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 A. Jaffe D. Szasz MTA MKI Dept. of Physics MTA MKI Realtanoda u. 13-15 Lyman Laboratory Realtanoda u. 13-15 H-1053 Budapest Harvard University H-1053 Budapest (Hungary) Cambridge, MA 02138 (USA) (Hungary) 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 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 v C 0 N T E N T S Preface vii Program of the Workshop ix List of Participants xv R.A. MINLOS: The Algebra of Many-Particle Operators 1 H. ARAKI and T. MATSUI: C*-Algebra Approach to Ground States of the XY-Model 17 M.Z. GUO and G. PAPANICOLAOU: Bulk Diffusion for Interacting Brownian Particles 41 L.A. PASTUR: Spectral Properties of Random and Almost Periodic Differential and Finite-Difference Operators 49 H. SPOHN: Equilibrium Fluctuations for Some Stochastic Particle Systems 67 A.G. SHUHOV and YU.M. SUHOV: Linear and Related Models of Time Evolution in Quantum Statistical Mechanics 83 P. COLLET: Systems with Random Couplings on Diamond Lattices 105 L.A. BUNIMOVICH: On the Diffusion in Dynamical Systems 127 A. KUPIAINEN: ~ 4 with Negative Coupling 137 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 153 YA.B. PESIN: A Generalization of Carateodory's Construction for Dimensional Characteristic of Dynamic Systems 191 M. MISIUREWICZ and A. ZDUNIK: Convergence of Images of Certain Measures 203 vi R.R. AKHMITZJANOV, V.A. MALYSHEV and E.N. PETROVA: Cluster EXpansion for Unbounded Non-Finite Potential 221 C. BARDOS, R. CAFLISCH and B. NICOLAENKO: Thermal Layer Solutions of the Boltzmann Equation 235 E. PRESUTTI, YA.G. SINAI and M.R. SOLOVIECHIK: Hyperbolicity and 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 303 C. MARCHIORO: Some Problems in Vortex Theory 317 P.M. BLECHER: The Maxwell Rule and Phase Separation in the Hierarchical Vector-Valued ~ 4-Model 329 R.L. DOBRUSHIN and S.B. SHLOSMAN: Constructive Criterion for the Uniqueness of Gibbs Field 347 R.L. DOBRUSHIN and S.B. SHLOSMAN: Completely Analytical Gibbs Fields 371 P.A. FERRARI, S. GOLDSTEIN and J.L. LEBOWITZ: Diffusion, Mobility and the Einstein Relation 405 B. SOUILLARD: Transition from Pure Point to Continuous Spectrum for Random Schrodinger Equations: Some Examples 443 M. AIZENMAN: Rigorous Studies of Critical Behavior II 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 ix PROGRAM Uonday 10.00 Opening Ceremony 1 0. 20 - 11 • 05 R. L. Dobrushin - s. B. Shlosman: Construc- tive criteria of uniqueness and analiticity in statistical mechanics 11.10- 11.55 J. L. Lebowitz: Mathematical and physical ideas in nonequilibrium statistical n.:!chanics 3.00- 3.45 E. Lieb: Various estimates for the eigen- values of the Laplacian 3.50- 4.35 H. Araki: C*-algebra approach to the ground states of the X-Y model coffee break 5.00 - 5.45 M. Aizenmann: Rigorous studies of critical behaviour 5.50 - 6.35 D. Kazhdan: c-representations 7.30 Welcome party Tuesday 9.00 - 9.45 A. Kupiainen: Non-trivial continuum limit for negative coupling ~4 4 9. 50 - 10.35 G. Papanicolau: Bulk diffusion and self dif- fusion for interacting Brownian motions coffee break 11.00 - 11.45 o. E. Lanford: Renormalization group methods for mappings with golden-ratio rotation num- ber 10.50- 12.00 H. Rost: Equilibrium fluctuations for a one- dimensional nearest neighbour model 1 2 • 02 - 1 2 • 1 2 A. Verbeure: States stationary for the de- tailed balance of reversible processes 12.14- 12.24 M. F. Chen: Jump Markov processes and inter- acting particle systems X 12.26- 12.36 Ra. Siegmund-Schulze: On existence of non- equilibrium dynamics of multidimensional infinite particle systems: the translation invariant case 12.38 - 12.48 J. Fritz: Interacting Brownian particles: existence and self-adjointness 3.00-3.10 A. Kramli - D. Szasz: The problem of recur- rence for Lorentz processes 3.12- 3.22 D. Dawson: Ensemble and multilevel models of critical behaviour 3.24 - 3.34 K. Fleischman: Occupation time processes at a critical point 3.36 - 3.46 J. A. Galves - E. Olivieri - M. E. Vares: Metastable behaviour of stochastic systems: a pathwise approach 3.48 - 3.58 J. R. Klauder: Langevin equations for sta- tistical computations 4.00 - 4.10 N. Ianiro: Stationary Boltzman-equation 4. 12 - 4.22 G. Jetschke: On stochastic nonlinear parabolic differential equations 4.24 - 4.34 P. Calderoni: On the Smoluchowski limit for simple particle systems 4.36 - 4.46 H. Cranel: Stochastic systems on manifolds coffee break 5. 14 - 5.24 C. A. Hurst: C*-algebra approach to the Pfaffian method for the Ising model 5.26 - 5.36 J. T. Lewis - J. V. Pule: Phase transitions and the weak law of large numbers 5.38 - 5.48 K. Kuroda: The Pirogov-Sinai theory of phase transitions for continuum systems 5.50- 6.00 Y. Higuchi: A weak version of Russo-Seymour- Welsh theorem for the two-dimensional Ising model 6.02 - 6.12 L. Laanait -A. Messager - J. Ruiz: Phases coexistence and surface tension for the Potts model 6.14 - 6.24 B. Toth: A lower bound for the critical probability of square-lattice site percola- tion 6.26 - 6.36 D. Merlini: On the Temperley conjecture for the two-dimensional Ising model 6.38 - 6.48 V. Warstat: A uniqueness theorem for systems of interacting polimers at low temperature 6.50- 7.00 M. Arato: The distribution of stochastic in- tegrals