mechanical particles on the line
which interact via a finite range even pair potential ~(r) , smooth for
r > 0 • In addition there is a hard core, i.e. they cannot cross. This is
416
irrelevant when ~(r)~~ as r ~ 0 but we shall keep it anyway. When
two particles meet, or collide, they exchange velocities. All particles
have the same mass and between collisions they undergo independent
Ornstein-Uhlenbeck (O.U.) processes with friction constants y/M and
reciprocal temperature B, i.e.
dv.=-(y/M)[v. dt -1/2/f,y dW.] - (1/M) E f-- ~(x.-x.)dt 1 1 1 j f.i xi 1 J
(3 .1)
where the Wi are independent standard Wiener processes.For a givenden
sity,the,. presumably unioue,stationary state of this system is a Gibbs -1
state at temperature 8 fen· the Hamiltonian H = (l/2)M v~ + E.< .. ~(x.-x.) 1 J_l 1 J
,P(r) = 0 but pre sum-
that the mean square
It is known rigorously for the case when
ably true also for the general non-crossing case
displacement of a particle in an infinite system
particle density p behaves asymptotically like
t , and so D = 0
in equilibrium with tl/2 [ 8 l rather than
The Einstein relation then predicts for this system (and other
one dimensional non-crossing systems we discuss later) that a should be
zero. As we shall see later this is true in a very strong sense with
u(E) = 0 for all values of E acting on a test particle.
We shall, however, first consider a system consisting of N O.U.
particles on a circle of length L . We show that the diffusion constant
DN and mobility oN are of O(N-1 ) and satisfy the Einstein rela
tion. The
preted as
displacement
Jtv(t'):!t' 0
x(t) of the tp is now of course to be inter-
, i.e. like an angular variable.
To investigate the diffusion of a tp
note two facts : a) since there is no crossing
diffusion constant of the "center-of-mass" x
1n such a system,we simply
DN must equal DN , the
of the system. b) The
motion of x is entirely independent of the forces between the particles
(as long as they satisfy Newton's law of action and reaction). Thus, set
ting
417
-1 N x • N l: x. • x1 < x. ~ x 1 +L
i•1 1 1
N-1 N
v • l: v.
(3.2)
i•l 1
We have from (3.1)
(3.3)
This gives immediately
(3.4)
When thereis a field E acting on a tp , say the first parti
cle, then (3.1) is modified for i = 1 by the addition of a term (E/M)dt
on the right side. The center of mass velocity now satisfies the equation
(3.5)
It follows that in the stationary state, which we shall discuss next
u(E)
Stationary States.
1 Ny
E (3. 6)
The stationary non-equilibrium state in the presence of a field
on the tp -is most easily obtained by considering the problem in the
frame of reference moving with the average velocity of the system E/yN •
In this frame the stationary distribution of the fluid particles relative
to the test particle is just the equilibrium measure in the presence of
an electrostatic potential E(xi-x1)/N . More precisely, setting
(3.7)
the stationary ensemble density will have the form
where
v1 = v. - E/yN l
N N U(y) = l: l:
i=2 j>i
418
(3.8)
$(y.-y.) +$(y.) +Ey./N} , l J l 1
(3. 9)
P is the density for the environment measure and Z is a normaliza-
tion constant. The stationarity of ~ rollows easily from the fact that
~ is a canonical Gibbs state periodic in x1 , of period L (in fact,
independent of x1).
N.B. In choosing the domain of the y. l
in (3.7) we have used
strongly the fact that there is a hard core between the tp and the
fluid particles. This permits the discontinuity in the electrostatic
potPntial and in P(~,~) between yi = 0 and yi = L which corres
pond simply to the right and left side of the tp .
To see more clearly what is happening, let us consider the case
¢(r) 0 , i.e. hard point particles. In this case P is a product mea-
sure
N P<x.~> = h8<v1> n [).(y.)h8<;.)J
i=2 l l
where (~hanging the domain of
nience) ).(y) has the form
y. l
to (-l/2L,l/2L)
{c exp[-pEy/N] , 0 < y ~(l/2)L
).(y) c
C exp[ -pE(y+L) /N] , (-l/2)L~ y < 0
-1 C • BE[l-exp(-BE/p)) /N , p s N/L
(3.10)
for future conve-
(3.11)
419
Put differently, the distribution of the fluid particles relative to the
position of the tp is Poisson with a density
p(y)he(vi-~/yN) p(y) _ (N-UA(y) •
Note that there is a discontinuity in p(y) at the position of
the tp Putting p(O+) = pR and p(O-) = .PL , the densities to the
right and left of the tp , we immediately f1nd
(3 .12)
(3.13)
The left side of (3 .12) represents the difference in pressure llp , exerted
on the tp by the fluid particlffi- the net remaining force on it
E -llp E/N = y u(E) (3.14)
is just enough to produce the average motion.
Infinite Volume Stationary States.
Taking now the thermodynamic limit N ~ ® , L + ® , N/L + p of
this stationary state, we obtain a Poisson field (for the environment
measure) with constant densities pR for y>O(<O) with
PR = PL +BE and on either side of the tp a Maxwellian velocity dis-
tributions h6 (v) . In this "blocked" state the electric force on the
tp is ~ntirely balanced by the difference in pressure exerted by the
fluid on its opposite sides and .u(E) = 0 as it should be.
It is useful to note here, and this will be important later when
we deal with other non-crossing systems, that this stationary measure
can also b~ obtained directly as a limit of constrained equilibrium
states in a box with rigid walls situated at ±L . Suppose we want to
obtain the infinite volume state wi:h some specified density to the
420
left of the tp . We then put NL ~ pL L particles to the left of the
tp and NR K [pL+SE]L particles to the right of it. Because of the hard
core interaction between the tp and the other particles NL and NR
are conserved quantities. The corresponding equilibrium canonical ensem
ble density will therefore be given by (with now the particles labelled
so that the o-th particle is the tp)
-1 ~(x_N ,v_N , ••• ,x1,v1, ••• ,xN ,V~=Z
L L R -R
• exp[SEx ]x(x;Nt,N ) o - R
(3.15)
where X is the characteristic function specifying that there be exactly
NL , NR particles on the left, right of x0 • It is now an easy exercise
to show that this state leads to the blocked state when L + m • (See
section 4 for the lattice version). The above construction will work also
when ~(r) ~ 0 . We will always end up with two semi-infinite Gibbs states
which, except near the boundary, will be uniform with densities and
pR determined by p(pR,B) -p(pL,a) = E where p(p,S) is the pressure
in the (infinite one dimensional) Gibbs state with density p at tempera
ture B
Remark. Going back to the stationary non-equilibrium state in the period
ic box we can also take there the limit L + m with N fixed. We obtain
then a stationary environment measure in which there are no particles on
the left and N independent particles with exponentially decaying density
on the right. The whole entity is moving to the right with a velocity
E/Ny • Such a stationary state cannot be produced by starting with a rigid
box and does not exist when E c 0 ; the particles would then disperse to
infinity. To the extent that one can define a diffusion constant here it
would correspond presumably to the center-of-mass of the block and would
satisfy the Einstein relation.
b. Brownian and Jump Dynamics.
The above analysis can be carried out almost verbatim for the
(positional part of) the distribution of interacting Brownian particles.
421
The results are also essentially identical. The problem is a bit more
complicated for particles on the 1-dimensional lattice which can only
jump to unoccupied nearest neighbor sites • The absence of complete
translational symmetry makes the computations awkward, particularly for
the periodic case. They are given in section 4.
c. Hamiltonian Dynamics.
When y is set equal to zero in (3.1), we are dealing with a con
servative system whose evolution is governed by classical mechanics. It
is clear that for such a system the velocity of the center of mass is a
constant of the motion and thus DN is infinite. Similarly, under the
action of a uniform (non-potential) field on the circle the finite systems
center of mass keeps on accelerating, no stationary state is possible and
is also infinite.
The situation is different for the infinite system. Here it was
shown by many authors [ 5) for the case ~ = 0 , i.e. for hard point par
ticles that D = <lvl>/p > 0 , where p is the uniform density and
<lvl> the expectation of the speed of the tp , which has the same mass
as the other particles, is given in equilibrium by (2/nMB) 112 • Einstein's
relation would now appear to say that when the field is put on the tp
there should result a non-equilibrium stationary state in which
u(E)/E ->-BD as E _,. 0 . On the other hand the constrained Gibbs state
constructed in (3.15) is stationary also for this system and as before,
gives rise in the limit L _,. ~ to the "blocked" state in which u(E) = 0
We believe but cannot prove that the resolution of this problem
lies in the fact that the blocked state is not the appropriate stationary
state for this system. Unlike the case of OU or Brownian particles, the
blocked state is not unique here. In particular we expect that there
exists an entirely different stationary stat~ one which would be obtained,
as t _,. "' , if we turned on the field at t = 0 when the sys tern is in equi lib
rium.!. Such a state should have quite a different velocity distribution
for "out going" particles to the right and the left of the tp . If such a
state exists it is presumably given by the construction (1.8) and satisfies
the Einstein relation as is expected from the considerations in section 5.
422
2. Higher dimensions, Crossing particles.
The existence of a diffusion constant D , 0 < D < m , for a tp
in a general interacting system of particles has been proven so far only
for : a) B-particles interacting via sufficiently soft superstable poten
tials [6,9].b) Particles on a lattice at infinite temperatures in dimension
greater than one (also in one dimension when the jumps extend beyond nearest
neighbor sites [ 10]) and for OU particles - with bounded potentials·· [ 11).
In none of these systems is there any rigorous information about the
existence of a stationary state in the presence of an electric field E act
ing on the tp • In fact, for a mechanical system with "soft" interactions,
the resistance of the fluid to the motion of the tp might be expected to
decrease at large speeds of the latter, as the cross section does, and there
will presumably not be any stationary state for E > 0 . The Einstein rela
tion might still hold however for some kind of "metastable" state in such
a system or when E is sealed properly : c.f. section 5 • We shall there
fore confine our discussion here to the case where the tp is OU or
Brownian while the fluid particles are Newtonian. It will be seen that for
these "mixed dynamics" the Einstein relation provides some interesting in
sights.
We begin with the o.u. mixed dynamics system, i.e. we set y = 0
1n (3.1) for all i -1 1 It is quite e; :y to see that the stationary state
of the system of N particles in a periodic box with a field E on the
tp is simply the \;ali lean transform of the canonical Gibbs state the
positional part remains canonical Gibbs while the velocity part is trans-
formed, h (v.) + h (v. -E/y), independent of N • It follows further-S 1 8 1
more from general arguments about spreading Markov processes,U2),thatfornon-
degenerate irrteraction~i.e. when the phase space cannot be decomposed in-
to separate components, this stationary state is unique.
The perturbation argument for the EGK relations given 1n section 1,
which there seems no reason to doubt, then leads to theresult that the diffus
ion constant for the tp in the equilibrium state, E = 0 , of this periodic -1 system is (By) ; the same as if there were no o~her particles present.
423
This result, while a little surprising at first sight, seems not
too unreasonable for the finite periodic system. After all the interac
tions between the particles conserve momentum and energy and the only
dissipation occurs via t'he tp • In fact, we shall now prove this expli
citely for the case where the tpis a B-particle.
For the mixed dynamics in which the tp is Brownian, the tp
has no velocity and in place of (3.l)for i = 1 we have
N Y-1 E- _a_ "' ( ) I dx1 = ~ "' cj> x1• -x1 dt + 2 By dW
oXl i=2
We can now compute the diffusion constant for E = 0 • Let
N v l: v.
i=2 1
Then dx1 + y -1 MdV V2/By dW .
and
I/21BY W(t) _ y -1M V(t)-V(O)
Vt 1/f
Since V(t) forms a stationary stochastic process (with
the last term on the right does not contribute in the limit t + w We
therefore obtain that the diffusion constant of the tp is the same as
if no other particles were present. (We also (more or less) immediately
obtain the invariance principle for the motion of the tp) .
