Introduction to Percolation Theory

Introduction to Percolation Theory Introduction to Percolation Theory Revised Second Edition Dietrich Stauffer and Amnon Aharony This edition published in the Taylor & Francis e-Library, 2010. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. UK Taylor & Francis Ltd, 1 Gunpowder Square, London EC4A 3DE. USA Taylor & Francis Inc, 325 Chestnut Street, 8th Floor, Philadelphia, PA 19106 1st edition copyright © Dietrich Stauffer 1985 2nd edition copyright © Dietrich Stauffer and Amnon Aharony 1991 Revised 2nd edition 1994 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, electrostatic, magnetic tape, mechanical, photo- copying, recording or otherwise, without the prior permission of the copyright owner. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data is available Cover design by Amanda Barragry ISBN 0-203-21159-6 Master e-book ISBN ISBN 0-203-26914-4 (Adobe ebook Reader Format) ISBN 0 7484 0027 3 (Print Edition) Contents Preface to the Second Edition viii P reface to the First Edition ix 1 Introduction: Forest Fires, Fractal Oil Fields, and Diffusion 1 1.1 What is percolation? 1 1.2 Forest fires 4 1.3 Oil fields and fractals 8 1.4 Diffusion in disordered media 10 1.5 Coming attractions 12 Further reading 13 2 Cluster Numbers 15 2.1 The truth about percolation 15 2.2 Exact solution in one dimension 19 2.3 Small clusters and animals in d dimensions 23 2.4 Exact solution for the Bethe lattice 26 2.5 Towards a scaling solution for cluster numbers 35 2.6 Scaling assumptions for cluster numbers 42 2.7 Numerical tests 45 2.8 Cluster numbers away from p c 56 Further reading 59 3 Cluster Structure 60 3.1 Is the cluster perimeter a real perimeter? 60 3.2 Cluster radius and fractal dimension 62 3.3 Another view on scaling 67 3.4 The infinite cluster at the threshold 68 Further reading 73 4 Finite-size Scaling and the Renormalization Group 74 4.1 Finite-size scaling 74 4.2 Small cell renormalization 79 4.3 Scaling revisited 86 4.4 Large cell and Monte Carlo renormalization 88 vi Content 4.5 Connection to geometry 91 Further reading 92 5 Conductivity and Related Properties 93 5.1 Conductivity of random resistor networks 93 5.2 Internal structure of the infinite cluster 98 5.3 Multitude of fractal dimensions on the incipient infinite cluster 100 5.4 Multifractals 104 5.5 Fractal models 109 5.6 Renormalization group for internal cluster structure 113 5.7 Continuum percolation, Swiss-cheese models and broad distributions 114 5.8 Elastic networks 118 Further reading 120 6 Walks, Dynamics and Quantum Effects 122 6.1 Ants in the labyrinth 122 6.2 Probability distributions 130 6.3 Fractons and superlocalization 132 6.4 Hulls and external accessible perimeters 135 6.5 Diffusion fronts 140 6.6 Invasion percolation 143 Further reading 145 7 Application to Thermal Phase Transitions 147 7.1 Statistical physics and the Ising model 147 7.2 Dilute magnets at low temperatures 149 7.3 History of droplet descriptions for fluids 150 7.4 Droplet definition for the Ising model in zero field 152 7.5 The trouble with Kertész 153 7.6 Applications 154 7.7 Dilute magnets at finite temperatures 155 7.8 Spin glasses 158 Further reading 159 8 Summary 160 A Numerical Techniques 163 A.1 Analysis of exact data 163 A.2 Analysis of Monte Carlo data 164 Content vii A.3 Computerized cluster counting 166 Further reading 178 B Dimension-Dependent Approximations 179 B.1 Upper critical dimension 179 B.2 Flory approximation 180 B.3 !-Expansion 181 Further reading 182 Exercises 183 Index 190 Preface to the Second Edition This edition is enlarged not only by a second author, but also by additional material, needed because of the progress in research since the first edition. In addition to correction of errors, omission of topics which are no longer active and updating of references, the new edition puts more emphasis on fractals and on the relations between the fractal geometry of percolation clusters and their physical properties. It also puts more emphasis on the crossover (as function of length scale) between the critical behaviour at and near the percolation threshold and other types of behaviour, and on the presentation of the renormalization group as a working tool. Dimensions above three are no longer ignored. Most of the new material in this edition is contained in Chapters 5, 6 and 7, and in Appendix B. New concepts which are introduced, at least briefly, include distribution functions, fluctuations, multifractals, minimal (chemical) paths, continuum percolation, elastic networks, superlocalization, fractons, hulls and perimeters, diffusion fronts, growth, self-affine curves, invasion percolation and self-organized criticality. We also discuss the role played by percolation theory in describing Ising model droplets, leading to the very fast modern Monte Carlo