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Chaotic Dynamics and Transport in Classical andQuantum Systems
NATO Science SeriesA Series presenting the results of scientific meetings supported under the NATO ScienceProgramme.
The Series is published by IOS Press, Amsterdam, and Kluwer Academic Publishers in conjunctionwith the NATO Scientific Affairs Division
Sub-Series
I. Life and Behavioural Sciences IOS PressII. Mathematics, Physics and Chemistry Kluwer Academic PublishersIII. Computer and Systems Science IOS PressIV. Earth and Environmental Sciences Kluwer Academic PublishersV. Science and Technology Policy IOS Press
The NATO Science Series continues the series of books published formerly as the NATO ASI Series.
The NATO Science Programme offers support for collaboration in civil science between scientists ofcountries of the Euro-Atlantic Partnership Council.The types of scientific meeting generally supportedare “Advanced Study Institutes” and “Advanced Research Workshops”, although other types ofmeeting are supported from time to time. The NATO Science Series collects together the results ofthese meetings.The meetings are co-organized bij scientists from NATO countries and scientists fromNATO’s Partner countries – countries of the CIS and Central and Eastern Europe.
Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advancesin a field.Advanced Research Workshops are expert meetings aimed at critical assessment of a field, andidentification of directions for future action.
As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO ScienceSeries has been re-organised and there are currently Five Sub-series as noted above. Please consultthe following web sites for information on previous volumes published in the Series, as well as details ofearlier Sub-series.
http://www.nato.int/sciencehttp://www.wkap.nlhttp://www.iospress.nlhttp://www.wtv-books.de/nato-pco.htm
Series II: Mathematics, Physics and Chemistry – Vol. 182
Chaotic Dynamics and Transport inClassical and Quantum Systems
edited by
P. ColletEcole Polytechnique,Paris, France
M. CourbageUniversité Paris 7-Denis Diderot,France
S. MétensUniversité Paris 7-Denis Diderot,France
A. NeishtadtSpace Research Institute,Moscow, Russia
and
G. ZaslavskyNew-York University,New York, NY, U.S.A.
Kluwer Academic Publishers
Dordrecht / Boston / London
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Study Institute onInternational Summer School on Chaotic Dynamics and Transport in Classical andQuantum SystemsCargèse, Corsica18–30 August 2003
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 1-4020-2946-2 (PB)ISBN 1-4020-2945-4 (HB)ISBN 1-4020-2947-0 (e-book)
Published by Kluwer Academic Publishers,P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
Sold and distributed in North, Central and South Americaby Kluwer Academic Publishers,101 Philip Drive, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributedby Kluwer Academic Publishers,P.O. Box 322, 3300 AH Dordrecht, The Netherlands.
Printed on acid-free paper
All Rights Reserved
No part of this work may be reproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical, photocopying, microfilming,recording or otherwise, without written permission from the Publisher, with the exceptionof any material supplied specifically for the purpose of being enteredand executed on a computer system, for exclusive use by the purchaser of the work.
Printed in the Netherlands.
