FRACTALS

PHYSICS OF SOLIDS AND LIQUIDS

Editorial Board: Jozef T. Devreese • University of Antwerp, Belgium Roger P. Evrard • University of Liege, Belgium Stig Lundqvist • Chalmers University of Technology, Sweden Gerald D. Mahan • The University of Tennessee, Knoxville, Tennessee Norman H. March • University of Oxford, England

AMORPHOUS SOLIDS AND THE LIQUID STATE Edited by Norman H. March, Robert A. Street, and Mario P. Tosi

CHEMICAL BONDS OUTSIDE METAL SURFACES Norman H. March

CRYSTALLINE SEMICONDUCTING MATERIALS AND DEVICES Edited by Paul N. Butcher, Norman H. March, and Mario P. Tosi

ELECTRON SPECTROSCOPY OF CRYSTALS V. V. Nemoshkalenko and V. G. Aleshin

FRACfALS Jens Feder

mGHLY CONDUCTING ONE-DIMENSIONAL SOLIDS Edited by Jozef T. Devreese, Roger P. Evrard, and Victor E. van Doren

MANY·PARTICLE PHYSICS Gerald D. Mahan

ORDER AND CHAOS IN NONLINEAR PHYSICAL SYSTEMS Edited by Stig Lundqvist, Norman H. March, and Mario P. Tosi

THE PHYSICS OF ACTINIDE COMPOUNDS Paul Erdos and John M. Robinson

POLYMERS, LIQUID CRYSTALS, AND LOW·DIMENSIONAL SOLIDS Edited by Norman H. March and Mario P. Tosi

SUPERIONIC CONDUCTORS Edited by Gerald D. Mahan and Walter L. Roth

THEORY OF THE INHOMOGENEOUS ELECTRON GAS Edited by Stig Lundqvist and Norman H. March

A Continuation Order Plan is available for this series. A continuation order will brio, delivery of each new volume immediately upon publication. Volumes are billed only upon actual shipment. For further informa­tion please contact the publisher.

FRACTALS Jens Feder

~pa~n1entofPhysics

University of Oslo Oslo, Norway

Springer Science + Business Media, LLC

Library of Congress Cataloging in Publication Data

Feder, Jens. Fractals.

(Physics of solids and liquids) Bibliography: p. includes index. 1. Fractals. I. Title D. Series.

QA447.J46 1988 516M3 87-31447 ISBN 978-1-4899-2126-0 ISBN 978-1-4899-2124-6 (eBook) DOI 10.1007/978-1-4899-2124-6

First Printing—May 1988 Second Printing—August 1988

Third Printing—April 1989 Fourth Printing—November 1989

© 1988 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1988

Softcover reprint of the hardcover 1st edition 1988

All rights reserved

No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming,

recording, or otherwise, without written permission from the Publisher

for Liv, Heidi, and Brummen

Foreword

This lovely little book will take off and fly on its own power, but the author has asked me to write a few words, and one should not say no to a friend. Specific topics in fractal geometry and its applications have already benefited from several excellent surveys of moderate length, and gossip and preliminary drafts tell us that we shall soon see several monographic treatments of broader topics. For the teacher, however, these surveys and monographs are not enough, and an urgent need for more helpful books has been widely recognized. To write such a book is no easy task, but Jens Feder meets the challenge head on. His approach combines the old Viking's willingness to attack many difficulties at the same time, and the modern Norwegian's ability to achieve fine balance between diverging needs. lowe him special gratitude for presenting the main facts about R/ S analysis of long-run dependence; now a wide scientific public will have access to a large group of papers of mine that had until this day remained fairly confidential.

Last but not least, we are all grateful to Jens for not having allowed undue personal modesty to deprive us of accounts of his own group's varied and excellent work. He did not attempt to say everything, but what he said is just fine.

Benoit B. Mandelbrot Physics Department, IBM Thomas J. Watson Research Center Yorktown Heights, New York 10598 and Mathematics Department, Yale University New Haven, Connecticut 06520

Vll

Preface

This book grew out of research on phase-transitions, on the aggregation of immunoglobulins and recently on the viscous fingering of fluid displace­ment in porous media. These research subjects represent examples of the general question: How is the microscopic behavior related to what we ob­serve on the macroscopic scale? I now feel that fractals, relating geometry on different scales, are essential for the description and understanding of this relation. My friend Torstein lfoSSSang and I have for many years at­tempted to gain insight into the connection between microscopic physics and macroscopic phenomena by means of experiments, theory and com­puter simulations. Much of what I know I have learned through association with Torstein. He also contributed to early lecture notes and reports on fractals. However, he felt that I should write a monograph. We have also had the benefit of working with many talented students.