Consider now the passage to the thermodynamic limit N + w, L + w,
N/L d -> p . \Je obtain then (for both the B and 0. U tp) a uniform Gibbs
state in the frame of reference moving with velocity E/y
It is clear however, that this stationary state is not the relevant
one for the Einstein relation. For starting with an infinite system in
equilibrium and putting the field on the tp will surely lead to a state
in which the velocity of the fluid particles "far away" will remain
424
essentially unaffected by the electric field in higher dimensions.
Also in one dimension for crossing particles, the fluid far away will be
moving relative to the tp • It is this state in which u(E) should be
computed. It will then presumably satisfy the Einstein relation, with the
correct equilibrium D for the infinite system.
We can say a little more about this model if we consider again
the non-crossing case. For the finite system in a periodic box the fluid
particles must have the same diffusion constant as the test particle, D =
(ay)-l . In the thermodynamic limit we will have the additional stationary
states obtained as the Galilean. transforms of the blocked states discus
sed earlier. This leads to the family of stationary states moving with
the velocity a and having a pressure jump 6p connected by the relation
For y = 0 , E
nical system.
a = (E-6p) /y •
6p and a is arbitrary as we found for the purely mecha-
The question now arises as to what is the diffusion constant of
the tp , again the same as that of the fluid particles because of non
crossing, in the infinite equilibrium system with E = 6p = 0 For the
O.U. tp, it is clearly not just (ay)-l since when y + 0 it should go
to < lvl >/p .
The origin of the problem with all these stationary states appears
to be the interchange of limits t + m , necessary to obtain the statio
nary state for E 'I 0 and diffusion constant for E .. 0 in the finite
periodic system, with- the thermodynamic limit L + m • We get the "wrong"
stationary state and diffusion constant. What then is the right answer ?
lV. Jump Processes on the Lattice.
a) Infinite one dimensional lattice gas.
We consider a one dimensional lattice gas in which all the parti
cles but one have a symmetric rate of jump (i.e. the rate of jumping to
425
the right • the rate of jumping to the left • 1/2). The test particle is
subjected to an external field : it jumps with the rate p(resp. q •1-p)
to the right (left), p > q • The relation between p , B and E is given -BE -
by q/p •e • (For this choice the (formal) Gibbs state satisfies de-
tailed balance). The interaction of the particles is merely simple exclu
sion, so when a particle attempts to jump to an occupied site the jump is
suppressed.
We describe the system directly as it is seen from the tagged
particle ("environment process"). The generator acting on cylindric func
tions f:{O,l}7l n{n:n(O)=l} isgivenby
L f(n) p
l: x,y'f'O x-y "1
0/2)[f(n )-f(n)] + xy
where (; n)(z) = n(x+z) X
and
Tl (z) xy -{
The semigroup
n(z)
n(x)
n(y)
if
if
if
z 'f x,y
z = y
Z = X
corresponding to the generator Lp deter-
mines a unique strong Markov p~ocess 7l
nt on {0,1} , in such a way that
denotes the expectation with respect to
the process with initial configuration n .
The set of extremal invariant measures Je is given by (see theo
rem Al in the appendix)
Je = tiJ : 0 < P < 1} U {';;' p - - n
n > O} (4. 2)
where ~P is a Bernoulli measure with parameters pi = p to the left of
pr = (1-q/p) + p(q/p) to the right of the origin. Thus the origin and
q/p 1-pr
= 1-p jJ n i
is concentrated on configurations with no particles to
426
the left of the origin and n particles to its right (cf. eq. (A.ll)
below).
The position X(t)of the tagged particle is given by the algebraic
number of shifts of the system (corresponding to the last two terms in the
generator (4.1)) in the interval [O,t] .consider that the initial configu
ration ~(0) is distributed accordingly with ~p{X(t)} = 0 .
The Einstein relation in this case follows trivially : It is known
(cf. [4)
U(E) = EX(t) t
p ) 0 .
) that when E = 0 , D = 0 for all p > 0 • On the other hand
1 . EX(t) 1m--
t p(l-pr)- q(l-pi) = 0 for all E and all
The Einstein relation is, in fact, satisfied for this model in a
somewhat stronger sense. In section c) below we show that for a sequence
of periodic approximations the Einstein relation is satisfied with non
zero diffusion constant DN and mobility oN . Moreover, these quantities
converge to their infinite volume values (0) . Finally, we will see that
the stationary states we have described in this section arise as limits
of the stationary state5 :or the periodic approximations.
In the next section, we show that the same thing is true for box
approximations except that here, of course, the diffusion constant and
drift are 0 •
b) Finite lattice gas.
We consider now finite approximations to the preceding model : the
particles now move on a finite lattice of length 2L with reflecting walls
at -L and L . Note filcst that the Einstein relation is now trivial.
Mrr.eover, stationary states for this model are easy to find, even for
E ; 0 , e.g. Gibbs states. The Gibbs states, however, don't have a good
limit as L-~ ~ • But if we condition the Gibbs states on the number of
particles to the left and right of the tp , in the appropriate way as
L ~ w , we obtain a sequence of stationary states (stationary because
particles cannot cross) converging to the mea~ure ~P of the infinite
case. This is based on the fact that for these constrained Gibbs states,
given the position x of the tp , the particles on its left (right) are
427
uniformly independently distributed with density p J. (pr) - whi~h depend upon x. What
must do is show that x is well localized as L + • at a position which
gives the correct value for p1/pr • This we proceed to do.
In the box -L,L put the tp and N additional particles,
M of them to the left of the tp . The position of the test particle
we denote by x • nie (average) density to the left is p1 • M/(L+x) ;
the density to the right of the tp is pr • l-p1 • N-M/(L-x)
Let M/L - a and N-M/L - b a~ L + m • Then, writing y • x/L ,
p1 - a/l+y: p1 (y) and pr-b/1-y = pr(y) •
The distribution of the tp corresponding to the constrained
Gibbs state (i.e. conditioned on there being M(N-M) particles to the
left (right)) when the field acts as before (q/p =e-BE) is given by
BEx(L+x)( L-x) f(x)- e M N-M
Using the approximation 1n n! ~ n 1nn , we obtain
where
f(x) F(x)
e
f (x) = e E X + (L+x) tn (L-+x) +
+ (L-x) tn (L-x)- (L+x-M) tn(L+Je-M)
- M tn M- (N-M)tn(M-N) -(L-x- (N-M~ tn(L-x-(N-M))
Under the change of variables y = x/L we obtain F (x) ~ L cj>(y)
where ¢(y) 6Ey + (l+y)~n(l+y)+(l-y)~n(l-y)-(l+y-a)~n(l+y-a)
-(1-y-b)tn(l-y-b)+g(N,M)
where g(N,M) arises from the terms of F(x) which don't depend upon x .
428
We thus have that the distribution of y - eL~(y) so that for
large L y is near the maximum y0 of ~(y) . Setting ~'(y0 ) • 0 we
find that
i.e.,
-BE e
q/p
(1 +y 0) ( 1-b-y 0)
(1-y )0-a+y) 0 0
l-pr(yo)
1-pR.(yo)
Moreover eL4> (y) -L/24>" (y ) (y-y ) 2 ~ e o o ' so that ly-y I - 1/V'L •
0
Thus 1n the thermodynamic limit y is localized at and hence we ob-
tain the state described in section 4a).
c) EGK relation for a periodic lattice model.
We here consider the symmetric lattice gas with the tp subjected
to an electric field as in section 4a but now with the particles moving in
a one dimensional periodic box of length L+l , L > l . Let X(t) be the
position of the tp in ~ induced by our process X(t) is the alge-
braic number of jumps performed by the tp up to time t (jumps to the
left make a negative contribution). Let 0 ; X0 (t) < x1 (t) < ... < ~(t) ~ L
be the positions of the particles relative to the tp and let
Yi(t); Xi(t) + X(t) , i = O, ... ,N , define the motion of the ith parti
cle in~ (Y0 (t) = X(t).) Then by considering the motion Y(t) of the center
of mass
N Y(t) = (1/N+l) E
- i=O y. (t)
1 (4.3)
we easily compute the diffusion constant for E = 0 , at least in the
limit L ~ ~ : It is easy to check that Y(t) is a martingale (with res
pect to the a-algebra generated by the motion of the entire system up
to time t) • Therefore, since IY (t)- Y(t) I< L for all t > 0 , we have 0
that
= lim t~
429
(4.4)
Here 1( is the indicator function and ~+l E L+l • The last equation
follows easily from the fact that each particle jumps to each unoccupied
neighboring site with rate 1/2 •
We are primarily interested in the situation in which N and L
are fixed. In this case the RHS of (4.4) is not as easy to compute as
it is when < >0 is Bernoulli with density p (grand canonical ensemble).
With this slight modification we find that
L ~ n(x)(l-n(x+l))
D = < ~x~=~O------~~--- > [~n(x)]2 o
p(l-p)L
(Lp)2
1-p 1 -p-L
since for large Lp we have that En(x) ~ Lo
We don't ~ish to make the approximation above more precise, since
we will compute D and u explicitly after again slightly modifying the
model. The real problem is with the computation of u(E): in order to com
pute u(E) we need detailed information on the stationary measure < >E
which is not so easy to obtain.
Nevertheless, if instead of fixing the length L+l of the box and
allowing N toberandom we fix N and allow L to be random in an appro
priate way the computations become much easier. Consider the process
~~(y) E NN+l , y = O, ... ,N where ~t(y) is the number of empty sites
to the right of the y-th particle. Let v = vp,E,N be a probability
measure on NN+l satisfying
a) < L > = N
< ~
y=O (~(y)+l) > = (N+l)
p
b) v is a product measure (i.e. ~(y) are independent random
variables) of geometric distributions with parameters a y (i.e
430
v{i,(y) = kl = ,/ (1-a )) y y
c) a y
satisfies the balance equations (4.5) below. One can prove
by direct computation that N v is stationary for the process 4t(y) , and
hence defines a stationary state < >E for our periodic system (with
rantl<'m length).
We want to compute
the quantity
lim u~E) . We find it convenient tc consider E->0
c _ c(p,E,N)
since BE ~ p-q as E ~ 0 ,
u(E) p-q
o = lim B. c . (p-q)~
Since a = v{~:((y) > O) ~he average velocity of the y-th y
part.icle, y -f 0 , is given by (l/2)a -(l/2)a 1 while that of particle y y-
zero is pa0 - qaN • (We are identifying N with -1). We thus have that -BE
~y , 0 < y ~ N must satisfy the following equations (q/p = e ) .
(l/2)a -(l/2) a 1 = c(p-q) y y-
(4. 5)
from which it follows that (for E > 0)
a c(2Nq+l) 0
aN = c(2Np+l)
a = (1-y/N)a +(y/N) aN y 0 0 < y < N
The relationship between p and c is then given by (cf. a) above) N .!: N
1/p = '{~·0(~(~)+1)} • (1/N+l) .!: 1
~ N+l 1-a y•O y
431
1 N-(N-y)c(2Nq+l)-yc(2Np+l)
and taking the limit (p-q) + 0 we obtain
Thus
N ! • ~ E (N-N(N+l) lim c)-l p N+1 y=O (p-q)+O
o • lim (p-q)+O
Be tl(l-p) .N+l
(1-(N+1) limc)-l p-q+O
On the other hand, (see eq. 4.4)
N m r
y=O
N r <l(x 1-x y=O y+ y
>1) > 0
D = "----=----.,.---(N+l)2
< 1(~(y)>O) > (N+l)2 o
(N+l)(l-p) 1-p (N+l):Z = N+l
This proves relation (1.1) for this model.
v. Macroscopic Formulation of Einstein Relation.
