Advances in SOVIET MATHEMATICS Volume 20 Probability Contributions to Statistical Mechanics R. L. Dobrushin Editor American Mathematical Society Titles in This Series 20 R. L. Dobrushin, editor, Probability contributions to statistical mechanics, 1994 19 V. A. Marchenko, editor, Spectral operator theory and related topics, 1994 18 Oleg Viro, editor, Topology of manifolds and varieties, 1994 17 Dmitry Fuchs, editor, Unconventional Lie algebras, 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 integral 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 theories 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 integrals in statistical physics 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. 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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 Ising Model J. ABDULLAEV and R. A. MINLOS 1 Central Limit Theorem for the Random Walk 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 Field 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 Dimension 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 Wiener process, the etalon process in the theory of random processes, is still often called Brownian motion, 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 ergodicity 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 theorems 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 random field has become immense: from pattern recognition 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.