Airways of the lung. The bronchi and bronchioles of the lung fonna "tree" that has multiple generations ofbranchings. The small-scale branching of

Transcription

1 Fractals in Science

2 Airways of the lung. The bronchi and bronchioles of the lung fonna "tree" that has multiple generations ofbranchings. The small-scale branching of the airways look like branching at larger scales. Courtesy of Christopher Burke, Quesada!Burke Studios, New York.

3 Armin Bunde Shlomo Havlin (Eds.) Fractals in Science With a Macintosh Program Diskette, 120 Figures and 10 Color Plates Springer-Verlag Berlin Heidelberg GmbH

4 Professor Dr. Armin Bun de Institut fur Theoretische Physik Universităt Giessen Heinrich-Buff-Ring 16 D Giessen Germany Professor Dr. Shlomo Havlin Department of Physlcs Bar-Ilan University RamatGan Israel ISBN ISBN (ebook) DOI / Additional material to this book can be downloaded from CIP data applied for. This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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 1994 Originally pnb1ished by Springer-Ver1ag Berlin Heide1berg New York in 1994 Softcover reprint of the hardcover 1 st edition 1994 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Please note: Before using the programs in this book, please consult the technical manuals provided by the manufacturer ofthe computer- and of any additional plug-in boards- tobe used. The authors and the publisher accept no legal responsibility for any damage by improper use of the instructions and programs contained herein. Although these programs ha ve been tested with extreme care, we can offer no formal guarantee that they will function correctly. The programs on the enclosed diskette are under copyright-protection and may not be reproduced without written permission by Springer-Verlag. One copy ofthe programs may be made as a back-up, but al! further copies offend copyright law. Camera-ready copy from the authors/editors using a Springer TEX macro package Production Editor: P. Treiber SPIN: / Printed on acid-free paper

5 Preface Applying fractal geometry to science is bringing about a breakthrough in our understanding of complex systems in nature that show self-similar or self-affine features. Self-similar and self-affine processes appear everywhere in nature, in galaxies and landscapes, in earthquakes and geological cracks, in aggregates and colloids, in rough surfaces and interfaces, in glassy materials and polymers, in proteins as well as in other large molecules. Fractal structures appear also in the human body; well known examples include the lung and the vascular system. Furthermore, fractal geometry is an important tool in the analysis of phenomena as diverse as rhythms in music melodies and in the human heartbeat and DNA sequences. Since the pioneering work of B.B. Mandelbrot, this interdisciplinary field has expanded very rapidly. The scientific community applying fractal concepts is very broad and ranges from astronomers, geoscientists, physicists, chemists and engineers to biologists and those engaging in medical research. The purpose of this book is to provide easy access to fractals in science and to bridge the gap between the different disciplines. Similar in style to the previous book Fractals and Disordered Systems in which the main emphasis was on fractals in materials science, all chapters are written in a uniform notation, and cross-references in each chapter to related subjects in other chapters are provided. In each chapter emphasis is placed on the various connections between theory and experiment. A special chapter (Chap. 9) entitled "Computer Exploration of Fractals, Chaos, and Cooperativity" presents computer demonstrations of fractal models. A diskette of these interactive programs, for either Macintosh or PC-compatible computers, is enclosed. The first chapter, by A. Bunde and S. Havlin, is for beginners in the field and serves to introduce the basic ideas and concepts of fractal geometry. The second chapter, by P. Bak and M. Creutz, deals with self-organized criticality, a process that may explain why fractals occur so widely in nature. In the third chapter, S.V. Buldyrev, A.L. Goldberger, S. Havlin, C.-K. Peng, and H.E. Stanley describe fractal processes in biology and medicine with particular emphasis on novel applications of fractal landscape analysis to DNA sequences and cardiac rhythms. In Chap. 4, J. Kertesz and T. Vicsek present an introduction