In our discussions so far the tp has been treated entirely on
a microscopic level - asking for a description of the stationary states
of the tp on thP. s~atial and temporal scale on which t~e basic dynamics
of the model is prescribed. While this level clearly gives the most de
tailed information the Einstein relation concerns quantiti~s. D and o
which are measured on very long, i.e. macroscopic, spatial and time scales.
Furthermore being a linear transport coefficient, o is calculated in the
limit E ~ 0 so ~~e system is really very close to equilibrium - hence,
of course, the EGK relations. It seens therefore sensible to formulate
the Einstein relation in a more macroscopi.: way. In fact this turns out
to b~ possiblP ;:1.;d ~as t~e (not so incide::atal) advantage that it is not
432
necessary to first find u(E) at finite E and then take the limit
E + 0 • Instead one goes to the appropriate macroscopic length and time
scale~ by setting 2 x' • ~x • t' • £ t
where £ is a small parameter related to the electric field E which is
assumed to change as E • tE' . A form of (1.1) is then the following
A) • If for E' • 0 , when the system is in equilibrium, the rescaled
trajectory of the test particle converges (weakly) to Brownian motion
with diffusion constant D , i.e.
then in the presence of a field gE
x£(t; E) - aE t +WD(t) , a £+0
BD
(5.1)
(5.2)
We believe that A can in fact be proven for all non-mechanical cases conside
red here and in oarticular for a B-particle in a random environment or inter-
acting with other B-particles [13] . It also seems to hold in
the case of an O.U. ~article in a periodic potential, recently shown by
Rodenhausen to satisfy (1.1) [14].
It should be noted that the scaling of the electric field is just
such that it remains effective, neither zero narinfinite, on the macrosco
pic scale a is then the nobility, as in Stoke's law, for a dilute concentration af B
particles in a fluid. It is presumably this situation which Einstein had
in mind. We are now in a position to present WQat we regard as the best
argument for the Einstein relation, a rigorous reformulation of Einstein's
original argument. We first observe that if we drop from equation (5.2) the
relation a = BD and replace oE in (5.2) by a more or less arbitrary
function u(E) of the field, what we obtain is a sort of regularity
condition for the macroscopic behavior of the tp • Moreover, if the field
is allowed to vary with x on a macroscopic scale
E (x) • &F(&x) • VxU(&x) £
433
the regularity condition (5.2) should become
t J u(F(x(s)))dB+WD(t) 0
(5.3)
(where the function u -+ u(F) does not depend on the particular field
x -+ F (x) under consideration, and the environment bas the same initial
distribution as for F • 0 , i.e. equilibrium).
Now suppose we have
tial U(x) -+CD, lxl -+CD ,
macroscopic regularity (5.3). Consider a paten
sufficiently rapidly. If the system is at tern--1
perature ll , so that it has a stationary state for which
distribution of x : x' ~ e-llU(x') , then the distribution E
the marginal -llU(x)
PB ~ e
must also be stationary for the limiting diffusion. Since the current in
this state must be zero, we have that
u(F(x))Ps(x) - DS F(x)ps(x)
so that u(F)/F = a does not in fact depend on F and o = llD • (It does
not much matter here whether we regard x as in ~ or ~d • Moreover
for d > 1 , it is sufficient to assume that the drift is a vector valued
function u (F of the local field. It then follows from the (Einstein)
argument that u(F ) = o!:_ = llDI , where o and D may be tensors).
P h d "ff . -llU(x) . . ut somew at ~ erently, 1f pll ~ e ~s to be stat1onary
for the limiting diffusion, whose (forward) generator is
Lp = -V·(u(F)p(x)) + Dl'.p
then o and D must be related by o = llD . Thus the Einstein relation
is more or less an immediate consequence of macroscopic
and the very meaning of a system's being at temperature
originally argued. (Interestingly enough this appears to
regularity (5.3) Q-1 . . " , as E1nste1n
be the case in the
Newtonian system of hard points when the initial state, before the field is
turned on, is a Gibbs state but not when the initial state is one in which
all the particles move with velocities !1[17].)
434
APPENDIX A.
In this appendix-we sketch the proof of equation (4.2) for the
process of section 4 in the infinite lattice. More precisely we prove the
following :
Theorem A.l. Let nt be the simple exclusion process as seen from the
test particle ("environment process"), for which the motion of all the
particles but the testparticlemoveswithrates p(resp. q) to the right (left),
p > q , i.e. the infinitesimal generator is given in equation (4.1). Then,
the set Je of extremal invariant.measuresfor the process is given by
where ~ a
J • {~ : 0 < a < l}L/{p : n > O} e a - - n -
is a Bernoulli measure with parameters
(A.l)
a ~a to the left of
the origin and a+= (l+q/p) + a(q/p) to the right of the origin, and
~n is concentrated on configurations with no particles to the left of
the origin and n-particles to its right (see equation A.ll) below for
a formal description).
of x
In order to prove theorem A.l we consider the following partition
X.,.~ {n: l: n(x) ~ l: n(x) ~ .. X:>O x<O
x+ • {n: l: n(x) .. x>O
l: n(x) < oo} x<O
x .. • {n: l: n(x) < .. ' l: n(x) = oo} x>O x<O
X+ • { n : l: n (x) = n. , l: n (x) < ""} n x>O x<O
Now one can prove [~] that
u[J n u n>O
(A.2)
(A.3)
435
where M(y) is the set of probability measures concentrated on y • We
prove theorem A.l by showing
- )J n n > 0
(A.4)
(A.S)
(A.6)
(A. 7)
Proof of (A.4). Let us introduce the zero range process naturally related
to the lattice gas as seen from the test particle we are discussing. A
given configuration n E XCID can be represented by a doubly infinite se-
i E 7Z , x. < xi+l ' X = 0 where x. denotes the site occu-
1 0 1 quence x. :
1
pied by the i-th particle (iE7Z). (In the same way configurations bel on-
ging to x: and
those belonging to X n
can be represented by semi-infinite sequences and
by finite sets of sites).
X u
Let x (t) u
The process
be the position of the particle initially at site
describing the evolution of
the number of sucessive empty sites to the right of the u-th particle,
is the so called zero range process
(A.8)
It is a process with state space y
rator Lzr given by (on h-cylindric)
:N7Z and infinitesimal gene-
(A.9)
+(1/2) E [h(~ 1) -h(~)] +(1/2) E [h(~ 1 )-h(~)] x#O x,x- x#-1 x' x+
where
436
f;(z)-1
{ <(•)•!
if z = x and f; (x) > 0
E; (z) x,y if z ~ y and f; (x) > 0
f;(z) if either z ; y , z;. x or E; (x) • 0
A set of invariant measures for this process is described in [16]:
where va is the geometric product measure defined by
r}n(x) k} (A.lO)
where ax a if x < 0 and ax =(q/p)a if x > 0 . We find {ax} by
solving the detailed balance equation
so 0 < a < 1 - X
k ; 0,1
(A.lO.b)
From equation (A.lO) one proves, using coupling techniques as in
[;15., 16], thaf the only extremal invariant measures for the nt -proces!> on
M(X~) are those described in equation (4.4).
Proof of (A.S) (A. G) (A. 7).
In the semi-infinite and finite case one has to look for solutions
of equation (A. lOb) allowing one (or two) of the ak to be one. The state
space for the corresponding zero range model is semi-infinite (or finite). + For X~ for instance, we consider ak = 1 for a fixed k < 0 . This
implies that in the corresponding zero range model at site
infinitely many particles" so, from site k to site k+l
with intensity 1/2 (respect q) if k < -1 (resp. k • -1).
« "there are
rarticles enter
the simple
437
exclusion picture ak • 1 must be read : "there are no particles to the
left of the (k+l)-th particle", so the rate of jumping to the left for
the (k+l)-th particle equals 1/2 if k < -1 and p if k • -1
In the semi infinite case we find solutions of the equations ana
logous to equation (A.lO.b) when k.• -1 and which imply that the density
of the invariant the measures must be that described in equation (A.5)
for the simple exclusion model. To prove that ~0 is the only invariant
measure requires coupling technicalities which we omit.
Equation (A.6) is studied similarly and the counterpart of equa
tion (A.lO.b) gives us that there are no solutions with An < 1 for
n < k and ~ c 1 for a fixed k > 0
Equation (A.7) is proven following the standard methods of finite
state markov chains, which imply that
j 1, ... ,j < n} = n a (i.)[l-a (i )b.]
t=l n .. n 1
where a (k) n
APPENDIX B.
z(p-q) k + (n+l)zq+l (n+l)zp+l (n+l)2p+l
(A. 11)
In this appendix, we indicate how from pw' the invariant non norma
lizable measure of section lib, we may easily obtain an invariant probabi-
lity measure for the "environment process". This is the process induced
by X(w,t) in the space n of environments by the relation
u (t) w
(or imply w(t) = 'x(w,t)w) where
envi,onments at time t = 0 , and T X
u w
In fact, the probability measure
is the configuration of
denotes translation by X
438
is invariant for the environment process. The key to this is the fact
that pw (x)
i.e. pw(x)
depends on x only through the environment seen from x
p1 ,..(0) . To see this note first that P,..(x)dx P(dw) is X
invariant for the process (X(t,w),w) , in which the environment does not
change. The second component of this process, wt - w , is of course not
the environment process. However, after the change of variables
(x,w)-+ (x,txw) we obtain the process (X(t,w),w(t)) , with invariant
measure d~ = P,..(O)dx P(dw) The w marginal for this measure should
be an invariant (probability) measure for the environment process, but
unfortunately this 1s not well defined, since ~ is not normalizable.
But we may regard x as a variable defined modulo L (for any L > 0)
both for the measure ~ and for the process (X(t,w),w(t)) since both
~ and the process (X(t,w)) don't essentially depend upon x ,
X(t,w)- X(O,w) depending only upon the autonomous process w(t) In
this way we obtain a norcalizable measure ~L which is stationary for
the process (X(t,w),w(t))
stationary for w(t) .
so that the w marginal p (O)P(dw) w is
The stationarity of P may also be seen directly : We may assume
without loss of generality that < pw(O) > = 1
sition probability for the environment process by
tion probability for the position x of the tp
f a "nice" function of the environment, that
Then, denoting the tran
Qt , and the transi
by- Pt , we have, for w
P(Qtf) = ff dx P(dw) p (0) Pt(O,x) f(t w) W W X
= JJ dx P(dw) p (O)Pt (O,x) f(w) t w t w -x -x
JJ dx P(dw) p (-x) Pt(-x,O)f(w) w w
• f P(dw) f(w) J dx p (x) Pt(x,O) w w
439
• f P(dw) f(w) pw(O) • P(f)
where we have used the translation invariance of P , the fact that
p...,(x) • pT ,}0) , the homogeneity of the X
and the stationarity of P...,(x) under
x-process
pt w
t t P (O,x) •P (y,y+x) T W W
y
Note that we didn't need to explicitly refer to this invariant
probability measure to obtain u(E) • In general, assuming ergodicity
(E) 1 . X(t,w) ~P h p · h · b b'l' u • 1m t a.s. were 1s t e stat1onary pro a 1 1ty
measuret~r the environment process. But here P is equivalent to P •
ACKNOWLEDGEMENTS :
We thank E.G.D. Cohen, D. Durr,c.Kipn~·E.Presutti, G. Papanicolaou
and H. Spohn for useful discussions. We also thank the organizers of the
Random Fields Conference in Koszeg, where a talk on the subject of this
paper was given by one of us (J.L.L.),for their patience in waiting for
the manuscript. P.A.F. thanks the Math. Dept. of Rutgers and J.L.L. for
their very friendly hospitality. Finally, we thank the IHES for their
hospitality while this was being written and (hopefully) clarified,and
Ms. F. Breiner, for typing it.