This book contains some of the topics I found particularly interesting and useful in teaching and in our research. Many more ideas and interesting facets of fractals are found in Mandelbrot's books and in the rapidly grow­ing research literature. I would like to apologize to all colleagues whose works have not been cited as my aim has been to write an introduction that may be useful for those who want to use fractals and not to write an exhaustive review.

The interesting phenomena that occur in the displacement of fluids in porous media became a focus of research in our cooperation with Den norske stats oljeselskap als (Statoil). This cooperation has exposed us to many questions of practical interest that have quickly evolved into questions of basic research. Our research effort and our students have benefited from the generous support of VISTA, a cooperative research effort between

ix

x PREFACE

the Norwegian Academy of Science and Letters and Statoil, initiated by director Henrik Ager-Hanssen.

Dr. Per Stokke at Statoil raised many interesting questions and asked for several reports on the possible applications of fractals relating to geo­logical, geochemical and other subjects of direct interest in oil exploration. These reports started the process of actually writing this book. Teaching a course to our students on the application of fractals pushed the project along and raised many new questions.

Amnon Aharony has encouraged me in writing this book and I have learned much from his many constructive remarks. Ivar Giaever read a preliminary version and made many penetrating comments and suggestions that I have tried to incorporate. I learned much from discussions with Paul Meakin. Jens Lothe has commented on several parts of the book, and I am also indebted to him as my thesis advisor. Harry Thomas made helpful comments on my first writings on the subject. During the summer of 1986, I visited Erling Pytte at the IBM Thomas J. Watson Research Center, and he suggested many improvements to the book. Benoit Mandelbrot read preliminary versions of my book during the summer and we had many interesting discussions. He pointed out, with humor and patience, errors and misunderstandings, made valuable suggestions and encouraged me in many ways. I am grateful to him for his inspiration and help.

Jan Fr{6yland has contributed much to the analysis of wave-height statistics. He has also generated many of the random translation surfaces shown in chapter 13. Many of the students J{6ssang and I have in our group have contributed directly to this book. Knut J{6rgen Mal{6y has with ingenu­ity done the experiments on viscous fingering in porous media. U nni Oxaal has made experiments on fluid displacement in micromodels of controlled geometry. Einar Hinrichsen carried out DLA simulations and also made many useful comments on the manuscript. Finn Boger has contributed to the analysis of experimental results. He has also developed programs that have generated the fractal landscapes and clouds shown in this book. Liv FUruberg simulated percolation processes and has contributed many of the figures in chapter 7.

Liv Feder created most of the illustrations that appear in the book. She has helped me in many ways, and this book would not have been written without her patience and support.

PREFACE .

Xl

Without doubt, this book can be improved. If you have comments or suggestions, I would be pleased to have them.

