Fractals in Science. Armin Bunde Shlomo Havlin (Eds.) Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest

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1 Armin Bunde Shlomo Havlin (Eds.) Fractals in Science With a MS-DOS Program Diskette, 120 Figures and 10 Color Plates Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest

2 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 Dürer Pentagon The Cantor Set The Mandelbrot-Given Fractal Julia Sets and the Mandelbrot Set Random Fractal Models Random Walks Self-Avoiding Walks Kinetic Aggregation Percolation How to Measure the Fractal Dimension The Sandbox Method The Box Counting Method Self-Affine Fracals Fractals in Nature 23 References 24 2 Fractals and Self-Organized Criticality By P. Bak and M. Creutz (With 10 Figures) 2.1 Introduction Simulations of Sandpile Models 29

3 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 45 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 81 References 83

4 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 114 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 151

5 X Contents Some Models Based on the CTRW The Effective-Medium Approximation 158 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 190 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 212

6 Contents XI Immobile Reactants; Equivalence to Catalysis, Kinetic Ising Models, and Branching Random Walks Heterogeneous Reaction Conditions Transient Response Steady-State Behavior Concluding Remarks 223 References 225 i 8 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 252 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 269

7 XII Contents 9.8 Chaos Logistic Map Double Pendulum More Collectivity Polymers Sandpiles Iterative Processes Affine Mappings Mandelbrot Set Summary 277 9A Appendix: Alphabetical Program List 277 9B Appendix: Mathematical Details 278 References 279 Subject Index 281