© 2005 Kluwer Academic Publishers
Contents
Preface vii
Part I : Theory
P. Collet : A SHORT ERGODIC THEORY REFRESHER 1
15
35
G.M. Zaslavsky, V. Afraimovich: WORKING WITH COMPLEXITY
87
P. Gaspard : DYNAMICAL SYSTEMS THEORY OF IRREVERSIBILITY 107
Walter T. Strunz: ASPECTS OF OPEN QUANTUM SYSTEM DYNAMICS 159
Eli Shlizerman, Vered Rom Kedar: ENERGY SURFACES AND HIERARCHIESOF BIFURCATIONS. 189
Part II : Applications
Ariel Kaplan, Mikkel Andersen, Nir Friedman and Nir Davidson:ATOM-OPTICS BILLIARDS 239
Fereydoon Family, C. Miguel Arizmendi, Hilda A. Larrondo: CONTROL OFCHAOS AND SEPARATION OF PARTICLES IN INERTIA RATCHETS 269
Participants ix
FUNCTIONS 73
v
M. Courbage: NOTES ON SPECTRAL THEORY, MIXING AND TRANSPORT
Valentin Affraimovich, Lev Glebsky: COMPLEXITY, FRACTAL DIMENSIONS AND TOPOLOGICAL ENTROPY IN DYNAMICAL SYSTEMS
Giovanni Gallavotti : SRB DISTRIBUTION FOR ANOSOV MAPS
SEMICLASSICAL TRACE FORMULAE 225Monique Combescure: PHASE-SPACE SEMICLASSICAL ANALYSIS. AROUND
F. Bardou: FRACTAL TIME RANDOM WALK AND SUBRECOIL LASERCOOLING CONSIDERED AS RENEWAL PROCESSES WITH INFINITEMEAN WAITING TIMES 281
Xavier Leoncini, Olivier Agullo, Sadruddin Benkadda, George M. Zaslavsky:ANOMALOUS TRANSPORT IN TWO-DIMENSIONAL PLASMA
Edward Ott,Paul So,Ernest Barreto,Thomas Antonsen: THE ONSET OFSYNCHRONISM IN GLOBALLY COUPLED ENSEMBLES OF CHAOTIC
SYSTEMS WITH PHASE SPACE STRUCTURES
S.V.Prants: HAMILTONIAN CHAOS AND FRACTALS IN CAVITY QUANTUM
M. Cencini,D. Vergni, A. Vulpiani: INERT AND REACTING TRANSPORT
Michael A. Zaks: ANOMALOUS TRANSPORT IN STEADY PLANE FLOWS OF
J. Le Sommer, V. Zeitlin: TRACER TRANSPORT DURING THE GEOSTROPHICADJUSTMENT IN THE EQUATORIAL OCEAN
Antonio Ponno: THE FERMI-PASTA-ULAM PROBLEM IN THE THERMODYNAMIC
A. Iomin, G.M. Zaslavsky: QUANTUM BREAKING TIME FOR CHAOTIC
vi
TURBULENCE 303
AND PERIODIC DYNAMICAL UNITS 321
333
ELECTRODYNAMICS 349
365
VISCOUS FLUIDS 401
413
LIMIT 431
Lectures 441
vii
Preface
From the 18th to the 30th August 2003 , a NATO Advanced Study Institute
(ASI) was held in Cargèse, Corsica, France. Cargèse is a nice small village situated
by the mediterranean sea and the Institut d'Etudes Scientifiques de Cargese provides∗a traditional place to organize Theoretical Physics Summer Schools and Workshops
in a closed and well equiped place.The ASI was an International Summer School* on
"Chaotic Dynamics and Transport in Classical and Quantum Systems". The main
goal of the school was to develop the mutual interaction between Physics and
Mathematics concerning statistical properties of classical and quantum dynamical
systems. Various experimental and numerical observations have shown new
phenomena of chaotic and anomalous transport, fractal structures, chaos in physics
accelerators and in cooled atoms inside atom-optics billiards, space-time chaos,
fluctuations far from equilibrium, quantum decoherence etc. New theoretical
methods have been developed in order to modelize and to understand these
phenomena (volume preserving and ergodic dynamical systems, non-equilibrium
statistical dynamics, fractional kinetics, coupled maps, space-time entropy, quantum
dissipative processes etc). The school gathered a team of specialists from several
horizons lecturing and discussing on the achievements, perspectives and open
problems (both fundamental and applied). The school, aimed at the postdoctoral
level scientists, non excluding PhD students and senior scientists, provided lectures
devoted to the following topics :
� Statistical properties of Dynamics and Ergodic Theory
� Chaos in Smooth and Hamiltonian Dynamical Systems
� Anomalous transport, fluctuations and strange kinetics
� Quantum Chaos and Quantum decoherence
� Lagrangian turbulence and fluid flows
� Particle accelerators and solar systems
More than 70 lecturers and students from 17 countries have participated to the ASl.
The school has provided optimal conditions to stimulate contacts between young
and senior scientists. All of the young scientists have also received the opportunity
to present their works and to discuss them with the lecturers during two posters
sessions that were organized during the School.
The proceedings are divided into two parts as follows:
I. Theory
This part contains the lectures given on basic concepts and tools of modern
dynamical systems theory and their physical implications. Concepts of ergodicity
and mixing, complexity and entropy functions, SRB measures, fractal dimensions
and bifurcations in hamiltonian systems have been thoroughly developed. Then,
∗http://www.ccr.jussieu.fr/lptmc/Cargese/CargeseMainPage.htm
viii
models of dynamical evolutions of transport processes in classical and quantum
systems have been largly explained.
