This book deals with the various thermodynamic concepts used for the analysis of nonlinear dynamical systems. The most important concepts are introduced in such a way as to stress the inter- connections with thermodynamics and statistical mechanics. The basic ideas of the thermodynamic formalism of dynamical systems are explained in the language of physics. The book is an elementary introduction to the subject and no advanced mathematical preknowledge on the part of the reader is required. Among the subjects treated are probabilistic aspects of chaotic dynamics, the symbolic dynamics technique, information measures, the maximum entropy principle, general thermodynamic relations, spin systems, fractals and multifractals, expansion rate and information loss, the topological pressure, transfer operator methods, repellers and escape. The more advanced chapters deal with the thermodynamic formalism for expanding maps, thermodynamic analysis of chaotic systems with several intensive parameters, and phase transitions in nonlinear dynamics. Written for students and scientists working in physics and related fields, this easily readable introduction can also be used as a textbook for special courses. Thermodynamics of chaotic systems Cambridge Nonlinear Science Series 4 Series editors Professor Boris Chirikov, Budker Institute of Nuclear Physics, Novosibirsk Professor Predrag Cvitanovic, Niels Bohr Institute, Copenhagen Professor Frank Moss, University of Missouri-St Louis Professor Harry Swinney, Center for Nonlinear Dynamics, The University of Texas at Austin Titles in this series 1 Weak chaos and quasi-regular patterns G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov and A. A. Chernikov 2 Quantum chaos: a new paradigm for nonlinear dynamics K. Nakamura 3 Chaos, dynamics and fractals: an algorithmic approach to deterministic chaos J. L. McCauley 4 Thermodynamics of chaotic systems: an introduction C. Beck and F. Schlogl Thermodynamics of chaotic systems An introduction Christian Beck School of Mathematical Sciences, Queen Mary and Westfield College, University of London and Friedrich Schlogl Institute for Theoretical Physics, University of Aachen CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521433679 © Cambridge University Press 1993 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993 First paperback edition 1995 Reprinted 1997 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Beck, Christian. Thermodynamics of chaotic systems : an introduction / Christian Beck and Friedrich Schlogl. p. cm. — (Cambridge nonlinear science series : 4) Includes bibliographical references. ISBN 0-521-43367-3 1. Chaotic behavior in systems. 2. Thermodynamics. 3. Nonlinear theories. I. Schlogl, Friedrich. II. Title. III. Series Q172.5.C45B43 1993 536'.7-dc20 92-33322 CIP ISBN-13 978-0-521-43367-9 hardback ISBN-10 0-521-43367-3 hardback ISBN-13 978-0-521 -48451 -0 paperback ISBN-10 0-521-48451-0 paperback Transferred to digital printing 2005 Contents Preface xiii Introduction xv PART I ESSENTIALS OF NONLINEAR DYNAMICS 1 1 Nonlinear mappings 1 1.1 Maps and trajectories 1 1.2 Attractors 2 1.3 The logistic map 4 1.4 Maps of Kaplan-Yorke type 11 1.5 The Henon map 14 1.6 The standard map 16 Probability in the theory of chaotic systems 20 2.1 Relative frequencies and probability 20 2.2 Invariant densities 22 2.3 Topological conjugation 28 3 Symbolic dynamics 32 3.1 Partitions of the phase space 32 3.2 The binary shift map 33 3.3 Generating partition 35 3.4 Symbolic dynamics of the logistic map 37 3.5 Cylinders 39 3.6 Symbolic stochastic processes 40 vii viii Contents PART II ESSENTIALS OF INFORMATION THEORY AND THERMODYNAMICS 44 4 The Shannon information 44 4.1 Bit-numbers 44 4.2 The Shannon information measure 46 *4.3 The Khinchin axioms 47 5 Further information measures 50 5.1 The Renyi information 50 5.2 The information gain 51 *5.3 General properties of the Renyi information 53 6 Generalized canonical distributions 56 6.1 The maximum entropy principle 56 6.2 The thermodynamic equilibrium ensembles 59 6.3 The principle of minimum free energy 61 *6.4 Arbitrary a priori probabilities 63 7 The fundamental thermodynamic relations 65 7.1 The Legendre transformation 65 7.2 Gibbs' fundamental equation 68 *7.3 The Gibbs-Duhem equation 71 7.4 Susceptibilities and fluctuations 73 7.5 Phase transitions 75 8 Spin systems 78 8.1 Various types of spin models 78 8.2 Phase transitions of spin systems 82 8.3 Equivalence between one-dimensional spin systems and symbolic stochastic processes 83 8.4 The transfer matrix method 85 PART III THERMOSTATISTICS OF MULTIFRACTALS 88 9 Escort distributions 88 9.1 Temperature in chaos theory 88 9.2 Two or