The work was supported in part by NSF grants DMRBl-14726 and
PRY 8201708,at Rutgers University.P.A.F. was supported by FAPESPGrant
021719-9 and CNPq Grant 201682-83.
440
REFERENCES
1. A. Einstein, Ann. of Physik 17 549 (1905); also L~. 371 (1906).-E. Nelson, Dynamical Theories-of Brownian Motion, Princeton University 1967. -Selected Papers on nois( and stochastic processes, J.L. Doob, L.S. Ornstein, G. E. Uhlenbeck, S.O. Rice, M. Kac, S. Chandrasekhar. Edited by Nelson Wax, Dover Publications New York (1954).
2. M.S. Green, J. Chern. Phys., 20 (1952) 1281- Kubo J. Phys. Soc. JaJ•. 12 (1957) 570 - R. Zwanzig, Ann-:-Rev. Phys. C1em. 16, .67 (1965) - D. Jorster, Hydrodynamic fluctuations, Broken symmetries an~Correlation Functions, Reading Mass : W.A. Benjamin Inc. 1975.
3 N.van~e.n,Phys. No:-v. ~ (1971) 279 -E.G.D. Cohen, Physica 118 A (1983) 17·-42.
4 T.E. Harris, J. App1. Probab. 2 (1965) 323-338 - R. Arratia Ann. Prob. 11 (1983) 362-373 ·- H. Rest, M.E.-Varo~, O•ai communication.
5. D.W. Jepsen, J. Math. Phys. 6 (1965) 405- J.L. Lebowitz and J.K. Percus, Ph:JS. Rev. 155 (l967) 122 -F. Spitzer, J. Math. Mech. 18 (1969) 973 -F. Spitzer,-cecture Notes in Math.~ (1969) 201, Springer, Berlin 1969.
6. A. De Masi, P.A. Ferrari, S. Gol~st~in and D. ~ick, An Invariance Principle for Reversible Markov Processes, with Applications to Random Motions in Random Environments, in preparation.
7 G Papanicolaou and S.R.S. Varadhan, Statistics and Probability : Essays in honor of C.R.. Rao, North Holland 1982 547-552.
3. D. Durr, S. Goldstein, J.L. Lebowitz, in preparation.
9. M. Guo, Limit Theorems for Interacting Particle systems, Phd. Thesis, Courant Institute 1984.
10. C. Kipnis, SRS Varadhan, to appear in Comm. Math. Phys.
11. S.R.S. Varadhan, private communica~ion
12. S.R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand, NY 1969.
13. To be written by •••
14. Rodenhausen, Communication at the Random Fields Conference Koszeg (1984).
15. P.A. Ferrari, preprint 1984.
~1
16. E.Andjel, Ann. Prob, 10 , 525-547 (1982)
17. P. Calderoni, D.Durr, Preprint.
INSTITUT DES HAUTES ETUDES SCIENTIFIQUES, 91440, Bures sur Yvette, France.
Permanent addresses
For S.G. and J. L.L. Dept. of Math. Rutgers University, New Brunswick, N.J. 08903, U.S.A.
For P.A.F. Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Cx Postal 20570, Sao Paulo, B~~SIL.
443
TRANSITION FROz.t PURE POINT TO CONTINUOUS SPECTRUM
FOR RANDOM SCHRODINGER EQUATIONS : SOME EXAMPLES
B. Souillard
Abstract
I review some recent results on Schrodinger equations
with random potentials, and specially discuss the known
examples wh~re a transition in the nature of the spectrum
occurs when varying the coupling constant or the energy :
a transition from pure point spectrum with power decaying
eigenfunctions to purely continuous spectrum has been pro
ven recently for two classes of diso~dered systems, a
transition which differs from the Mott-Anderson transition
proven in certain Anderson models.
I will also discuss relevance of localization theory
to hydrodynamics and plasma physics.
444
The week before the present meeting in Koszeg was held
in Braunschweig (Germany) a conference on "localization in
impure metals". At this conference were present the theore
ticians of localization and experimentalists from many
domains : 3-d doped semiconductors, metallic glasses, 2-d
accumulation layers such as field effect transitors or
heterojunctions with modulated doping, very narrow wires ...
What is this subject called "localization" which has been
worth the Nobel prize for Anderson and for Mott and can
attract the interest of people from so different fields ?
Let us consider a particle, an electron, in a poten
tial AV(x). This potential will be taken random in order
to model the disorder present in the sample. If we treat
this electron as a classical particle, it will remain loca
lized in a bounded region if its energy E is smaller
than the percolation threshold Ec of the random field
AV(x); in contrast if E >Ec it will move in an infinitely
extended region and propagate to infinity.
If we now treat this electron as a quantum particle,
which we must do for example in order to discuss the elec
tric conductivity of disordered systems at low temperatu
res, we are led in the common approximation of independant
electrons to study a Schrodinger Hamiltonian
H -I'> + AV ( 1)
acting on L2 (IRd) where V is a random potential : if
~1e take into account the well known tight-binding approxi
mation we are in contrast led to study the finite
445
difference version of this Hamiltonian defined as
(HljJ) (x) = ) ljJ (y) + AV (x) ljJ (x) y: !y-x!-1
( 2)
acting on £ 2 (Zd). The continuou~ and the discrete cases
are expected to yield the same results, besides the obvious
differences. The random potential in equation (2) is often
considered as given by independant and identically distri
buted random variables {V(x)}x EZd :we will also suppose
here for simplicity, that these random variables have zero
expectation and posses a density which is L00 and with
moments of order 2 +€; these conditions in many of the
results quoted are not necessary and we refer to the papers
for complete results.
What do we want to know ? One of the basic questions
is to determine for almost-all realization of the potential
V(x) the nature of the spectrum of H near some energy E :
is it pure point, singular continuous, absolutely conti
nuous or a superposition ofthese types. If the spectrum
is pure point near some energy E, then a wave packet built
up with energies near E will basically remain in a finite
region during time evolution and a solid with Fermi level
at E should be an insulator. In contrast if the spectrum
is continuous, the particle will go to infinity as t ~ oo
and a solid with Fermi level atE may be a conductor [1][2]
Now when E <<Ec' it is clear on the physical ground
that quantum tunneling will not spoil the localization of
the classical particle this is because randomness of the
potential does not allow constructive tunneling interfe
rences. This fact is however not necessarily easy to prove!
Some progress have been done recently going in the good
direction : it has been proven that almost-surely the dif
fusion constant vanishes [3][4],and that there is no abso
lutely continuous spectrum [5] • One expects that in such
situations the spectrum is only pure point.
446
From the pioneering works of Anderson and Mott, two
remarquable facts have been discovered which are responsi
ble for the emergence of localization as a full domain of
physics :
1. The quantum localized regime persists much farther
than the classical one.
2. There exists a sharp transition from extended states
to localized states when varying the coupling constant or
the energy at least for d~3 : this Mott-Anderson transi
tion explains a certain number o.f metal-insulator transi
tions observed in various materials at low temperatures
when varying the concentration of impurities.
The most spectacular result connected with point 1 is
the complete exponential localization of all states for
one-dimensional systems, for arbitrary small disorder. As
for mathematical results, it was proven that for any A and
almost-all realization of the potential, the spectrum of H
is pure point and the wave functions are exponentially
decaying. Other proofs applying to various cases have been
also given since [7-11] • We do not discuss these results
in greater detail, since our main purpose here is to discuss
the transition from pure point to continuous spectrum. Let
us just mention a fact which is useful in the proof of one
of the results below : among bhe present various proofs of
localization for one-dimensional systems, the one of refe
rence [11] is, up to now the strongest one in the sense
that it is the only one which allows to study non homoge
neous random potentials, that is random potentials which
ao not posess translation invariance properties.
Let us come now to the second point mentionned above :
the theory predicts for d~3 a transition from exponential
ly localized to extended states in appropriate domains of
E and A; this is expected to correspond to a transition
from a situation with only pure point spectrum and exponen
tially decaying eigenfunctions to a situation with only
continuous spectrum. As far as mathematical results are
447
concerned the Mott-Anderson transition is only proven on
the Bethe lattice, and we discuss the corresponding results
below. On the other hand, a new type of transition, namely
a transition from pure point spectrum with power law loca
lized states to purely continuous spectrum, has been disco
vered recently in other situations, and we will discuss it
later on.
So let us come now to the discussion of the results
on the Mott-Anderson transition on the Bethe lattice. A
Bethe lattice is a connected graph with no closed loops
and a constant coordination number at each lattice site;
we can consider a model of type (2) associate to this graph.
Why are we interested in such a model ? In the statistical
mechanics of phase transitions, one knows the crucial role
played by the mean field theory : it was the basis of our
understanding of phase transitions and it allows to compute
the critical exponents for systems in large enough dimen
sion d ; unfortunately no mean field theory is known for
the localization problem. More precisely the models which
would be natural candidates for a mean field theory of
localization do not exhibit a transition. It is thus natu
ral to look for the next simple model which should exhibit
a transition : it should be the model on the Bethe lattice.
Unfortunately this model for localization is still very
hard. Abou-Chacra, Anderson and Thouless [12] could esta
blish a limit of stability of localized states and numeri
cally investigate the region of extended states. Recently
we have studied this problem again [13] . Although we have
not obtained a complete solution of it, we could at least
get a certain number of exact results for a large class
of distributions of random potentials we prove that almost
surely
1. The spectrum of H is pure point for A >A , and is 1
pure point for lEI >E 1 (A) •
2. The spectrum of H is purely absolutely continuous
for A <A 2 in lEI< E2 (A).
448
Thus we have here the first model where the Matt-Anderson
transition is proven. The next results are concerned with
the critical properties of the model, and for them we need
to require stronger assumption on the distributions of ran
dom variables Vx and in particular we need that their den
sity probability r is an analytic function. Then
3. the density of states is an analytic function of E
and A ; in particular it is analytic at the transition
points.
4. the localization length, which governs the rate of
decay of the eigenfunctions in the pure point spectrum,
diverges as IE -E 1-l at the threshold, yielding a critic cal exponent v= 1 for the localization problem on the
Bethe lattice.
This is the first critical exponent exactly computed for
the localization problem. This result implies that the cri
tical exponent v is equal to 1/2 for dimension d large
enough. This fact, and possible consequences on the question
of finding the upper critical dimension for the localization
transition are discussed in [14] •
Let us turn now to the other type of transition which
has been found recently and which we alluded to above. Let
us first consider a continuous one-dimensional Schrodinger
equation in an electric field : we have thus to study the
Hamiltonian
H d2 -- - Fx +V(x) dx 2
(3)
When F =0 and V(x) is a random potential, then almost-
surely H has only pure point spectrum and exponentially
decaying eigenfunctions as mentionned previously. What
happens for F 1- 0 ?
1. if the random potential in V(x) is almost-surely
sufficiently smooth (e.g. has two bounded derivatives) then
for any F f-0, H has only absolutely continuous spectrum
and in fact an initial state is uniformly accelerated [15].
449
2. if in contrast the random potential V(x) is given as
a Kronig-Penney potential V(x) ~ V(n) o(x-n) I V(n) n E :11:
independant random variables, a model which is the natural
one for several physical situations, then [16]
if 0 <F <F1 , H has almost-surely only a pure point
spectrum with power low decaying wave functions at + oo
if F >F 2 , H has almost-surely only a continuous
spectrum.
Some of these results were anticipated in the physical lit
terature by perturbation treatment or numerical calcula
tions [17][18] .