II. Applications
In this part, many specific applications in physical systems have been presented. It
concerns transport in fluids, plasmas and reacting media. On the other hand, new
experiments of cold optically trapped atoms and electrodynamics cavity have been
thoroughly presented. Finally, several papers bears on synchronism and control of
chaos.
We also provide most recent references of the other given lectures at the school.
These lecture notes represent, in our views, the vitality and the diversity of the
research on Chaos and Physics, both on fundamental and applied levels, and we ff
hope that this summer-school will be followed by similar meetings.
The summer-school was mainly supported by NATO and the staff of the University
Paris 7. M. Courbage was a coordinator of the Summer-School, G. Zaslavsky was af
director of the NATO-ASI and A. Neishtadt was a co-director.
We would like to thank all the institutions who provided support and
encouragements, namely, NATO ASI programm, the European Science Foundation
through Prodyn programm, the Collectivités Territoriale Corse, the Laboratoire de
Physique Théorique de La Matière Condensée (LPTMC) and the Présidence of the
University Paris 7. Thanks also to the Centre de Physique Théorique de l'Ecole
Polytechnique de Paris.
The meeting was an occasion for a warm interactive atmosphere beside the scientific
exchanges. We want to thank those who contributed to its success: the director and
the staff of the Institut d'Etudes Scientifiques de Cargèse and, the team of the
Université Paris 7, especially Evelyne Authier, Secretary of the LPTMC, who
provided essential help to organize this ASI.
P . Collet, M. Courbage, S. Metens, A. Neishtadt, G.W. Zaslavsky
PARTICIPANTS
Valentin AfraimovichIICO UASLP. Communication Optica. Universitet Autonomny de San Luis. Karakorum 1470, Iomas, 4eme section.CP78200 San Luis Potosi. SLP [email protected]
Omar Al HammalInstitute "Carlos I" for Theoretical and Computational Physics. Universidad de Granada Avda. Fuente Nueva, s/n, 118071 Granada [email protected]
Athanasios ArvanitidisDept. of Mathematics Aristotle University ofThessaloniki 54006 Thessaloniki, [email protected]
Francois BardouIPCMS-CNRS, 23, rue du Loess, BP 43 67037 Strasbourg Cedex 2 [email protected]
Maxim BarkovSpace Plasma Physics-Space Research InstituteProfsoyuznaya str. 84/32 117997 Moscow [email protected]
Jacopo BellazziniUniversita' di Pisa - Dipartimento Ingegneria AerospazialeVia Caruso 120 56100 Pisa [email protected]
Viatcheslav BelyiTheoretical department-Russian Academy of Sciences IZMIRAN142190 Troitsk, [email protected]
Cristel ChandreCPT— CNRS-Centre de Physique Theorique Campus de Luminy - case 90713288 Marseille, Francechandre@cpt. univ-mrs. fir
Hugues ChateSPEC - CEA - Saclay, 91191 Gif-sur-Yvette, Francechate@drecam. saclay. cea.fr
Guido CiraoloCentre de Physique Theorique.CNRS case 90713288 Marseille cedex 9, [email protected]
Steliana CodreanuDepartment of Theoretical Physics-Babes-Bolyai University Kogalniceanu str. 1 2400 CLUJ-NAPOCA [email protected] .ro
Pierre ColletCentre de physique theorique-CNRS Ecole Polytechnique, route de Saclay 91128 Palaiseau cedex [email protected]. ft
Pieter CollinsControl and computation-CWIKruislaan 413 1098 SJ Amsterdam [email protected]
ix
M unique CombescureInstitut de Physique Nucleaire de hyon, CNRS-Bat Paul Dirac, 4 rue Enrico Fermi 69622 VILLEURBANNE France
Maurice CourtageLPTMC case 7020-Universite de Paris 7, 2place Jussieu 75231 Paris Cedex 05 France
