Fractals and disordered systems

Author(s)

Bibliographic Information

Fractals and disordered systems

Armin Bunde, Shlomo Havlin (eds.)

Springer, c1996

2nd revised and Enlarged ed.

  • : pbk.

Available at  / 44 libraries

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Note

Includes bibliographical references and index

1st ed. published in 1991

Description and Table of Contents

Volume

ISBN 9783540562191

Description

A discussion of the effects of disorder on mesoscopic scales (fractures, aggregates, colloids, surfaces and interfaces, glasses and polymers) including a presentation of tools to describe them in mathematical language. This text introduces the reader to basic concepts and techniques in disordered systems whilst a large part of the work is devoted to the development of scaling theories based on fractal concepts.

Table of Contents

1. Fractals and Multifractals: The Interplay of Physics and Geometry 2. Percolation I 3. Percolation II 4. Fractal Growth 5. Fractures 6. Transport Across Irregular Interfaces: Fractal Electrodes, Membranes, and Catalysts 7. Fractal Surfaces and Interfaces 8. Fractals and Experiments 9. Cellular Automata 10. Exactly Self-similar Left-sided Multifractals
Volume

: pbk. ISBN 9783642848704

Description

Fractals and disordered systems have recently become the focus of intense interest in research. This book discusses in great detail the effects of disorder on mesoscopic scales (fractures, aggregates, colloids, surfaces and interfaces, glasses and polymers) and presents tools to describe them in mathematical language. A substantial part is devoted to the development of scaling theories based on fractal concepts. In ten chapters written by leading experts in the field, the reader is introduced to basic concepts and techniques in disordered systems and is led to the forefront of current research. This second edition has been substantially revised and updates the literature in this important field.

