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Advances in SOVIET

Volume 20

Probability Contributions to

R. L. Dobrushin Editor

American Mathematical Society Titles in This Series

20 R. L. Dobrushin, editor, contributions to statistical mechanics, 1994 19 V. A. Marchenko, editor, Spectral and related topics, 1994 18 Oleg Viro, editor, of manifolds and varieties, 1994 17 Dmitry Fuchs, editor, Unconventional Lie , 1993 16 Sergei Gelfand and Simon Gindikin, editors, I. M. Gelfand seminar, Parts 1 and 2, 1993 15 A. T. Fomenko, editor, Minimal surfaces, 1993 14 Yu. S. Il'yashenko, editor, Nonlinear Stokes phenomena, 1992 13 V. P. Maslov and S. N. Samborskil, editors, Idempotent analysis, 1992 12 R. Z. Khasminskii, editor, Topics in nonparametric estimation, 1992 11 B. Ya. Levin, editor, Entire and subharmonic functions, 1992 10 A. V. Babin and M. I. Yishik, editors, Properties of global attractors of partial differential equations, 1992 9 A. M. Vershik, editor, Representation theory and dynamical systems, 1992 8 E. B. Vinberg, editor, Lie groups, their discrete subgroups, and invariant theory, 1992 7 M. Sh. Birman, editor, Estimates and asymptotics for discrete spectra of and differential equations, 1991 6 A. T. Fomenko, editor, Topological classification of integrable systems, 1991 5 R. A. Minlos, editor, Many-particle Hamiltonians: spectra and scattering, 1991 4 A. A. Suslin, editor, Algebraic ^-theory, 1991 3 Ya. G. Sinai, editor, Dynamical systems and statistical mechanics, 1991 2 A. A. Kirillov, editor, Topics in representation theory, 1991 1 V. I. Arnold, editor, Theory of singularities and its applications, 1990

Probability Contributions to Statistical Mechanics

Advances in 10.1090/advsov/020 S oviet Mathematics

Volume 20

Probability Contributions to Statistical Mechanics

R. L. Dobrushin Editor

American Mathematical Society Providence, Rhode Island A d v a n c es in So v ie t M a t h e m a t ic s

E d it o r ia l C o m m it t e e V. I. ARNOLD S. G. GINDIKIN V. P. MASLOV

Translation edited by A. B. SOSSINSKY

1991 Mathematics Subject Classification. Primary 82Bxx.

Abstract. This collection contains papers written by representatives of the Moscow school of mathematical statistical mechanics, and illustrates certain as­ pects of the developing interaction between statistical mechanics on the one hand and the of probability and of dynamical systems on the other. It includes papers on random walks, on phase transition phenomena for Gibbs random fields, on the existence of nonstandard motion in statistical models, and on the analysis of the Frenkel-Kontorova model.

Library of Congress Catalog Card Number: 91-640741 ISBN 0-8218-4120-3 ISSN 1051-8037

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this pub­ lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Ser­ vices, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-peraissionQ m ath. ams .org. The appearance of the code on the first page of an article in this book indicates the copyright owner’s consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, Massachusetts 01923. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale.

© Copyright 1994 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. © The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. C* Printed on recycled paper. This publication was typeset using the American Mathematical Society’s TgX macro system. 10 9 8 7 6 5 4 3 2 1 98 97 96 95 94 Contents

Foreword ix An Extension of the J. ABDULLAEV and R. A. MINLOS 1 Central Limit for the of One and Two Particles in a Random Environment, with Mutual Interaction C. BOLDRIGHINI, R. A. MINLOS, and A. PELLEGRINOTTI 21 Random Walk of a Particle Interacting with a Random R. A. MINLOS 77 Large and Moderate Deviations in the Ising Model ROLAND L. DOBRUSHIN and SENYA B. SHLOSMAN 91 Asymptotically Additive Integrals of Motion for Particles with Nonpairwise Interaction in One B. M. GUREVICH 221 On a Ground State in the Frenkel-Kontorova Model and Metric Properties of Mappings of Standard Type L. D. PUSTYL'NIKOV 277

vii

Foreword

The title of this volume, Probability Contributions to Statistical Mechanics, reflects its intent and contents only in part. It would have been more precise to call it Contributions of Probability and Dynamical Systems Theories to Statistical Mechanics and Contributions of Statistical Mechanics to Probability and Dynamical Systems Theories, but this, of course, is too unwieldy for a title. The relationship between probability theory and the theory of dynamical systems on one hand and statistical mechanics on the other has not always been cloudless. In the period when the modem theories of probability and of dynamical systems were being created (at the turn of the century), many of the main ideas came from statistical mechanics. It suffices to recall that a , the etalon process in the theory of random processes, is still often called , despite the fact that the description of ac­ tual Brownian motion requires more complicated models, taking into account the inertia of the particle, while the notion of arose from efforts to give meaning to the main postulates of statistical mechanics. However, by the middle of the century, probabilists and statistical physicists practically lost their mutual understanding as well as their interest in each other. One should recall with warmth the heroic efforts undertaken by M. Kac and A. Khinchin to reestablish the lost contacts between probability theory and statistical me­ chanics, but the seeds that they planted were to sprout only when the time for it had come. At the end of the sixties, a new branch of science, which might well be called “mathematical statistical mechanics”, came into being. Its aim was to clarify the ideas of statistical mechanics from the point of view of rig­ orous mathematical methods and of the latest mathematical achievements. Since then hundreds of articles and several books in this field have been published. To summarize, one may say that now all the main notions of statistical mechanics have been reformulated in a rigorous mathematical lan­ guage, and statements that were long established on a level of rigor usual in theoretical physics have been reinterpreted as or as mathemati­ cal conjectures awaiting their proofs. There are also new results, or, at least, mathematical theorems that allow one to make a unique choice among several points of view appearing in the physics literature, although in fact there aren’t too many such cases. The opinions of various representatives of traditional