Giampaolo CristadoroCenter for Nonlinear and Complex Systems and Dipartimento di Scienze Chimiche, Fisiche e Matematiche-Universita' dell'Insubria (sededi Como)-via Valleggiojl 22100 Comoltaliegiampaolo. cristadoro@uninsubria. it
Nir DavidsonDept. of Physics of Complex Systems- Weizmann Institute of Science Rehovot 76100 Israel
Filippo DeLilloDipartimento di Fisica Generale-University ofTorinoVia GiuriaJ 10125 Torino [email protected]
Jean-Pierre EckmannDepartement de Physique Theorique and Section de Mathematiques Universite de Geneve-32, Bid D 'Yvoy 1211Geneva 4 Suisse
Massimiliano EspositoService de Chimie Physique CP 231-Universite Libre de Bruxelles Boulevard du Triomphe B-1050 BruxellesBelgique
Fereydoon FamilyPhysics Department-Emory UniversityPhysics Department, Emory University GA Atlanta USA
Stefano GalatoloDipartimento di matematica applicata-Universita di PisaVia Bonanno Pisano 25b 56126 Pisa Italie
Govanni GallavottiFisica -Univ. Roma IP.le Mow 2 00185 Roma Italie
gallavotti@roma 1. infn. it
Pierre GaspardCenter for Nonlinear Phenomena and Complex SystemsCampus Plaine, CP 231 B-1050 Brussels Belgique
Alessandro GiulianiPhysics department-Universita' di Roma "La Sapienza"- Via Ivanoe Bonomi, 92, 00139 Roma, Italy
alessandro.giuliam'@romal .infn.it
Vasiliy GovorukhinRostov State University - Computational mathematics Zorge str. 5 344090 Rostov-on-Don Russie
xi
Seiichiro HonjoUniversity of Tokyo, Graduate School of Arts and Sciences-Department of Basic Science, Kaneko LaboratoryKomaba3-8-1, Meguro-ku 153-8902 Tokyo Japon
Alexander IominDepartment of Physics, Technion 32000 Haifa Israel
phr92ai@phy sics. technion. ac. il
Alexander ItinLab. 627 Space Research Institute-Profsoyuznaya str. 84/32 117997 Moscow Russia
alx_it@y ahoo. com
Brunon KaminskiFaculty of Mathematics and InformaticsNicholas Copernicus University, ul Chopina 12/18, 87-100 Torun, POLAND
Janina KotusWarsaw Univ. of Technology-Department of MathematicsPlac Politechniki 1 00-661 Warsaw Pologne
Alexandra LandsmanPrinceton University, 235 Thunder Circle PA 19020 Ben Salem USA
landsman@princeton. edu
Jacques LaskarAstronomie et Systemes Dynamiques-IMC77, Av. Denfert-Rochereau F-75014 PARIS France
Xavier LeonciniPIIM-Universite de Provence, Centre de St Jerome, case 32113396 Marseille Cedex 20 France
Fabio LepretiSection of Astrophysics, Astronomy, and Mechanics-Department of Physics, Aristotle University ofThessalonikiDepartment of Physics, Aristotle University ofThessaloniki 54124 Thessaloniki [email protected]
Emanuel LimaInstituto de Fisica de Sao Carlos-Universidade de Sao PauloMajor Julio Salles 870, Vila Pureza, 13561-010 16 SaoCarlos Brazil
Helen MakarenkoDepartment of Physics-V.Karazin National University4 Svobody square 61077 Kharkiv Ukraine
Amar MakhloufUniversite de Annaba-Labo de Mathematiques, 14 Rue Zighoud Youcef, DREAN 36 ELTARF, ALGERIE
makhloufamar@yahoo. fr
Miguel MannaPhysique Mathematique et Theorique CNRS-UMR5825-Universite Montpellier 2Place Eugene Bataillon 34095Montpellier, France
manna@lpm. univ-montp2.fr
xii
Paul MannevilleLaboratoire d'Hydrodynamique-CNRS-Ecole PolytechniqueEcole polytechnique 91128 Paiaiseau, France
Stephane Mi-tensLPTMC, case 7020-Universite Paris 7-Denis Diderot, 2place Jussieu 75231 Paris Cedex 05, France
Stavros MuronidisDept. of Mathematics Aristotle University ofThessaloniki 54006 Thessaloniki, Grece
Stefano MusacchioDipartimento diFisica Generalevia Pietro Giuria 1 10125 Torino, ltalie