The previous transition is strongly related to the
following one [16] : let us consider a one-dimensional
discrete Schrodinger equation of the type
(HljJ)(n) =ljJ(n+1) +ljJ(n-1) +).lni-CLV(n)ljJ(n) (4)
The V(n) are again random, and the only difference with
equation (2) is that here the random potential V(n) is
multiplied by lni-CL and thus the disorder decreases as lnl
increases. For a = 0, we have the usual one-dimensional
case and the spectrum is pure point with exponentially
decaying eigenfunctions. What happens for a>O? Then almost
surely [16] the essential spectrum of H is [-2,2] and
1. if 0 <a <1/2, it is pure point and the eigenfunc
tions decay as exp {- lnl (1- 2a)} •
2. if a = 1/2, it is pure point with power low decaying
eigenfunctions if). >). 1 , and it is purely continuous
if ). < ,\ 2 in 1 E 1 < E 1 ( ,\) < 2 .
3. if a > 1/2, it is purely continuous. In fact,
S. Kotani [19] has proven it is purely absolutely continuous
(scattering theory shows it is purely absolutely continuous
only for a > 1).
In all these statements we have always supposed that
450
V(n) has a distribution with a finite moment of order 2+ £
It is curious to note that if we choose instead for the
V(n) a Cauchy distribution (the so called Lloyd model),
then we can prove [16] that, in the electric field case
with a Kronig Penney potential, the spectrum is almost-su
rely pure point for all F, and in the discrete model des
cribed above the essential spectrum is almost-surely pure
point now for any 0 <a <1; in both cases the eigenfunctions
decay as exp {-jnja}, 0 <a < 1. It is the first case where
one can rigorously prove a difference of behaviour between
Cauchy and distributions \•lith moments of order 2 + c:.
As a conclusion, I would like to stress that these
phenomenons are not specific to quantum problems. The basic
mechanism at work is interference : is it constructive or
destructive ? So localization and Anderson-Matt transition
can be exhibited by other wave propagation problems. Let
us consider for example the propagation of shallow water
waves on a rough bottom [20] : the waves will be partly
neglected and partly transmitted by the fluctuations of the
bottom, and localization will arise in various situations;
this can be observed experimentaly. Let us also consider
the propagation of electromagnetic waves in a plasma [21]
here also they will be partly reflected and partly transmit
ted, this time because of the density fluctuations inside
the plasma; we have shown that localization is relevant for
real plasmas in various circumstances and moreover that
localization phenomenon has consequences on the instabili
ties inside the plasma, yielding a new mechanism to turn
the convective ones into absolute ones.
Other aspects of disordered systems
We have focuss here mainly on the transitions from
pure point to continuous spectrum. However disordered sys
tems present many other interesting aspects and some pro
gress have been also achieved in these directions. Many
aspects complementary to this talk can be found in the talk
of Pastur [22] or in [23] • A very large bibliography
on the topic can be found in [24].
451
References
[1] D. Ruelle, Nuovo Cirnento A61 (1969) 655
[2] W. Amrein, V. Georgescu, Helv. Phys. Acta 46 (1973) 633 [3] J. Frohlich, T. Spencer, Commun. Math. Phys. 88 (1983)
151.
[4] H. Holden, F. Martinelli, Commun. Math. Phys. (in press) •
[5] F. Martinelli, E. Scoppola, "A remark on the absence of absolutely continuous spectrum in the Anderson model for large disorder or low energy". Preprint Bielefeld.
[6] Y. Gold'sheid, S. Molchanov, L. Pastur , Funct. Anal. i Pril. 11 (1977) 1; S. Molchanov, Math. USSR Izv. 12 (1987) 69."
[7] H. Kunz, B. Souillard, Commun. Math. Phys. 78 (1980) 201. -
[8] G. Royer, Bull. Soc. Math. France 110 (1982) 27.
[9] R. Carmona, Duke Math. J. 49 (1982) 191.
[10] J. Lacroix, Ann. Inst. Elie Cartan N°7, Nancy (1982).
[11] F. Delyon, H. Kunz, B. Souillard, J. Phys. A16 (1983) 25.
[12] R. Abou-Chacra, P.W. Anderson, D.J. Thouless, J. Phys. C6 (1973) 1734.
[13] H. Kunz, B. Souillard, J. Physique (Paris) Lett. 44 (1983) 411, and to be submitted to J. Stat. Phys.
[14] H. Kunz, B. Souillard, J. Physique (Paris) Lett. 44 (1983) 503.
[15] F. Bentosela, R. Carmona, P. Duclos, B. Simon,
[16]
B. Souillard,R. Weder, Commun. Math. Phys. 88 (1983) 387.
F. Delyon, B. Simon, B. Souillard, Phys. Rev. Lett. 52, (1984) 2187 and "From power pure point to continuousspectrum in disordered systems", Preprint Ecole Polytechnique (1984)
[17] V.N. Prigodin, ZH. Eksp. Teor. Fiz. 79 (1980) 2338.
[18] C.M. Soukoulis, J.V. Jose, E.N. Economou, Ping Sheng, Phys. Rev. Lett. 50 (1983) 754.
[19] s. Kotani, Private communication.
452
[20) E. Guazzelli, E. Guyon, B. Souillard, J. Phys. (Paris) Lett. _!i (1983) 837.
[21) D. Escande, B. Souillard, Phys. Rev. Lett. 52 (1984) 1296.
[22) L. Pastur, in these Proceedings.
[23) B. Souillard, in Proceeding of the "Les Houches Workshop on Corrunon Trends in Particle and Condensed Hatter Physics, Phys. Rep. 103 (1984) 41.
[24) B. Simon, B. Souillard, Franco-American Meeting on the Uathematics of Random and Almost-Periodic Potentials, to appear in J. Stat. Phys.
B. Souillard
Centre de Physique Theorique Ecole Polytechnique
F-91128 Palaiseau Cedex - France.
453
RIGOROUS STUDIES OF CRITICAL BEHAVIOR II
1. Introduction
Michael Aizenman*
The Institute for Advanced Study Princeton, New Jersey 08540, USA
The task addressed in the talk is the analysis of the critical
behavior in statistical mechanical systems, and related problems in the
theory of random fields. While it would have been very useful to have
mathematical means of approaching this subject at the level of gener
ality presented here in Professor Dobrushin's analysis of the high
temperature phases, we find that the richness of the behavior exhibited
at low temperatures calls for more restricted treatments of special
models. Nevertheless, it was found that the somewhat delicate issues
of the phase structure and the critical behavior may be approached by
methods which apply to a number of systems. In particular, the picture
offered by random walk representations led to very useful insights for
the study of ferromagnetic systems, q,4d field theory, and percolation
models. (It is hoped that random surface expansions might play a some
what similar role for gauge fields and some models of spin glasses.)
Since most of the talk given at the Conference is covered by a
paper which appears in the Proceedings of the Sitges 1984 meeting, I
shall focus here on the newest results which are not discussed there.
(Their presentation here is self-contained.)
A brief outline of [1] is given in Section 2. The results which
are derived in this sequel to it deal with the phase structure of ferro
magnetic systems of one-component spin variables with the Hamiltonian
H J > 0 z
(1)
Our methods apply to systems with single spin distributions, p ( cr) do ,
which belong to the Griffiths-Simon class. This includes the following
454
standar·d examples: 1) Ising spins p (a) = o (a 2 -1) , ii) "<I> 4n variables
p (a) = exp (-!-a 4 + ba2) A > 0 , iii) the "classical n spin" p (a) =
L~no(a- k) , and iv) the uniformly distributed p(a) = I[-1, 1] • Since
at one point we need the "Gaussian domination bounds" whose deriva
tion required reflection positivity, the strongest results are re
stricted to the nearest-neighbor interactions. For such systems we
prove here the following:
a) Absence of an intermediate phase. The above one-component
systems, in d > 2 dimensions, exhibit a direct transition from the
high-temperature phase, at which the correlations decay exponentially,
to the low-temperature phase, characterized by spontaneous magnetiza
tion. (It is known that the mass vanishes as T+Tc+O). What is
ruled out is an intermediate regime, in which the correlation functions
could, for example, behave like in the Kosterlitz-Thouless phase of the
two-dimensional, two-component, plane rotor system.
b) Mean-field bound on the critical exponent of the spontaneous
magnetization. It is shown here that for T < Tc the spontaneous mag
netization satisfies:
M(T) > const.(T -T) 112 (1.2) - c
i.e., quite generally e < 1/2 • where e stands for the critical
exponent.
c) Convergent upper and lower bounds on the critical
temperature. A method is derived for the use of the values of the
magnetic susceptibility of finite systems (which are in principle
calculable) for rigorous upper and lower bounds on the critical
temperature, whose discrepancy can be made arbitrarily small by per
forming the calculation for a system of a large, yet predetermined,
size. This complements the previous upper bounds of Simon (based on
the Simon-Lieb inequality).
While the convergence is only by a fractional power law, such
bounds can be used for theoretical results, such as:
d) ~~Restricted Continuity" of Tc -- as a function of the
interaction. The restriCtion is to regimes in which the systems are uniformly
regular, in a sense introduced below. Thus, for the nearest neighbor models
in d > 2 dimensions we can prove that Tc depends continuously on the spin
distribution. Another example is discussed in Section 6.
The above results are derived from a differential inequality
which was proven, and can be understood, within the context of a random
walk representation. I shall not discuss its derivation here;
455
however in the next section I recapitulate that general scheme -- and
the content of the talk given in Kosheg.
2. A Brief Outline of Part 1.
Random walk representations have turned out to be quite useful for
the analysis of statistical mechanical models with the Hamiltonian (1.1)
and of the closely related ¢4d field theory. Such representations led
to both heuristic insights and to rigorous arguments. For the latter,
the R. W. representations were applied as means of derivation of various
relations between the physical quantities of the given model. The new
relations were used in the form of either direct inequalities, like a
bound on the truncated four-point function in terms of the two-point
function, or differential inequalities in which the bound is on the
derivative of some quantity with respect to a ("bare-") parameter (such
an inequality is used in the next section). Analogies with random walk
problems suggest relations which may be both valid and useful.
Some of the results which were arrived at by this approach (by a
number of authors) are referred to in [1]. Current ideas for further
progress call for the derivation of inequalities involving partial
derivatives of one dynamical quantity with respect to another (e.g.,
the correlation length). Such an approach has been implemented in the
analysis of an analogous Brownian motion problem mentioned later in
this section.
Two random walk representations for the Ising model and the ¢4d
field theory were presented in the works of the author [2], and Brydges,
Frohlich and Spencer [3]. While the two are different, one offering
some advantages for the strongly-coupled, and the other for the pertur
bative, regimes they share a number of basic properties (introduced in
[2,4]). These are quoted in the first part of this report, which in
cludes also a description of a random walk representation for Ising
spin systems, whose derivation is simpler than that of [2].
The relevance of our understanding of the intersection properties
of simple random walks, or the paths of independent Brownian motions,
spurred the reconsideration of some classical results. This led to a
new method for the analysis of the probability that the paths of two
independent Brownian motions are within a short distance of each other.
In recent, parallel, works of Frolich and Felder [7] (which follows an
456
earlier suggestion of Frohlich [5], and the author [6], a "renormal
ization group" differential equation was set for this quantity. It was
shown there that simple bounds on the corresponding 8 function are quite
effective, leading to a unified treatment of the probability of inter
section (which vanishes for d.:_ 4) above, below, and at the critical
dimension d = 4 • (In fact, since the linear part of the 8 function
was determined exactly, the above mentioned bounds provided new infor
mation -- on the sign and the magnitude of the •·second order" terms).