IX X FOREWORD

statistical mechanics about the new rigorous approaches to the field are di­ verse. Some admit that the mathematical results give a deeper understanding of the essense of the problems. The other extreme viewpoint is that they are merely a very complicated way to present simple and well-known facts. Who is right? Only time will tell. But there are also other aspects. Physics has always been an important source of new mathematical notions and ideas, and in the past decade this stream of ideas and notions from physics to mathematics has increased dras­ tically. If at the beginning, the majority of experts in probability theory were rather skeptical about what they felt were not very understandable ex­ cursions into statistical mechanics, now no doubts seem to remain. It is clear that this interaction with statistical mechanics gave rise to a new field of probability theory - the theory of random fields, i.e., random functions of many variables. Such notions as Gibbs random fields, Markov processes with interaction, the hydrodynamical limit approach have found their math­ ematical formalization and have firmly established themselves in probability theory. Moreover, it turned out that a natural path of physical ideas to other sciences passes through mathematics. While a quarter century ago it seemed that the notion of Gibbs distribution is something intimately connected with the specifics of systems appearing in statistical mechanics, and that this no­ tion is useless outside this specific environment, it is now clear that the Gibbs approach is simply a natural way of describing a large class of random fields, i.e., random functions of many variables. Since it was necessary to give mathematical descriptions of such random functions in many fields of sci­ ence, the spectrum of developing applications of the notion of Gibbs has become immense: from and the theory of com­ munication networks with queues to the description of systems of economic or even sociological nature. The same path from physics through rigorous mathematics into other sciences was taken by the notion of phase transition behaviour. Now one can claim that the possibility of phase transition is a common property of all large multidimensional systems. The present collection, written by representatives of the Moscow school of mathematical statistical mechanics, illustrates certain aspects of the develop­ ing interaction of statistical mechanics on the one hand and the theories of probability and of dynamical systems on the other. The series of three articles written by R. Minlos and coauthors is devoted to the study of an important and natural probabilistic object - random walks in a random medium, for different definitions of this object. The results are obtained by applying the method of cluster expansions. This deep analytic method has become the foundation for obtaining many rigorous results both in statistical mechanics and in , but it seems that it is still far from being appreciated at its true value and has not been adopted by the experts in probability theory. FOREWORD xi

The long joint article by S. Shlosman and the editor of this collection deals with the theory of large deviations for Gibbs random fields. Its main goal is to demonstrate the cardinal changes in the familiar probabilistic picture that occur when phase transitions are allowed. It turns out that this probability theory question is closely connected with the natural physical question of the level of concentration at which condensation of droplets of one phase within another begins. Conceivably, there is something new about this even from the physical point of view. The proofs are based on a combination of probabilistic methods from the theory of large deviations and the use of a special version of the method of cluster decompositions, applicable to the contour ensembles that appear in the case of multiple phases. Chapter 3 of the article, devoted to the exposition and development of the method of cluster decompositions, can be read separately. The large article by B. Gurevich is devoted to the development of an im­ portant series of studies of his, related to certain fundamental problems of statistical mechanics. Along the road to establishing its main postulate, claim­ ing that the statistical properties of a system in equilibrium are described by a Gibbs distribution with potential determined by the physical energy of the system, there is a difficult mathematical problem. One must prove that the classical integrals of motion (the number of particles, the energy, and the total momentum) are the only asymptotically additive translation-invariant integrals for classical one-dimensional mechanical systems. However, this is not always so. It has been established that in dimension d = l for pair potentials of the form a U(r) = (sinh(br))2 ’ where a > 0 and b > 0 are constants, there exists an infinite series of addi­ tional first integrals. The study of such integrable one-dimensional systems has become a pretty fashionable affair: here certain deep relationships with modem mathematical theories arise, but from the point of view of statistical mechanics, this is an ugly case. Indeed, in such situations the main postulates of statistical mechanics break down. The only hope is that such ugliness can only occur in exceptional, unstable situations. In the works of Gurevich it is shown that this is indeed the case. When the dimension d is 2 or more, there are no exceptions whatsoever, when d = 1, they are limited to the class of potentials described above. The corresponding mathematical study requires certain a priori restrictions on the class of interactions under con­ sideration. The main restriction used by Gurevich in his previous work was the assumption that only pairwise interaction takes place. In the present ar­ ticle he rejects this harsh restriction and considers three-particle interactions. This case is already quite difficult and requires the introduction of original analytic techniques. xii FOREWORD

L. Pustyl’nikov’s article deals with the ground states for a well-known one-dimensional model in statistical mechanics, the Frenkel-Kontorova model. Ya. Sinai reduced this problem to the study of a special . This is the object considered by the author, who finds new cases when phase transitions for it exist. R. L. Dobrushin Translated by A. S. SOSSINSKY

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