Anatoly NeishtadtSpace Research Institute -Russian Academy of Sciences, Profsoyuznaya 84/32 Moscow 117997, Russia
Tali OlikerTechnionyuvalim 56 20142 d.n. misgav, Israel
otali@technion. ac.il
Edward OttI.R.E.A.P.-University of Maryland, City COLLEGE PARK 207 42 MARYLAND, USA
Siverine PacheITP-EPFL, Ecublens 1015 Lausanne, Suisse
Antonio PolitiISTITUTO NAZIONALE DIOTTICA APPLICATA, LARGO E. FERMI 6 50125 FIRENZE, Italy
Antonio PonnoDipartimento di Matematica-Universita' di Milano, Via Saldini 50 20133 Milano, ltalie
Serguei PrantsInstitution Pacific Institute of the Russian Academy of Sciences 43 BALTIISKAYA St. 690041 VLADIVOSTOKRUSSIA
Saar RahavTechnion-Department of Physics 32000 Haifa Israel
Vered Rom-KedarWeizmann InstituteP.O. Box 26-Department of Computer science and applied mathematics 76100 Rehovot, Israel
vered@ wisdom, weizmann. ac.il
Majid SaberiLPTMC, case 7020-Universite Paris 7-Denis Diderot, 2 place Jussieu 75231 Paris Cedex 05,France
Marc SenneretLPTMC, case 7020-Universite de Paris 72place Jussieu 75251 PARIS CEDEX05 France
xiii
Michael ShlesingerOffice of Naval Research, 800 n'Quincy Str., Arlington, VA22217-5660, [email protected]
Eli ShlizermanComputer Science and Applied Mathematics - Dynamical Systems-Weizmann Institute of Science 76100 RehovotIsraeleli. [email protected]. il
Dominique SimpelaereUniversite Paris 6-Pierre et Marie Curie, Lab. De Probabilie, 4, Place Jussieu 75252 Paris cedex 05, [email protected]
Tom SolomonBucknell University Lewisburg, PA 17837, [email protected]
George StilogiannisDept. of Mathematics Aristotle University of Thessaloniki 54006 Thessaloniki, [email protected]
Ion StroeMechanics-POLITEHNICA University of Bucharest Splaiul Independentei 313 RO-77206 Bucharest ROMANIAion.stroe@,rosa.ro
Walter StrunzPhysikalisches Institut, Universitat Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, [email protected]
Vladimir TenDept of Mathematics and Mechanics Moscow State University-MSU, Vorobevy gory 119899 Moscow [email protected]
Sandro VaientiUniversity of Toulon and Centre de Physique TheoriqueCase 907 3288 Marseille Cedex 09 Francevaienti(a),cpt.univ-mrs.fr
Jiri VanicekJefferson Physical LaboratoryMSRI, 1000 Centennial Drive CA 94720 Berkeley [email protected]
Alexei VasilievLaboratory of Chaotic Dynamics Space Research Institute-Profsoyuznaya 84/32 117997 Moscow [email protected]
Sebastien ViscardyUniversite Libre de Bruxelles-Service de Chimie-PhysiqueCampus Plaine, CP 2311050 Brussels [email protected]
Therese VivierUMR5584 (CNRS) - Institut Mathematiques de Bourgogne BP47970 21078 Dijon Cedex [email protected]
Angelo VulpianiDipartimento di Fisica, Universita' di Roma La Sapienza Pie A. Moro 21-00185 Roma Italieangelo.vulpiani@romal .infh.it
Piotr WazCenter for Astronomy-Nicholaus Copernicus University, Gagarina 11 87-100 Torun [email protected]
xiv
Tatiana YeleninaNumerical Simulation of Electrodynamics and Magnetohydrodynamics-Keldysh Institute of Applied Mathematics ofRAS4, Miusskaya sq. 125047 Moscow Russie
L.S. YoungCourant Institute, New-York University, 251, Mercer Street, NY-10012-1182 New-York, USA
Michael ZaksHumboldt University of Berlin-Dept of Stochastic Processes, Newtonstr. 15D-I2489 Berlin A llemagne
Georges ZaslavskyCourant Institute, New-York University, 251, Mercer Street, NY-10012-1182 New-York, USA
zaslav(a!cims.nyu.edu
Vladimir ZeitlinLaboratoire de Meteorologie Dynamique, ENS, 24Rue Lhomond 75231 PARIS CEDEX05 France
zeitlin @lmd.ens.fr