Table of Contents

1 Fractals and Multifractals: The Interplay of Physics and Geometry (With 30 Figures).- 1.1 Introduction.- 1.2 Nonrandom Fractals.- 1.3 Random Fractals: The Unbiased Random Walk.- 1.4 The Concept of a Characteristic Length.- 1.5 Functional Equations and Fractal Dimension.- 1.6 An Archetype: Diffusion Limited Aggregation.- 1.7 DLA: Fractal Properties.- 1.8 DLA: Multifractal Properties.- 1.8.1 General Considerations.- 1.8.2 "Phase Transition" in 2d DLA.- 1.8.3 The Void-Channel Model of 2d DLA Growth.- 1.8.4 Multifractal Scaling of 3d DLA.- 1.9 Scaling Properties of the Perimeter of 2d DLA: The "Glove" Algorithm.- 1.9.1 Determination of the l Perimeter.- 1.9.2 The l Gloves.- 1.9.3 Necks and Lagoons.- 1.10 Multiscaling.- 1.11 The DLA Skeleton.- 1.12 Applications of DLA to Fluid Mechanics.- 1.12.1 Archetype 1: The Ising Model and Its Variants.- 1.12.2 Archetype 2: Random Percolation and Its Variants.- 1.12.3 Archetype 3: The Laplace Equation and Its Variants.- 1.13 Applications of DLA to Dendritic Growth.- 1.13.1 Fluid Models of Dendritic Growth.- 1.13.2 Noise Reduction.- 1.13.3 Dendritic Solid Patterns: "Snow Crystals".- 1.13.4 Dendritic Solid Patterns: Growth of NH4Br.- 1.14 Other Fractal Dimensions.- 1.14.1 The Fractal Dimension dw of a Random Walk.- 1.14.2 The Fractal Dimension dmin ? 1/?? of the Minimum Path.- 1.14.3 Fractal Geometry of the Critical Path: "Volatile Fractals".- 1.15 Surfaces and Interfaces.- 1.15.1 Self-Similar Structures.- 1.15.2 Self-Affine Structures.- 1.A Appendix: Analogies Between Thermodynamics and Multifractal Scaling.- References.- 2 Percolation I (With 24 Figures).- 2.1 Introduction.- 2.2 Percolation as a Critical Phenomenon.- 2.3 Structural Properties.- 2.4 Exact Results.- 2.4.1 One-Dimensional Systems.- 2.4.2 The Cayley Tree.- 2.5 Scaling Theory.- 2.5.1 Scaling in the Infinite Lattice.- 2.5.2 Crossover Phenomena.- 2.5.3 Finite-Size Effects.- 2.6 Related Percolation Problems.- 2.6.1 Epidemics and Forest Fires.- 2.6.2 Kinetic Gelation.- 2.6.3 Branched Polymers.- 2.6.4 Invasion Percolation.- 2.6.5 Directed Percolation.- 2.7 Numerical Approaches.- 2.7.1 Hoshen-Kopelman Method.- 2.7.2 Leath Method.- 2.7.3 Ziff Method.- 2.8 Theoretical Approaches.- 2.8.1 Deterministic Fractal Models.- 2.8.2 Series Expansion.- 2.8.3 Small-Cell Renormalization.- 2.8.4 Potts Model, Field Theory, and ? Expansion.- 2.A Appendix: The Generating Function Method.- References.- 3 Percolation II (With 20 Figures).- 3.1 Introduction.- 3.2 Anomalous Transport in Fractals.- 3.2.1 Normal Transport in Ordinary Lattices.- 3.2.2 Transport in Fractal Substrates.- 3.3 Transport in Percolation Clusters.- 3.3.1 Diffusion in the Infinite Cluster.- 3.3.2 Diffusion in the Percolation System.- 3.3.3 Conductivity in the Percolation System.- 3.3.4 Transport in Two-Component Systems.- 3.3.5 Elasticity in Two-Component Systems.- 3.4 Fractons.- 3.4.1 Elasticity.- 3.4.2 Vibrations of the Infinite Cluster.- 3.4.3 Vibrations in the Percolation System.- 3.4.4 Quantum Percolation.- 3.5 ac Transport.- 3.5.1 Lattice-Gas Model.- 3.5.2 Equivalent Circuit Model.- 3.6 Dynamical Exponents.- 3.6.1 Rigorous Bounds.- 3.6.2 Numerical Methods.- 3.6.3 Series Expansion and Renormalization Methods.- 3.6.4 Continuum Percolation.- 3.6.5 Summary of Transport Exponents.- 3.7 Multifractals.- 3.7.1 Voltage Distribution.- 3.7.2 Random Walks on Percolation.- 3.8 Related Transport Problems.- 3.8.1 Biased Diffusion.- 3.8.2 Dynamic Percolation.- 3.8.3 The Dynamic Structure Model of Ionic Glasses.- 3.8.4 Trapping and Diffusion Controlled Reactions.- References.- 4 Fractal Growth (With 4 Figures).- 4.1 Introduction.- 4.2 Fractals and Multifractals.- 4.3 Growth Models.- 4.3.1 Eden Model.- 4.3.2 Percolation.- 4.3.3 Invasion Percolation.- 4.4 Laplacian Growth Model.- 4.4.1 Diffusion Limited Aggregation.- 4.4.2 Dielectric Breakdown Model.- 4.4.3 Viscous Fingering.- 4.4.4 Biological Growth Phenomena.- 4.5 Aggregation in Percolating Systems.- 4.5.1 Computer Simulations.- 4.5.2 Viscous Fingers Experiments.- 4.5.3 Exact Results on Model Fractals.- 4.5.4 Crossover to Homogeneous Behavior.- 4.6 Crossover in Dielectric Breakdown with Cutoffs.