6 VI Preface to the new and fascinating field of self-affine fractal surfaces generated by natural processes like fractures, erosion, imbibition, and burning. In Chap. 5, G. H. Weiss introduces the reader to the theory of diffusion and random walks, which represent the basic mechanisms for disorder in nature, and describes several applications to disordered media, semiconductors, and ecology. M. Daoud reviews, in Chap. 6, fractal applications to polymer science, with emphasis on single polymer chains, polymer solutions, melts, branched polymers, and gels. Chapter 7, by S. Redner and F. Leyvraz, introduces the reader to the recent developments in chemical reactions controlled by diffusion, a study relevant to a wide range of processes including electron-hole recombination in semiconductors, and catalytic reactions. In Chap. 8, D. Avnir, R. Gutfraind and D. Farin discuss the use of fractal analysis in heterogeneous chemistry and demonstrate the importance of fractal geometry to relevant chemical processes, including the fundamental pharmacological problem of controlled drug release. In Chap. 9, D. Rapaport and M. Meyer present interactive computer demonstrations of basic fractal models. We wish to thank first and foremost the authors, and also our colleagues H. Bolterauer, H. Brender, L. Lam, R. Nossal, S. Rabinovich, H. E. Roman, and H. Taitelbaum for useful discussions. We kindly acknowledge the help of S.V. Buldyrev, S. Glotzer, S. Harrington, M. Meyer, M. Sernetz, and P. Trunfio, who contributed the color figures. We also wish to thank H.J. Kolsch and P. Treiber from Springer-Verlag Heidelberg for their continuous help during the preparation of this book. We hope that Fractals in Science can be used as a textbook for graduate students, for teachers at universities preparing courses or seminars and for researchers in a variety of fields who are about to encounter fractals in their own work. Armin Bunde Shlomo Havlin Giessen, Ramat-Gan, February 1994

7 Contents 1 A Brief Introduction to Fractal Geometry By A. Bunde and S. Havlin (With 22 Figures) 1.1 Introduction Deterministic Fractals The Koch Curve The Sierpinski Gasket, Carpet, and Sponge The Durer Pentagon The Cantor Set The Mandelbrot-Given Fractal Julia Sets and the Mandelbrot Set.... Random Fractal Models Random Walks Self-A voiding Walks Kinetic Aggregation Percolation How to Measure the Fractal Dimension The Sandbox Method The Box Counting Method Self-Affine Fracals Fractals in Nature References Fractals and Self-Organized Criticality By P. Bak and M. Creutz (With 10 Figures) 2.1 Introduction Simulations of Sandpile Models

8 VIII Contents 2.3 Abelian Sandpile Models The Abelian Group An Isomorphism A Burning Algorithm and the q = 0 Potts Model Real Sandpiles and Earthquakes The Dynamics of Sand Earthquakes and SOC /f Noise On Forest Fires and Turbulence References 47 3 Fractals in Biology and Medicine: From DNA to the Heartbeat By S.V. Buldyrev, A.L. Goldberger, S. Havlin, C.-K. Peng, and H.E. Stanley (With 17 Figures) 3.1 Introduction Fractal Shapes Long-Range Power Law Correlations Information Coding in DNA Conventional Statistical Analysis of DNA Sequences The "DNA Walk" Graphical Representation Correlations and Fluctuations Other Methods of Measuring Long-Range Correlations Differences Between Correlation Properties of Coding and Noncoding Regions Long-Range Correlations and Evolution Models of DNA Evolution Long-Range Correlations and DNA Spatial Structure Other Biological Systems with Long-Range Correlations The Human Heartbeat Physiological Implications Human Writings Dynamics of Membrane Channel Openings Fractal Music and the Heartbeat Fractal Approach to Biological Evolution References 83

9 Contents IX 4 Self-Affine Interfaces By J. Kertesz and T. Vicsek (With 10 Figures) 4.1 Introduction Roughness and Pinning in Equilibrium Dynamic Scaling and Growth Models Continuum Equations, Directed Polymers, and Morphological Transitions Effects of Correlated, Power-Law, and Quenched Noise: Nonuniversal Roughening and Pinning Correlated Noise Noise with Power-Law Distributed Amplitudes Quenched Noise Summary References A Primer of Random Walkology By G.H. Weiss (With 11 Figures) 5.1 Introduction Basic Formalism Jump Probabilities Characteristic Functions The Continuous-Time Random Walk (CTRW) The Characteristic Function and Properties of the Lattice Random Walk Asymptotic Properties The Central-Limit Theorem and Some Generalizations The Diffusion Approximation A Mathematical Excursion: Abelian and Tauberian Theorems Asymptotic Properties of the CTRW in an Unbounded Space Asymptotic Properties of Random Walks on a Lattice: Recurrent and Transient Behavior The Expected Number of Distinct Sites Visited by an n-step Random Walk Random Walks in Disordered Media Introductory Remarks The Trapping Model