In view of the correspondence of the renormalized coupling con
stant to the probability of intersection of certain (self-interacting)
random walks, these works suggest a new approach to the study of the
¢4d field theory, and of issues like the hyperscaling in the three-dimen
sional Ising model. The sights are now set on non-perturbative bounds
on 8 functions; however, this still requires a considerable sharpening
of our methods.
3. Absence of an Intermediate Phase
We turn now to the results stated in the Introduction. The main
result is stated in this section as Theo~em 3. 1. Its proof is given in
Section 4.
We consider here one-component spin systems with
invariant Hamiltonian (1.1), for which [J[ L JO,x X
the translation
< oo The single
site spin distribution p(a) is assumed to belong to the Griffiths
Simon class [8,9], and to decay (in a) faster than the Gaussians, i.e., ba2
! p(da) e < oo for any bEJR . The G.-S. class consists of spin distributions for which the
variable a can be represented by means of a sum of ferromagnetically
coupled Ising spins, or a distributional limit of such sums. Included
are some of the most frequently referred to one-component spin variables
(some of them continuous) which are listed in the Introduction.
In particular, the following analysis applies to the standard Ising
models. We shall later require another, dynamical, hypothesis, and
prove it for the nearest neighbor models in d > 2 dimensions.
i) Preliminaries
Let me first describe the problem and some preliminary results,
most of them well known. In general, the spins described above (which
may be represented as "block variables") inherit most of the properties
which were initially derived for Ising spins; like various
457
"dimensionally balanced" correlation inequalities. Likewise, the
Lee-Yang theory [10] applies and shows that phase transitions can occur
only at zero magnetic field (h = 0) •
For a finite system, in a region AL = [-L,L]dC zd with periodic
boundary conditions, we denote by XL= xL(a) the magnetic suscepti
bility at h = 0 :
where (f(cr))L = tr e-aH f(cr)/tr e-aH • a is the inverse temperature,
"tr" stands for r II p(crx) dcrx , and "periodic boundary conditions" xEAL
means that the interaction J is replaced by L d Jx,y+2Ln • nEZ
The corresponding
denoted by ( cro cr ) f. b. XL
quantities for the free boundary conditions are
and XLf.b.(~ XL). At the infinite volume
limit, L ~ ~ , we drop the subscript L •
The f.b. correlation functions are monotone non-decreasing as a
function of L, and thus converge to a limit. At high temperatures
(= small a) the magnetic susceptibility at h = 0 , / • b • = L (cr0 cr ) , X X
is finite and the following result applies.
PJtopo-6-Lt.ton 3.1: 16 xf.b. < ~ , 6011. a.n .<.n-teJr.a.c.:Uon wLth h = 0 and IJI = L J0 < ~ • .the.n ,y
(3.2)
and .<.6, 6wr..the.JunoJte., .the. .<.n-te.Jta.e.tton de.ea.y-6 e.xpone.n:Ua.Uy, .the.n .the. Urn.i.:Ung eo!Vl.e.la..tto n 6u.ne.tto n -6 a..tt-6 fr{.e.-6 :
(3.3)
wLth a. -6tJUcfty pa-6-i.:ti.ve. "maM" m(B)
For finite range models this follows easily from Simon's analysis
[11]. The proof of the generalization stated here is in the Appendix.
The spontaneous magnetization M(a) is defined as:
458
(3.4)
The limit exists and is independent of the boundary conditions since
e(a0 ) = aP(e,h)/ah where P(e,h) (the bulk-free energy) is a convex function of h •
The non-vanishing of M(e) at h = 0 is the manifestation of a
spontaneous symmetry breaking. This phenomenon occurs at low tempera
tures (for d ~ 2) , and is implied by the long-range order, which refers
to the non-vanishing of (a0 ax) for lx I +.. • The latter does not
occur in the high temperature phase described in Proposition 2.1,
however, the stronger statement is also true.
PJtOpo-6-Uion 3. 2: 16 x <"' , a.6 in PJtopo-6-Uion 1, then
M<e> = o (3.5)
The proof (which may be well known) is given in Appendix II.
Following is a general criterion for the opposite phase.
lim sup xL(e)/IALI = D > o (3.6) L+oo
:then theJte .i.-6 both a non-vani-6h.i.ng -6pontaneou& magne:tLzation:
M(e) > n112 (3.7)
and iong-Jtange-oJtdeJt, in :the Mn-6e that any .e..tmi;Ung 6unc:Uon de6.{.ned with p. b. c.., and a -6ub-6equenc.e o6 volumu, -6at.i.-6 6.{.u
(3.8)
The above criterion for the spontaneous magnetization was
derived by Griffiths [12], and (3.8) is obtained by the Schwarz
inequality. For the understanding of the above statements it is use
ful to express them in terms of the quantities .SA= L a /IAI xEA x
In the absence of more information, one could define three
critical temperatures (we regard T and e-l as synonymous):
T h. t. = inHT I x< e) <"' } (h. t. -- for the high temperature phase;
if J ~ecays exponentially, then so does (a0a ) for T > T h · t · • ) O,x x c
T s.m. = sup{TiM(8) > 0} c
459
T l.r.o. c
sup{TI(3.8) holds with some D > 0}
From what is said above
T l.r.o. < T s.m. < T h.t. c c c
(3. 9)
(3 .10)
Examples where the last is a strict inequality, are provided by the two
component plane rotor and its approximating discrete clock models in
two dimensions. In particular, the latter have an intermediate
Kosterlitz-Thouless-type phase at which M = 0 -- yet the correlations
decay only by a power law [13,14,15].
Our main result is that for a general class of one-component
systems in d > 2 dimensions, all the critical temperatures in (3.10)
are equal.
ii) The Regularity Condition
Since our analysis leads also to a re-derivation of the existence
of a phase transition, one might expect that it should require some
additional hypothesis which, in particular, should rule out the one
dimensional finite-range systems. We shall now introduce such a
condition.
Let us denote by B1 the sum:
B = L
(3.11)
where (--) is with the periodic boundary conditions. Removing from
this "bubble diagram" its "zero momentum component" we obtain the
quantity
B T L
where the truncated function is defined as
(3.12)
(3.13)
460
Since o is not restricted to be an Ising spin, its scale, as
well as those of X and B , are quite arbitrary at the level of gener
ality considered here. However, the combinations
(3.14)
where !JI = L J0 , are invariant under the simple "field strength X ,x renormalization."
Vep.,Lrz,.Uton: We MIJ that the -61J-6.tem -W negu£a1t -i6 6oJt aLe. 8 and 1
, T , 2 , 1 + B1 _.'5. x1 f(x1 ) (3.15)
f f(s) ds < oo (3.16) 1
Remarks:
i) It is easy to see that
be assumed that
l/s2 _2 f(s) < 1 + l/s2
Hence, it may always
(3.17)
ii) In the one-dimensional Ising model at the infinite volume
limit (where (o0 ox) = e-mlxl) , B::::: x/2 . Thus (3.15) can hold
-1 there only with f(s) ~ (2s) In this case (3.16) fails, but barely
so. One may expect that above its lower critical dimension, the model
is regular. We shall see that this is the case in dimensions d > 2 •
iii) For another insight into the notion of regularity, let us
note that if (3.16) holds, and f(•) is monotone, then f(s) =s-1 o(l).
An innnediate implication is that when a regular system exhibits "long
range order," i.e., (3.6) holds, then in the 12 sense the correlation
function strongly condenses into its zero momentum component. The ratio
B1T/ (x12/]1\l> is bounded by l'\l!x1 o(l}=D-lo(l) which vanishes
as 1 + oo •
461
iv) For systems endowed with reflection positivity, regularity is
essentially implied, up to considerations of discretization effects, by
the familiar condition (which has already played a key role in proofs
of the existence of a phase transition [16, 17]):
where
I dp Elp)
[-rr,rr]d
E(p)
< "" (3.18)
(3.19)
(3.18) and (3.16) are related via the "Gaussian domination" (or
"equipartition") bound of Fr15hlich, Simon and Spencer [16], which states
that in RP systems
obeys
d at the dual momenta p E (II/L)~ n [ -rr ,rr) d
By the Plancherel identity:
B T = ( 2rr) -d L
(L)
J d dp [ -rr ,rr)
p#O
(3. 20)
(3.21)
(3.22)
(L)
where " J dp " represents the "Riemann sum" over the discrete set
of momenta specified above, with the weights (rr /L) d , excluding p = 0
Combining (3.21) with the obvious inequality GL(p) ~XL , and
using (3.22), one gets
(3.23)
with the functions:
462
II I (L} dpll mini (2EI/pl)s)2 ' 11
[ -71. 71) d piO
for which
J 00
f(L) (s) ds _2 1 + (271) -d
1
II
hl E(p)
(3.24)
(3. 25)
In particular, we can establish regularity for the following im
portant class of models.
P!topo~.>Ltion 3. 4: In d > 2 citmen~.>ion~.>, non-degenella-te nea.Jtel.>:t
nughbO!t modeu [ 6M whic_h J I 0 iff I x-y I = 1 ) • aJLe JtegutaJL. x,y
Proof: These models are reflection positive [16], and the above discus
sion applies, with
E(p)
if J.=l ~
(3. 27)
Furthermore, the discretization effects in (3.24) and (3.25) are easy
to control here. (3.24) -- for whose integration it is convenient to
use: min{a- 2 ,1} ::_ 2J(a2 +1] , implies that (3.15) is satisfied with
c s-min{d/2,2} d I 4
f(s) ( 3. 28) -2
c s [l+ln+ s] d 4
This function is integrable for d > 2 , as suggested already by
(3.25) .•
Other examples of reflection positive interactions are [17]
and (3.29)
463
in d = 1 dimension. Using the analysis of Ref. [17] one can show that
these are regular for 1 < a. < 2 •
It should be emphasized that (3.18) is only a sufficient condition
(for RP interactions). I would conjecture that finite-range one-com~
ponent systems, of the type considered here, are regular even in d = 2
dimensions -- where (3.18) fails.
Statement of the Main Result
TheOJtem 3. 1 : In any Jteg ultvL <> y<> .tem, o 6 .the .type de<> c!Ub ed a.t .the
beg..inn.i.ng o6 :tkU. <>ection (w.Lth 6eJtJtOmagne.tic. pa.iJr. in.teJtaction
jJj <"", and <>pin<> in .the GJti..66Uh<>-Simon cl.a&<>),
T l.r.o. = T s.m. = T h.t. c c c
(3.30)
(3.31)
Whelte llcm.f. = (jJj(cr2 )0)-l -<-<> .the cJU;t;[c.a.f. value in a mean-6-i_e.td
appltouma.tion.
obey<>
T<T c
.the <>ponta.neou& magnetiza.:Uon
and, moJteoveJt, .thelte-<-<> iong-Jtange-oJtdeJt and (3.8) hold<> wi.th
(3.32)
D = (lljJj)-l ln(ll/ll) (who<>e <>qua.Jte Mo.t-<-<> de<>c!Ubed in .the R.H.S. c
06 (3.32)).
FoJt T > Tc .the c.ondi..tion in PMpo<>Won 1 -<-<> <>a.ti<>Med,
and
(3.33)
wi.th .the nunction g(t) de6ined by .the 6oiiowing equation
t = r J
f(s) ds (3. 34)
g (t)
464
4. Proof of the Main Result
Our analysis is based on the following inequality, derived in
Aizenman and Graham [18] (on p. 273) for the class of systems described
at the beginning of the preceding section, at zero magnetic field
(h = 0).
(4.1)
In terms of the dimensionless quantities introduced in (3.14),
we obtain simply:
> (4. 2)
(4.2) supplements the upper bound
(4.3)
which follows from the Lebowitz inequality [19].