- 4.7 Is Growth Multifractal?.- 4.8 Conclusion.- References.- 5 Fractures (With 18 Figures).- 5.1 Introduction.- 5.2 Some Basic Notions of Elasticity and Fracture.- 5.2.1 Phenomenological Description.- 5.2.2 Elastic Equations of Motion.- 5.3 Fracture as a Growth Model.- 5.3.1 Formulation as a Moving Boundary Condition Problem.- 5.3.2 Linear Stability Analysis.- 5.4 Modelisation of Fracture on a Lattice.- 5.4.1 Lattice Models.- 5.4.2 Equations and Their Boundary Conditions.- 5.4.3 Connectivity.- 5.4.4 The Breaking Rule.- 5.4.5 The Breaking of a Bond.- 5.4.6 Summary.- 5.5 Deterministic Growth of a Fractal Crack.- 5.6 Scaling Laws of the Fracture of Heterogeneous Media.- 5.7 Hydraulic Fracture.- 5.8 Conclusion.- References.- 6 Transport Across Irregular Interfaces: Fractal Electrodes, Membranes and Catalysts (With 8 Figures).- 6.1 Introduction.- 6.2 The Electrode Problem and the Constant Phase Angle Conjecture.- 6.3 The Diffusion Impedance and the Measurement of the Minkowski-Bouligand Exterior Dimension.- 6.4 The Generalized Modified Sierpinski Electrode.- 6.5 A General Formulation of Laplacian Transfer Across Irregular Surfaces.- 6.6 Electrodes, Roots, Lungs,.- 6.7 Fractal Catalysts.- 6.8 Summary.- References.- 7 Fractal Surfaces and Interfaces (With 27 Figures).- 7.1 Introduction.- 7.2 Rough Surfaces of Solids.- 7.2.1 Self-Affine Description of Rough Surfaces.- 7.2.2 Growing Rough Surfaces: The Dynamic Scaling Hypothesis.- 7.2.3 Deposition and Deposition Models.- 7.2.4 Fractures.- 7.3 Diffusion Fronts: Natural Fractal Interfaces in Solids.- 7.3.1 Diffusion Fronts of Noninteracting Particles.- 7.3.2 Diffusion Fronts in d = 3.- 7.3.3 Diffusion Fronts of Interacting Particles.- 7.3.4 Fluctuations in Diffusion Fronts.- 7.4 Fractal Fluid-Fluid Interfaces.- 7.4.1 Viscous Fingering.- 7.4.2 Multiphase Flow in Porous Media.- 7.5 Membranes and Tethered Surfaces.- 7.6 Conclusions.- References.- 8 Fractals and Experiments (With 18 Figures).- 8.1 Introduction.- 8.2 Growth Experiments: How to Make a Fractal.- 8.2.1 The Generic DLA Model.- 8.2.2 Dielectric Breakdown.- 8.2.3 Electrodeposition.- 8.2.4 Viscous Fingering.- 8.2.5 Invasion Percolation.- 8.2.6 Colloidal Aggregation.- 8.3 Structure Experiments: How to Determine the Fractal Dimension.- 8.3.1 Image Analysis.- 8.3.2 Scattering Experiments.- 8.3.3 Sacttering Formalism.- 8.4 Physical Properties.- 8.4.1 Mechanical Properties.- 8.4.2 Thermal Properties.- 8.5 Outlook.- References.- 9 Cellular Automata (With 6 Figures).- 9.1 Introduction.- 9.2 A Simple Example.- 9.3 The Kauffman Model.- 9.4 Classification of Cellular Automata.- 9.5 Recent Biologically Motivated Developments.- 9.A Appendix.- 9.A.1 Q2R Approximation for Ising Models.- 9.A.2 Immunologically Motivated Cellular Automata.- 9.A.3 Hydrodynamic Cellular Automata.- References.- 10 Exactly Self-similar Left-sided Multifractals with new Appendices B and C by Rudolf H. Riedi and Benoit B. Mandelbrot (With 10 Figures).- 10.1 Introduction.- 10.1.1 Two Distinct Meanings of Multifractality.- 10.1.2 "Anomalies".- 10.2 Nonrandom Multifractals with an Infinite Base.- 10.3 Left-sided Multifractality with Exponential Decay of Smallest Probability.- 10.4 A Gradual Crossover from Restricted to Left-sided Multifractals.- 10.5 Pre-asymptotics.- 10.5.1 Sampling of Multiplicatively Generated Measures by a Random Walk.- 10.5.2 An "Effective" f(?).- 10.6 Miscellaneous Remarks.- 10.7 Summary.- 10.A Details of Calculations and Further Discussions.- 10.A.1 Solution of (10.2).- 10.A.2 The Case ?min = 0.- 10.B Multifractal Formalism for Infinite Multinomial Measures, by R.H. Riedi and B.B. Mandelbrot.- 10.C The Minkowski Measure and Its Left-sided f(?), by B.B. Mandelbrot.- 10.C.1 The Minkowski Measure on the Interval [0,1].- 10.C.2 The Functions f(?) and f?(?) of the Minkowski Measure.- 10.C.3 Remark: On Continuous Models as Approximations, and on "Thermodynamics".- 10.C.4 Remark on the Role of the Minkowski Measure in the Study of Dynamical Systems. Parabolic Versus Hyperbolic Systems.- 10.C.5 In Lieu of Conclusion.- References.

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