Fractals and Disordered Systems

1. Fractals and Multifractals: The Interplay of Physics and Geometry.- 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.- 1.9 "Phase Transition" in DLA.- 1.10 The Void-Channel Model of DLA Growth.- 1.11 Applications of DLA to Fluid Mechanics.- 1.11.1 Archetype 1: The Ising Model and Its Variants.- 1.11.2 Archetype 2: Random Percolation and Its Variants.- 1.11.3 Archetype 3: The Laplace Equation and Its Variants.- 1.12 Applications of DLA to Dendritic Growth.- 1.12.1 Fluid Models of Dendritic Growth.- 1.12.2 Noise Reduction.- 1.12.3 Dendritic Solid Patterns: "Snow Crystals".- 1.12.4 Dendritic Solid Patterns: Growth of NH4Br.- 1.13 Other Fractal Dimensions.- 1.13.1 The Fractal Dimension dw of a Random Walk.- 1.13.2 The Fractal Dimension dmin = 1/v of the Minimum Path.- 1.13.3 Surfaces and Interfaces of Fractal Structures.- 1.13.4 Fractal Geometry of the Critical Path: "Volatile Fractals".- 1.14 Possible Origin of Fractality and Multifractality.- 1.A Appendix: Analogies with Thermodynamics and Multifractal Scaling.- References.- 2. Percolation I.- 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 Invasion Percolation.- 2.6.4 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.- References.- 3. Percolation II.- 3.1 Introduction.- 3.2 Anomalous Transport on Fractals.- 3.2.1 Normal Transport in Ordinary Lattices.- 3.2.2 Transport in Fractal Substrates.- 3.3 Transport on Percolation Clusters.- 3.3.1 Diffusion on the Infinite Cluster.- 3.3.2 Diffusion in the Percolation System.- 3.3.3 Conductivity in the Percolation System.- 3.3.4 Transport in General 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.- 3.5 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 Transport in the Presence of Additional Physical Constraints.- 3.8.1 Biased Diffusion.- 3.8.2 Dynamic Percolation.- 3.8.3 Trapping and Diffusion Controlled Reactions.- References.- 4. Fractal Growth.- 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.- 4.5 Aggregation in Percolating Systems.- 4.5.1 Computer Simulations.- 4.5.2 Viscous Finger Experiments.- 4.5.3 Exact Results on Model Fractals.- 4.5.4 Crossover to Homogeneous Behaviour.- 4.6 Crossover in Dielectric Breakdown with Cutoffs.- 4.7 Is Growth Multifractal?.- 4.8 Conclusion.- References.- 5. Fractures.- 5.1 Introduction.- 5.2 Some Basic Notions of Elasticity.- 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 Conclusion.- References.- 6. Fractal Electrodes, Fractal Membranes, and Fractal Catalysts.- 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 The Response of Ideally Polarizable Electrodes.- 6.6 Scaling Length in the Blocking Regime.- 6.7 Electrodes, Roots, Lungs.- 6.8 Fractal Catalysts.- 6.9 Summary.- References.- 7. Fractal Surfaces and Interfaces.- 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 of Interacting Particles.- 7.3.3 Diffusion Fronts in d = 3.- 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.- 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 Scattering Formalism.- 8.4 Physical Properties.- 8.4.1 Mechanical Properties.- 8.4.2 Thermal Properties.- 8.5 Outlook.- References.- 9. Cellular Automata.- 9.1 Introduction.- 9.2 A Simple Example.- 9.3 The Kauffman Model.- 9.4 Classification of Cellular Automata.- 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.- 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 (a).- 10.6 Miscellaneous Remarks.- 10.7 Summary.- 10.A Appendix.- 10.A.1 Solution of (10.2).- 10.A.2 The case ?min= 0.- References.