simulation techniques in statistical physics. Many of the concepts listed here are not limited to percolation clusters. They arise in many modern branches of statistical physics and other disciplines which involve the interplay of geometry, statistics and physics. Examples include polymer science, aggregation, etc. Percolation theory is still the simplest context in which all these tools can be introduced and explained. Another new element in this edition is the set of exercises. Many of these were originally research problems, and their solutions are hidden in the ‘further reading’ lists or in references therein. However, any student who follows the text should be able to solve these exercises, and use them to test his or her ability to actively use the tools offered here in research. Obviously, this book still represents our biased view of percolation. One cannot do justice to all the topics which have appeared in the literature, and it is also beyond the scope of such an introductory text. Since this book is not a review monograph, we also avoided giving a full list of references. In many cases we have preferred to list a recent review or research article which contains many of the earlier references. In any case, the responsibility for these omissions, as well as for the remaining and new errors, rests fully with my coauthor. Joan Adler, Preben Alstrom, Etienne Guyon, Shlomo Havlin, Rudolf Hilfer, Ury Naftaly and Gene Stanley made comments on the first version which were helpful in improving the book. Dietrich Stauffer Amnon Aharony Tel Aviv, January 1991 Preface to the First Edition This book is an attempt to introduce the reader to a research field which is already more than forty years old but which has become very fashionable in the research publications of the last ten years. More than a hundred publications are printed each year where ‘percolation’ or similar words appear in the title. But in contrast to many other modern research fronts, percolation theory is a problem which is, in principle, easy to define. It is, however, not so easy to solve. Thus percolation theory gives the reader the opportunity to enter current research without having to hear many specialized courses or to read voluminous textbooks. Percolation theory has been taught to first year undergraduates and even younger students by this author and by others, but it has also been utilized in courses for more advanced students on disordered systems or on computer simulations. The present book tries to be useful for all these purposes. Since percolation is not, and in my opinion should not be, a standard subject for university examinations, it is addressed mainly to readers interested in research. But it is aimed at one who is just starting research in this field, not to readers who have already worked in percolation theory. These experts will doubtless notice my biased selection of material. The prerequisites are familiarity with computer programming (e.g. Fortran), integration and differentiation of functions of one variable, and probabilistic elements like averages or statistical independence. Therefore the book should be understandable to students in computer science, mathematics, chemistry or biology who might be interested in stretching a big computer to its limits by efficient programming, in simple applications of probability theory, in theories for the gelation of branched macromolecules, or in the spread of epidemics in an ensemble of living beings. But presumably the book will be read mostly by physics students since the methods by which it attacks the percolation problem are taken from the theory of phase transitions like ferromagnetic Curie points. For these we will mention throughout the book analogies between the geometrical aspects of percolation and the physical aspects of thermal phase transitions. Any modern textbook on Statistical Physics will give the background to understand these analogies, and students not interested in the analogies may simply ignore them. Some readers may be acquainted with the scaling theory of phase transitions developed during the last twenty years and honoured by the 1982 Nobel prize for Kenneth G.Wilson. They will notice that many aspects of percolation theory are simply borrowed from the physics of phase transitions, as will be mentioned in the text. On the other hand, a reader who has not yet learned scaling theory or renormalization group methods for general phase transitions, but who wants to know something about them, may use percolation theory as a starting point. In many respects percolation is the simplest not exactly solved phase transition and thus may serve as an introduction to the sometimes more difficult review articles or books on phase transitions and critical phenomena.

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