more intensities 90 *9.3 Escort distributions for general test functions 91 Contents ix 10 Fractals 94 10.1 Simple examples of fractals 94 10.2 The fractal dimension 97 10.3 The Hausdorff dimension 100 10.4 Mandelbrot and Julia sets 102 *10.5 Iterated function systems 109 11 Multifractals 114 11.1 The grid of boxes of equal size 114 11.2 The Renyi dimensions 115 11.3 Thermodynamic relations in the limit of box size going to zero 117 11.4 Definition of the Renyi dimensions for cells of variable size 123 *12 Bit-cumulants in multifractal statistics 127 12.1 Moments and cumulants of bit-numbers 127 12.2 The bit-variance in thermodynamics 129 12.3 The sensitivity to correlations 131 12.4 Heat capacity of a fractal distribution 132 *13 Thermodynamics of finite volume 136 13.1 Boxes of finite size 136 13.2 Thermodynamic potentials for finite volume 139 13.3 An analytically solvable example 143 PART IV DYNAMICAL ANALYSIS OF CHAOTIC SYSTEMS 146 14 Statistics of dynamical symbol sequences 146 14.1 The Kolmogorov-Sinai entropy 146 14.2 The Renyi entropies 149 14.3 Equivalence with spin systems 155 14.4 Spectra of dynamical scaling indices 156 15 Expansion rate and information loss 158 15.1 Expansion of one-dimensional systems 158 15.2 The information loss 160 *15.3 The variance of the loss 161 15.4 Spectra of local Liapunov exponents 164 15.5 Liapunov exponents for higher-dimensional systems 167 15.6 Stable and unstable manifolds 172 x Contents 16 The topological pressure 178 16.1 Definition of the topological pressure 178 16.2 Length scale interpretation 182 16.3 Examples 184 *16.4 Extension to higher dimensions 187 *16.5 Topological pressure for arbitrary test functions 188 17 Transfer operator methods 190 17.1 The Perron-Frobenius operator 190 17.2 Invariant densities as fixed points of the Perron-Frobenius operator 193 17.3 Spectrum of the Perron-Frobenius operator 196 17.4 Generalized Perron-Frobenius operator and topological pressure 197 *17.5 Connection with the transfer matrix method of classical statistical mechanics 199 18 Repellers and escape 204 18.1 Repellers 204 18.2 Escape rate 207 *18.3 The Feigenbaum attractor as a repeller 208 PART V ADVANCED THERMODYNAMICS 211 19 Thermodynamics of expanding maps 211 19.1 A variational principle for the topological pressure 211 19.2 Gibbs measures and SRB measures 214 19.3 Relations between topological pressure and the Renyi entropies 218 19.4 Relations between topological pressure and the generalized Liapunov exponents 221 19.5 Relations between topological pressure and the Renyi dimensions 222 20 Thermodynamics with several intensive parameters 226 20.1 The pressure ensemble 226 *20.2 Equation of state of a chaotic system 229 20.3 The grand canonical ensemble 233 20.4 Zeta functions 237 *20.5 Partition functions with conditional probabilities 240 Contents xi 21 Phase transitions 243 21.1 Static phase transitions 243 21.2 Dynamical phase transitions 248 21.3 Expansion phase transitions 251 21.4 Bivariate phase transitions 254 *21.5 Phase transitions with respect to the volume 256 21.6 External phase transitions of second order 260 *21.7 Renormalization group approach 262 21.8 External phase transitions of first order 267 References 273 Index 282 Preface In recent years methods borrowed from thermodynamics and statistical mechanics have turned out to be very successful for the quantitative analysis and description of chaotic dynamical systems. These methods, originating from the pioneering work of Sinai, Ruelle, and Bowen in the early seventies on the thermodynamic formalism of dynamical systems, have been further developed in the mean time and have attracted strong interest among physicists. The quantitative characterization of chaotic motion by thermodynamic means and the thermodynamic analysis of multifractal sets are now an important and rapidly evolving branch of nonlinear science, with applications in many different areas. The present book aims at an elucidation of the various thermodynamic concepts used for the analysis of nonlinear dynamical systems. It is intended to be an elementary introduction. We felt the need to write an easily readable book, because so far there exist only a few classical monographs on the subject that are written for mathematicians rather than physicists. Consequently, we have tried to write in the physicist's language. We have striven for a form that is readable for anybody with the knowledge of mathematics a student of physics or chemistry at the early graduate level would have. No advanced mathematical preknowledge on the part of the reader is required. On the other hand, we also tried to avoid serious loss of rigour in our presentation.