Jens Feder Oslo, Norway

Contents

Color Plates XVll

1 Introduction 1

2 The Fractal Dimension 6

2.1 The Coast of Norway . 6

2.2 The Schwarz Area Paradox 9

2.3 The Fractal Dimension . 11

2.4 The Triadic Koch Curve . . . . 15

2.5 Similarity and Scaling . 18

2.6 Mandelbrot-Given and Sierpinski Curves . 22

2.7 More on Scaling. . . . . . . . . . . . . 26

2.8 The Weierstrass-Mandelbrot Function 27

3 The Cluster Fractal Dimension 31

36 3.1 Measurements of Cluster Fractal Dimensions

4 Viscous Fingering in Porous Media 41

4.1 Fluid Flow in the Hele-Shaw Cell . . 41

4.2 Viscous Fingers in Hele-Shaw Cells . 45

4.3 Viscous Fingers in Two-Dimensional Porous Media 49

4.4 Viscous Fingering and DLA . . . . . . . . . . . . . 53

4.5 Viscous Fingers in Three-Dimensional Porous Media 56

xiii

· XIV CONTENTS

5 Cantor Sets 62

5.1 The Triadic Cantor Set 62

5.2 Scaling with Unequal Ratios. . 64

6 Multifractal Measures 66

6.1 Curdling and the Devil's Staircase 67

6.2 The Binomial Multiplicative Process 70

6.3 Fractal Subsets . . . . . . . . . . . 73

6.4 The Lipschitz-Holder Exponent a . 75

6.5 The I(a) Curve. . . . . . . . . . . . 76

6.6 The Measure's Concentrate . . . . . 78

6.7 The Sequence of Mass Exponents T(q) 80

6.8 The Relation between T(q) and I(a) . 83

6.9 Curdling with Several Length Scales 85

6.10 Multifractal Rayleigh-Benard Convection 89

6.11 DLA and the Harmonic Measure . . . . 92

6.12 Multifractal Growth of Viscous Fingers 96

7 Percolation 104

7.1 Site Percolation on a Quadratic Lattice 105

7.2 The Infinite Cluster at Pc ....•... 109

7.3 Self-Similarity of Percolation Clusters 112

7.4 Finite Clusters at Percolation . . . . 117

7.5 The Cluster Size Distribution at Pc . 120

7.6 The Correlation Length e . . . . . 122

7.7 The Percolation Cluster Backbone 126

7.8 Invasion Percolation . . . . . 131

7.9 The Fractal Diffusion Front 139

8 Fractal Records in Time 149

8.1 Hurst's Empirical Law and Rescaled Range Analysis 149

8.2 Simulations of Random Records. . . . 154

8.3 Simulations of Long-Term Dependence 157

CONTENTS

9 Random Walks and Fractals

9.1 Brownian Motion . . . . . .

9.2 Random Walk in One Dimension ........... . 9.3 Scaling Properties of One-Dimensional Random Walks

9.4 Fractional Brownian Motion ........ .

9.5 Definition of Fractional Brownian Motion . .

9.6 Simulation of Fractional Brownian Motion ..

9.7 RIB Analysis of Fractional Brownian Motion

9.8 Successive Random Addition .

10 Self-Similarity and Self-Affinity

10.1 The Strategy of Bold Play.

11 Wave-Height Statistics

11.1 RIS Analysis of the Observed h,

11.2 RIB for Seasonally Adjusted Data

12 The Perimeter-Area Relation

12.1 The Fractal Dimension of Clouds

12.2 The Fractal Dimension of Rivers

13 Fractal Surfaces

13.1 The Fractal Koch Surface

13.2 Random Translation Surfaces 13.3 Generating Fractal Surfaces .

13.4 Random Addition Surfaces .

13.5 Comments on Fractal Landscapes ..

14 Observations of Fractal Surfaces

14.1 Observed Surface Topography ..

14.2 D for Landscapes and Environmental Data 14.3 Molecular Fractal Surfaces.

References

Author Index

Subject Index

xv

163 163

164 166

170

172

173

178

180

184

189

193

194

195

200

202

208

212

212

214

216

221

228

229

229

234

236

244

258

265

Color Plates

FIGURE C.l: The effect of increasing the occupation probability p on a 160 x 160 quadratic lattice. From top to bottom we have p = 0.58, 0.6 (see facing page) and 0.62 above. In each of the three figures the largest cluster is white. The other clusters are colored according to decreasing size with the colors cyan J red, orange J yellowJ light green J green, turquoise and blue in the shades light to dark. The smallest clusters are not visible with this coloring scheme.

FIGURE C.2: (a) Air displacing glycerol at a high capillary number on the percolation cluster shown in figure 7.13. (b) The results of numerical simu­lation of fluid displacement on the same percolation cluster. The different colors represent pores invaded by air observed at successive time steps. The number of pores invaded by air is 30 - white, 86 - red, 213 - green and finally at breakthrough 447 - yellow, for both the experiment and the simulation (Oxaal et al., 1987).

FIGURE C.3: A system of diffusing particles at large scale. The screen is 150 x 150. The particles connected to the source are green, empty sites connected to the sink are turquoise. The particles in isolated clusters (is­lands) are dark green, isolated empty sites (lakes) are dark blue. Sites in the hull are black.

FIGURE C.4: Fractal landscape with H = 0.75 generated using a scale factor r = 112 on a 2049 x 2049 lattice. (a) The central 1024 x 800 portion seen from above. (b) The landscape with perspective and curvature. (c)

The complete landscape seen from above. (d) The complete landscape presented as 'clouds.' For a discussion see section 13.4 (Boger et al., 1987).

b

FIGURE C.5: Fractal landscapes presented as 'clouds.' (a) H = 0.5 land­scape (see figure 13.11). (b) H = 0.7 landscape (see figures 13.11 and 13.13) (Boger et al., 1987).

FRACTALS