The bounds (4.1) and (4.3) have already been used for the study of
the high temperature (= low 8) phase. The new idea which is applied
here is to analyze the transition from the high to the low temperature
phases (defined within the infinite volume limit) without first re
moving the "finite size cutoff." Since XL ( 8) are real analytic
functions of S , one may wonder how is the phase transition manifested
in a finite system. We shall see a clear answer to that question.
Proof of Theorem 3.1
Combining (4.2) with (3.12) and (3.15) we get:
> 1 (4.4)
465
For an intuitive grasp of the situation, let us note that if SL
is defined by the condition
then
IIALI/2
[2f(xL)J-1
(4.5)
(4.6)
Integrating this inequality in both directions from SL one can see
that for S>SL' xL(I3-SL) _?_ (13-SL)IALI/2 i.e., X acquires the
size of the volume; whereas for 13 < aL' XL (13) is less than a finite
valued function of the difference (13- SL), which is independent of L. (For
the integration of (4. 6) into the regime 13 < SL it is convenient to view
13 as a function of XL .) We shall now make this argument in a
slightly more efficient way, and also establish some uniform bounds on
the value of SL (and other finite-volume approximants of 13c ).
i) Upper and Lower Bounds on xL(I3) •
For each L , XL is a smooth and strictly increasing function
of 13 This allows us to reverse their functional dependence, and
express (4.6) in the following form:
Integrating it we learn that for each pair 0 < 131 < 13 2
xL c 62) - xL c sl)
I ALl +
I
XL (132)
I f(s) ds
f(s) ds (4.8)
Let SL be some value chosen according to a rule which will be
466
specified below. Regardless of the choice of f\ , (4. 8) implies
the following pair of hounds:
for (4.9)
and
for S < f\ ( 4 .10)
where g(•) is the function defined by (3.24), the E and 6 are:
EL f f(s) ds
XL (BL)
(4.11)
and (4.10) is proven via the inequality
(4.12)
ii) The Choice of S1
We shall now choose s1 so that as L->- oo both
( 4 .13)
By the integrability of f(•), the former is equivalent to:
whereas the second requirement is
( 4 .15)
467
TO accomplish both, we choose BL by the condition
(4 .16)
(One can also take SL, defined by (4.5).)
iii) Bounds on BL
Let us now establish some uniform upper and lower bounds on BL ,
which are needed for a compactness argument.
For a lower bound we use a familiar argument (see references in
[2oD based on <4.2). Re-writing it as lax~1 t ael 2. IJI • and
integrating from B = 0 , one gets
-1 -1 I I XL (13) ~ XL (0) - 13 J
Since
we have
To get an upper bound, let us assume
with B1 = e2 and e2 = a~· f. , implies
B < 13m. f. L- c
(4.17)
(4.18)
(4.19)
Then (4.8),
(4.20)
However, using the Griffiths inequality in the obvious way,
1 (4.21)
Substituting this in (4.20), we get
468
(4.22)
iv) Conclusion
The uniform bounds (4.19) and (4.22) imply that there is a sub
sequence of volumes for which ~L converges to a limit, which clearly
obeys (3.31). Restricting L to such a subsequence, we define
(4. 23)
(Our conclusions imply that ~c is, in fact, independent of the sub
sequence, i.e., the limit exists in the usual sense.)
Since ~L were chosen so that EL , o1 + 0 , and the function
g(•) is continuous, (4.9) and (4.10) imply that
a) for
lim sup 1->-oo
b) for
~ > ~ c
~ < ~ c
13-13 [ (~-13 )] .::_ ln :c = ~ c c 1 + 0 ~ c c
Applying now Propositions 2.1 and 2.3, we see that for
(4.24)
(4.25)
13>13 c there is spontaneous magnetization and long-range order, whereas for
~ > 13 c the correlations decay exponentially. Combined with Proposition
2.2, this leads to the claimed equality of the critical points, (3.30).
The bounds established above imply all the inequalities stated in
Theorem 2.1, with the exception of the lower bound in (3.33). That
however is a known consequence of (4.2), derived by integrating it
down from Sc (as opposed to (4.17)). A careful argument is presented
in Ref. [21]. •
Notice that the only characteristics of ~L which were used are
those described by (4.14) and (4.15). Since the critical point to
which s1 converge has an independent characterization, we can make
469
the following observation:
CoJtoUalty 4.1: Any .6equenc.e ofi invCMe :tempe!taWJLC-6, f\ , 60Jt w!Uc.h
both
lim XL (SL) = oo L+oo
(4. 26)
and
0 (4.27)
An interesting question which has some bearing on the theory of
finite-size scaling is to determine the actual behavior of xL(Bc) .
For example, the inequality (6.3) of Ref. [2] implies that if Ref. [22]
is correct, then
(4. 2 8)
i.e., our BL are lower bounds:
(4.29)
5. Convergent Upper and Lower Bounds on Be
In this section it is shown how finite size calculations may be
used for rigorous bounds on the critical temperature. The first result
of this type was the convergent sequence of lower bounds (on Be) of
Simon [11]. We supplement it here with upper bounds. Furthermore, we
complete the result stated in Corollary 4.1, where a sufficiency cri
terion was established for the convergence of finite volume approximants
to Be , by presenting explicit error bounds on Be- SL . Although the
convergence is not very fast, such results are useful for theoretical
purposes, as is demonstrated in the next section.
The new upper bounds are expressed by the following inequality:
P!topo.6ilion 5.1: In a JtegutM .6y.6:tem ofi :the :type dCJ..c.JribedinSec.Uon 3,
470
6oft any S and K ,
(5.1)
In order to extract from (5.1) good numerical bounds, one should
f . d f 1 K . 1 1 f Q for wh•ch x'Kf.b. ( 0 ) ~n , or as arge as pract~ca , va ues o ~ • ~
is very large, yet small, compared with 1~1
Proof: The inequality is obviously true for S ~ S c For S _::_ S , the
upper bound (3.33) --in its stronger version of (4.25), implies that
f(s) ds
< J f(s) ds < f(s) ds (5. 2)
x< s>
where in the first step we used the definition of g(•) ((3.34)), and
in the last one the Griffiths inequality. Exponentiating (5.2) one
gets (5.1). • Our lower bounds are based on an extension of Simon's result [11].
To formulate it we define the mean range of an interaction as
with l!xll max { lx.l} , for x = (x1 , ••• , xd) • l<i<d ~
We shall now assume that R(J) <"" .
(5.3)
Pftopo . .6Wovz 5.2: In a .6lf!.dem on :the above type (Le., .6yXn.6 -Lvz the S.G.
claM and the Ham-LUon-Lan I 1. 1) wdh h = 0 and I J I <"" ) , -L6 6oft .60me
Sand L>O
(5.4)
471
then the c.ondUion o6 P!topoJ.>-i.tion 3. 1 -i.-6 J.>a.:ti..!.>Med, L e., x(8) <"',and
8 < 8 c (5.5)
AUeJtnativety J.>tated, at the th!teJ.>hotd o6 the h-igh tempeJtatwte phMe,
nOll. eac.h 1 > 0
The Proof of Proposition 5.2 is given in Appendix I. (The
restriction imposed here on the spin distribution may be relaxed.)
Combining it with the previous result, we get:
(5.6)
CoMil.a.Juj 5. 1: Let B~ be deMned by the 6oUow-i.ng gene!talization on Si..mon' !.> c.ondUion [11)
1 / R(J)
Then, -in a Jteguta.Jr. J.>yJ.>te.m, 6oJt each
!"' f(s) ds
8s < 8 < 8s 1/R(J) 1-c-1e
(5. 7)
1>0
(5. B)
Proposition 3.4 shows that for the nearest neighbor models, one may
-min{1, 2} use f(s) = C s [1 + I[d=4)jlns !] In particular, in
d = 3 dimensions, we have:
Of course, better upper bounds may be obtained by using (5.1)
with higher values of 8 than 8~ ; minimizing empirically the right
hand side of (5.1). Combining (5.1) with (4.8), where we take s
81 = 8 82 = 81 , we have the following error bounds for such estimates:
CoJtoUa.Jty 5.2: In any Jte.guta.Jt J.>yJ.>te.m o6 the. type. deJ.>c.Jt-i.bed above., 6oJt
any 8 and L :
472
13 expl-[x1 (l3) + J f(s) Jl2. 13 < 13 IALI L/R(J) J c
(5 .10)
Summary: Each set of values of {13, x1 (13), x~·b.c. (13)} yields both
upper and lower bounds on 13c However, the above results may be best
utilized by a separate use of (5.1) and (5.8). Before engaging in such
a calculation, one would like to know the accuracy it would yield with
a given L . Our a priori over-estimate for d = 3 is O(L-l/ 2) •
This may be slightly too pessimistic, although the inequality (4.4)
suggests that the actual discrepancy between the bounds would still not
decay faster than a power of L . For a better method, one should have
a more complete theory of the finite-size scaling.
In the next section, we shall see examples of theoretical
applications of the bounds derived here.
6. Convergence of the Critical Temperatures of Finite-Width Slabs
We shall now consider ferromagnetic systems with a fixed coupling
J , on the infinite slabs : [-1, L] X zd with periodic boundary x,y
conditions in the first component of the position vector.
If d > 2 , then for each L there is a positive critical tempera-
ture whose inverse we denote by Sc(d; L) One may expect that
lim 13c(d; L) = 13c(d+l) L+oo
(6.1)
where the right-hand side is the critical point for the system on
Zd+l However, (6.1) is not totally obvious. In particular, it
fails for d = l , since for each L , while
Nevertheless, (6.1) "should" be true once d is above the lower
critical dimension.
<co
The conjecture that an analog of (6.1) (with d= 2) holds for the
473
bond percolation model, was used in Ref. [23] for the derivation of
rigorous block-variable techniques which allowed one to study a random
surface problem up to its critical point.
Ptropo~.>-ition 6. 7: La tic (d; L1 , .•. , ~) denote the C)ut;icai -inveMe
tempe!I.atWLe-6 6oJt 6eMomagvze.Uc -61j-6tern6 ovz the "th-ici2 J.J£ab-6" k d ~[-Li, Li]@Z , wUh the neaJte~.>t ne-ighboJt inteJtaction J 0 , 2 =olzl,l
and a J.J,£vzg£e-J.Jp1vz fu:t!tibu.tiovz in :the G.S. daM. I 6 d > 3 , :then
(6.2)
60Jt eve.Jty 61xed k •
The proof is by a simple application of the bounds of the pre
ceding section, and the uniform regularity of the slab systems, which
is expressed in the next Lemma.
Lemma 6. 7 : The!te eUJ.J.t-6 a con-6tavzt C <"" J.Ju<:h that 60Jt each d > 3
k, L and L1 , ... , Lk ~ L , the above -61j-6tem -ivz the 6-in,ite vo£ume
k 0[-L., L.] 0 [-L, L]d MW6ie.J.J (3.75) wdh f(s) = Cs-112 1 ~ ~
The proof is left as an exercise in the technique used in the
Proof of Proposition 3.4. It requires also some fairly simple bounds
on Riemann sums. The value of C to which we refer above, is the one
which ( 3. 28) yields for d = 3 .
Proof of Proposition 6.1
The main idea of the proof is to approximate the critical tempera
tures of both the thick d-dimensional slabs, and the d + k dimensional ' -1
system by the temperature SL defined by (4 .16) for the d + k dimen-
sional cube [-L, L]d+k , where
(6. 3)
Since the bounds (5.10) apply to all those systems with the same
function, described in Lemma 6.1, we learn that
with
474
- d+k /2 exp i f ( s) ds + ( 2L) ( ) + [
00
( 2L)(d+k) /2 J f(s)
L
exp[2C(2L)-(d+k)/4 + (2L)-(d+k)/ 2 + 2CL-l/ 2 J
(6.4)
(6.5)
(6.5) not only implies (6.2), but it also shows that the convergence
is not slower than O(L-l/ 2) . •
Remark: A similar argument shows that in d > 3 c!imensions, S c
depends continuously on p(•) .
