Statistical Mechanics and Fractals
This book is composed of two texts, by R.L. Dobrushin and S. Kusuoka, each representing the content of a course of lectures given by the authors. They are pitched at graduate student level and are thus very accessible introductions to their respective sub
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P.l.
Advisor: S. S. Chern, R-j. Jiang
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg R Eckmann, Zurich F. Takens, Groningen Subseries: Nankai Institute of Mathematics; Tianjin, China (vol. 10)
P.l.
Advisor: S. S. Chern, R-j. Jiang
1567
R.L. Dobrushin S. Kusuoka
Statistical Mechanics and Fractals
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Roland Lvovich Dobrushin Institute for Problems of Information Transmission Ermolovoj 19 103051 Moscow, Russia Shigeo Kusuoka Research Institute of Mathematical Sciences Kyoto University 606 Kyoto, Japan New address: Department of Mathematical Sciences University of Tokyo 113 Tokyo, Japan
Mathematics Subject Classification (1991): 82BXX, 82B31, 60K35, 60160
ISBN 3-540-57516-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57516-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights oftranslation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany 2146/3140-543210 - Printed on acid-free paper
FOREWORD
The Nankai Institute of Mathematics held a special Year in Probability and Statistics during the academic year 19881989. We had over 150 specialists, professors and graduate students, who participated in this Special Year from August 1988 to May 1989. More than twenty outstanding probabilists and statisticians from several countries were invited to give lectures and talks. This volume contains two lectures, one is written by Professor R. L. Dobrushin, and the other one by Professor S. Kusuoka. We would like to express our gratitude to Professors Dobrushin and Kusuoka for their enthusiasm and cooperation.
ZePei Jiang ShiJian Yan Ping Cheng Rong Wu
TABLE OF CONTENTS Part 1. On the Way to the Mathematical Foundations
Statistical Mechanics
by R. L. Dobrushin
§O. Introduction
1
§1. Realization of the Classical Fluid Model
5
§2. Dynamics of a Finite System
7
§3. Dynamics of an Infinite System
14
§4. Random Evolution
19
§5. Gibbsian States in Finite Volumes
21
§6. Gibbsian Measures in an Infinite Volume
26
§7. Random Evolution (revisited)
28
§8. Hydrodynamical Equations
31
References
36
Part 2. Diffusion Processes on Nested Fractals
by S. Kusuoka
Introduction
39
§1. Self-similar Fractal
41
§2. Nested fractals and their geometrical properties
47
§3. Transition probability of Markov chain
54
§4. Dirichlet form on nested fractal
62
§5. Probability measure induced by random matrices
72
§6. E
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