10 X Contents Some Models Based on the CTRW The Effective-Medium Approximation References Polymers By M. Daoud (With 14 Figures) 6.1 Introduction Linear Chains and Excluded Volume The Random Walk The Self-Avoiding Walk Dilute Solutions Semi-Dilute Solutions Dynamics Adsorption The Single Chain The Plateau Regime Branched Polymers and Gels The Sol-Gel Transition The Flory Approximation Dilute Solutions Semi-Dilute Solutions and Swollen Gels Dynamics References Kinetics and Spatial Organization of Competitive Reactions By S. Redner and F. Leyvraz (With 6 Figures) 7.1 Introduction Irreversible Homogeneous Reactions Decay of the Density Interparticle Distances The Domain-Size Distribution in One Dimension The Domain Profile The Interparticle-Distance Distribution Reactions with Particle Input Steady Input and Diffusing Reactants The Approach to Asymptotic Behavior

11 Contents XI Immobile Reactants; Equivalence to Catalysis, Kinetic Ising Models, and Branching Random Walks Heterogeneous Reaction Conditions Transient Response Steady-State Behavior Concluding Remarks References Fractal Analysis in Heterogeneous Chemistry By D. Avnir, R. Gutfraind, and D. Farin (With 12 Figures) 8.1 Introduction The Reaction Dimension Surface Morphology Effects on Drug Dissolution Size Effects in Catalysis Multifractal Analysis of Catalytic Reactions The Accessibility of Fractal Surfaces to Derivatization Reactions Conclusion References Computer Exploration of Fractals, Chaos, and Cooperativity By Dennis C. Rapaport and Martin Meyer (With 10 Figures) 9.1 Introduction The Software Collection Fractals Deterministic Fractals Stochastic Landscapes Cellular Automata Cell Arrays Cluster Growth Diffusion-Limited Aggregation Invasion Percolation Cooperative Phenomena Ising Model Percolation Many-Body Systems Soft-Disk Fluid

12 XII Contents 9.8 Chaos Logistic Map.... Double Pendulum More Collectivity Polymers Sandpiles Iterative Processes Affine Mappings Mandelbrot Set Summary.... 9A Appendix: Alphabetical Program List.... 9B Appendix: Mathematical Details.... References Subject Index

13 List of Contributors David Avnir Department of Organic Chemistry, The Hebrew University of Jerusalem Jerusalem 91904, Israel Per Bak Department of Physics, Brookhaven National Laboratory Upton, NY 11973, USA Sergey V. Buldyrev Center for Polymer Studies, Boston University Boston, MA 02215, USA Armin Bunde Institut fiir Theoretische Physik, Justus-Liebig-Universitat D Giessen, Germany Michael Creutz Department of Physics, Brookhaven National Laboratory Upton, NY 11973, USA Mohamed Daoud Service de Physique Theorique de Saclay F Gif-Sur-Yvette Cedex, France

14 XIV List of Contributions Dina Farin Department of Organic Chemistry, The Hebrew University of Jerusalem Jerusalem 91904, Israel Ary L. Goldberger Cardiovascular Division, Harvard Medical School Beth Israel Hospital, Boston, MA 02215, USA Ricardo Gutfraind Department of Organic Chemistry, The Hebrew University of Jerusalem Jerusalem 91904, Israel Shlomo Havlin Department of Physics, Bar-Ilan University Ramat-Gan 52100, Israel Janos Kertesz Technical University of Budapest, Budafoki lit 8 H-1521 Budapest, Hungary Francois Leyvraz Instituto de Fisica, Laboratorio de Cuernavaca UNAM, Mexico Martin Meyer Institut ffu Theoretische Physik, Justus-Liebig-UniversiUit D Giessen, Germany C.-K. Peng Center for Polymer Studies, Boston University Boston, MA 02215, USA Dennis C. Rapaport Department of Physics, Bar-Ilan University Ramat-Gan 52100, Israel

15 List of Contributions XV Sidney Redner Center for Polymer Studies, Boston University Boston, MA 02215, USA H. Eugene Stanley Center for Polymer Studies, Boston University Boston, MA 02215, USA Tamas Vicsek E6tv6s University, Department of Atomic Physics Puskin u. 5-7, 1088 Budapest, Hungary George H. Weiss Physical Sciences Laboratory, Division of Computer Research and Technology National Institutes of Health, Bethesda, MD 20205, USA

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