Appendix I: Extension of Simon's High Temperature Criterion to
Interactions of Unbounded Range
In Propositions 3.1 and 5.2, reference was made to an extension
of Simon's analysis [11] in which the condition that the interaction
be of finite range is replaced by the weaker requirement of exponential
decay. In this Appendix we prove the necessary results. An interesting
outcome is that the relevant length scale is not the minimal distance be
yond which Jx vanishes, but rather the mean range R(J) which is the
first moment of l!xll with respect to J --defined by (5.3). X
Our starting point is the following inequality:
(cr0crx)- (cr0crx)' < L (cr0cry)' SJy,z (crycrz) (I.l)
{y,z}EB
in which ( -} refers to the correlation functions in some finite sys
tem and (-}' denotes the correlations in a weakened system, which is
obtained by setting Jy,z to zero on a collection of pairs {y,z}EB
For the models considered here, (I.l) is a consequence of the
most elementary of the techniques used in Refs. [1, 18]. Such an
inequality was derived also for a somewhat different class of spins,
475
for which p (o) -V(o2) e with a convex function V(•) [24].
The primes in (I.l) represent a significant improvement of the
inequality obtained by Simon [11] on the basis of the Lebowitz inequal
ity [19]. (The effect is somewhat similar to that of Lieb's strength
ening of the original Simon "site" inequality [11,25]).
We shall now apply (I.l) to compare the correlation function for
a finite volume system with the periodic boundary conditions on
A C zd with the one obtained with the free boundary condition in L ~C A1 The p.b. interaction, in A1 , is given by the couplings
/L) y,z L d J
nez y,z+2Ln ; whereas for the f.b. J =I[y,zEAk]J • y,z y,z
(I[-] denotes here the {0,1} valued indicator function.)
An immediate implication of (I.l) is that for each L>K
+ L nezd \{0}
For a clearer view of this inequality, let us sum it over xe 11 1
One gets
(I. 2)
(I. 3)
(I.2) is the basis for the considerations of this section. The
results dealing with the exponential decay of·the correlation function in
the high temperature phase are restricted to exponentially decaying
interactions, i.e., those for which there is some A> 0 such that
L J e'-lxl < X O,x
(I.4)
476
A simple way to obtain exponential decay is hy considering the
weighted sums,
e ll[x[
and -- which are defined analogously.
Let us also define
Notice that if (I.4) is satisfied, than
and is continuous in the limit 11 +0
Lemma I. 1: FOIL eac.h L > K and ll.:':. 0
is finite for
/ 11 ) < 1 , ;t}um fio!t att L < K K
(I. 5)
(I. 6)
(I. 7)
(I.8)
Proof: To prove (I. 7) we sum (I. 2) over xE AL , with the weight
ll[x[ ll[z[ 11[x-z[ e , which for the R.H.S. is replaced by e e
(.:=:_ ell[xl).
(I.8) is an obvious consequence of (I.7), assuming (as an observant
reader might note) that XL< oo • That, however, is implied by the
stability assumption we made about p(a) ·•
We are now ready to prove Proposition 3.1. Its first claim, (3.2),
requires only the finiteness of [J[. In the corresponding part of the
proof we use the above lemma only with 11 = 0
Proof of Proposition 3.1: First let us note that if f. b.
X <oo , then
as K + co (I. 9)
477
That follows, by the monotone convergence theorem, from the bound:
l: d y,zE Z
(1.10)
Assumenowthat Xf.b.<oo. Letting K bethefirstvaluefor
which yiO) _2 1/2 , we learn from (1.8) that for each L > K
X < 2 f.b. < 2xf.b. L- XK
Substituting (I.ll) in (1.7) we get
(I.ll)
f. b. < 2xf.b. Y(O) _,_ o , XL - XL - L
as L-+oo
(1.12)
(1.12) clearly implies both of the statements made in (3.2).
Let us now add the assumption that the interaction satisfies (I.4),
and nrove that the correlation function decays exponentially. First we
note that by the continuity of Yi~) , there is some ~ > 0 for which,
with the above value of K :
(!.13)
Using it in (I.8), one gets the following uniform bound, which holds
for all x , and L > K
( !.14)
This clearly proves (3.3).•
The proof of the following result shows that the value of K used
above is not greater than 2xf.b. R(J)
478
Lemma I. 2: I6 f,of[ Mme M '
f. b. XM < MIR(J) ( 1.15)
then
/0) < 1 ( 1.16) K
6of[ Mme K E [1 , M]
Proof: Averaging over the interval [1, ... , M], we obtain
(where use was made of the Griffiths inequality)
< l: y,z
< a 0a >f. b • BJ II z- Y II I M = x~ · b • B L J 0 II x II I M y M y,z x ,x
X~.b. R(J) IM ( 1.17)
If (I.l5) holds, then the R.H.S. of (I.l7) is less than one, and there
fore, so is y~O) for at least one value of K in [1 , M]. •
Proof of Proposition 5.2: The condition (5.5) is identical to (I.l5).
If it is satisfied for some L (here denoted by H ) , then, by Lemma I.2,
(I.l6) holds for some KE [1, L], and thus, by (I.8), /.b. <oo ·•
Appendix II: Vanishing of the Spontaneous Magnetization for T > Th. t. c
The following argument proves that M(T) = 0 at the high tempera
ture ~base which is characterized by the finiteness of X
~roo~of Proposition 3.2: The free energy mentioned after (3.4) may be
represented as
479
P(8,h)
where S is the magnetization per site:
S=-1- L lA I xEAL
cr X
( 1!.1)
(11.2)
The Newman inequality [26], applied to the terms obtained by the power
expansion of the exponential, leads to the following bound:
< L h2n lAin x~ (2n-1)(2n-3) ... l/(2n)! n=O
(II. 3)
Substituting (11.3) in (11.2), we get
(0 2_) P(8, h) - P(8, 0) ::_ h 2 x/2 (11.4)
(where the first bound is by the Jensen inequality.) Therefore, as
long as x(B) < oo ,
M(S) lim [P(S, h) - P(S, 0) ]/h 0 ·• (II.S) h+O+
Acknowledgements: I would like to thank here K. Binder for a very
stimulating discussion of finite-size scaling theories. It is also
a pleasure to thank D. Jasnow and B. Widom for useful comments, the
Aspen Center for Physics for providing the uplifting environment in
which this work was done, and the organizers of the Conference in
KOszeg for a very enjoyable and stimulating meeting.
480
* J.S. Guggenheim Foundation Fellow. On leave from Departments of
Mathematics and Physics, Rutgers University. Supported by N.S.F.
Grant PHY-8301493 AOl.
References
[1] M. Aizenman, "Rigorous Studies of Critical Behavior," in ~plications of Field Theory in Statistical Mechanics, Proceedings , Sitges 1984, L. Garrido (ed.), Springer Verlag (Lecture Notes in Physics), in press.
[2] M. Aizenman, Phys. Rev. Lett.~. 1 (1981); and Commun. Math. Phys. ~. 1 (1982).
[3] D. Brydges, J. Fr6hlich and T. Spencer, Commun. Math. Phys. 83, 123 (1982).
[4] J. Frohlich, Nucl. Phys. B200 [FS 4], 281 (1982).
[5] J. Frohlich, in Progress in Gauge Theory, Carg~se 1983, G. 't Hooft et al. (eds.), Plenum (New York), in press.
[6] M. Aizenman, "The Intersection of Brownian Paths as a Case Study of a Renormalization Group Method for Quantum Field Theory," to appear in Commun. Math. Phys.
[7] G. Felder and J. Frohlich, "Intersection Properties of Simple Random Walks: a Renormalization Group Approach," to appear in Commun. Hath. Phys.
[8] R. Griffiths, J. Math. Phys. 10, 1559 (1969).
[9] B. Simon and R. Griffiths, Co·mmun. Math. Phys. 33, 145 (1973).
[10]
[11]
[12]
T.D. Lee and C.N. Yang,
B. Simon, Commun. Math.
R. Griffiths, Phys. Rev.
Phys. Rev. 83, 410 (1952).
Phys. 77, 111 (1980).
152, 2l,Q (1966).
[13] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6, 1181 (1973).
[14] S. Elitzur, R. Pearson and J. Shigemitsu, Phys. Rev. Dl9, 3698 (1979).
[15] J. Frohlich and T. Spencer, Commun. Math. Phys. 81, 527 (1981).
[16] J. Frohlich, B. Simon and T. Spencer, Commun. Math. Phys. 50, 79 (1976) 0
[17] J. Frohlich, R. Israel, E.H. Lieb, B. Simon, Commun. Math. Phys. 62' 1 (1978) 0
[18] M. Aizenman and R. Graham, Nucl. Phys. B225 [FS 9], 261 (1983).
[19] J.L. Lebowitz, Commun. Math. Phys. ~. 87 (1974).
[20] M. Aizenman, in Scaling and Self-Similarity (Renormalization in Statistical Mechanics and Dynamics), J. Frohlich, ed., Birkhauser (Boston, 1983).
[21] M. Aizenman and C.M. Newman, J. Stat. Phys. 1&_, 107 (1984).
481
[22] K. Binder, M. Nauberg, V. Privman and A.P. Young, "Finite-Size Tests of Hyperscaling," UCSC preprint, 1984.
[23] M. Aizenman, J.T. Chayes, L. Chayes, J. Frohlich and L. Russo, Commun. Math. Phys. 21• 19 (1983).
[24] D. Brydges, in Gauge Theories: Fundamental Interactions and Rigorous Results, P. Dita, V. Georgescu and R. Purice (eds.), Birkh~user (Boston, 1982).
[25] E.H. Lieb, Commun. Math. Phys. 22• 127 (1980).
[26] C.M. Newman, Z. Wahr. 12• 75 (1975).
PROGRESS IN PHYSICS Already Published
PPH 1 Iterated Maps on the Interval as Dynamical Systems Pierre Collet and Jean-Pierre Eckmann ISBN 3-7643-3026-0 256 pages, hardcover
PPH2 Vortices and Monopoles, Structure of Static Gauge Theories Arthur Jaffe and Clifford Taubes ISBN 3-7643-3025-2 294 pages, hardcover
PPH 3 Mathematics and Physics Yu. I. Manin ISBN 3-7643-3027-9 112 pages, hardcover
PPH 4 Lectures on Lepton Nucleon Scattering and Quantum Chromodynamics W. B. Atwood, J. D. Bjorken, S. J. Brodsky, and R. Stroynowski ISBN 3-7643-3079-1 574 pages, hardcover
PPH 5 Gauge Theories: Fundamental Interactions and Rigorous Results P. Dita, V. Georgescu, R. Purice, editors ISBN 3-7643-3095-3 406 pages, hardcover
PPH6 Third Workshop on Grand Unification, 1982 P. H. Frampton, S. L. Glashow, H. van Dam, editors ISBN 3-7643-3105-4 388 pages, hardcover
PPH7 Scaling and SelfSimilarity in Physics (Renormalization in Statistical Mechanics and Dynamics) J. Frolich, editor ISBN 3-7643-3168-2 ISBN 0-8176-3168-2 440 pages, hardcover
PPH 8 Workshop on NonPerturbative Quantum Chromodynamics K. A. Milton, M. A. Samuel, editors ISBN 3-7643-3127-5 ISBN 0-8176-3127-5 284 pages, hardcover
PPH9 Fourth Workshop on Grand Unification H. A. Weldon, P. Langacker, P. J. Steinhardt, editors ISBN 3-7643-3169-0 ISBN 0-8176-3169-0 415 pages, hardcover