An Introduction to Anomalous Diffusion and Relaxation

Description

This book provides a contemporary treatment of the problems related to anomalous diffusion and anomalous relaxation. It collects and promotes unprecedented applications dealing with diffusion problems and surface effects, adsorption-desorption phenomena, memory effects, reaction-diffusion equations, and relaxation in constrained structures of classical and quantum processes. The topics covered by the book are of current interest and comprehensive range, including concepts in diffusion and stochastic physics, random walks, and elements of fractional calculus. They are accompanied by a detailed exposition of the mathematical techniques intended to serve the reader as a tool to handle modern boundary value problems. This self-contained text can be used as a reference source for graduates and researchers working in applied mathematics, physics of complex systems and fluids, condensed matter physics, statistical physics, chemistry, chemical and electrical engineering, biology, and many others. Preface Contents About the Authors Symbols Sets Integral Transforms Special Functions of Fractional Calculus Fractional Operators 1 Integral Transforms and Special Functions 1.1 Integral Transforms 1.1.1 Fourier Transform 1.1.2 Laplace Transform 1.1.3 The Cauchy Integral Formula 1.2 Gamma and Related Functions 1.2.1 The Mellin Transform 1.3 Special Functions of Fractional Calculus 1.3.1 The Functions of the Mittag-Leffler Type 1.3.2 The Wright Function 1.3.3 The H-Function of Fox References 2 Concepts in Diffusion and Stochastic Processes 2.1 Molecular Diffusion 2.1.1 The Mean Square Displacement 2.2 Pioneering Studies on Diffusion Problems 2.3 Brownian Motion 2.3.1 The Works of Einstein, Smoluchowski, and Langevin 2.4 Generalized Langevin Equation 2.4.1 Derivation of a Diffusionlike Equation 2.5 Anomalous Diffusion: Basic Concepts 2.5.1 Superdiffusion 2.5.2 Lévy Flights 2.5.3 Subdiffusion References 3 Random Walks 3.1 Elementary Approach 3.1.1 Stochastic Variables 3.2 Random Walks: Nonlinear Fokker-Planck Equations 3.2.1 Nonlinear Random Walk 3.2.2 Generalized Random Walks 3.3 Continuous-Time Random Walk 3.3.1 Different Diffusive Regimes 3.3.2 Linear Reaction Dynamics 3.3.3 Coupled Jump-Length and Waiting-Time Distributions 3.3.4 Intermittent Continuous-Time Random Walk References 4 Elements of Fractional Calculus 4.1 Introduction 4.2 Early Definitions 4.3 The Riemann-Liouville Operators 4.4 The Grünwald-Letnikov Operators 4.5 The Caputo Operator 4.6 Some Space-Fractional Derivatives References 5 Fractional Anomalous Diffusion 5.1 A Space-Time Fractional Diffusion Equation 5.2 Generalized Space-Time Fractional Diffusion Equation 5.2.1 Composite Time Fractional Operator 5.2.2 Fractional Diffusion Equation: Singular Term 5.3 Tempered Fractional Diffusion Equation 5.4 Fractional Diffusion Equation in Heterogeneous Media 5.4.1 Free Case 5.4.2 External Force 5.5 Heterogeneous Media and Fractional Spatial Operator 5.6 Time Derivative Operators: A Comparison 5.6.1 Diffusion and Time Derivative Operators 5.6.2 Predicted Distributions: A Balance References 6 Adsorption Phenomena and Anomalous Behavior 6.1 Kinetic Equation: Normal and Fractional 6.1.1 Electrical Impedance in Liquid Crystals 6.2 Anomalous Diffusion in Complex Fluids 6.2.1 Diffusion and Surface Dynamics 6.2.2 Time-Dependent Solutions 6.3 Memory Kernels 6.3.1 Chemisorption and Physisorption Processes 6.3.2 Confined Systems: Periodically Varying Medium References 7 Reaction-Diffusion Problems 7.1 Diffusion and Kinetics 7.1.1 Reversible Linear Reaction 7.1.2 Irreversible Linear Reaction 7.2 Fractional Diffusion of Two Species 7.3 Subdiffusion and Linear Reaction 7.4 Hyperbolic Diffusion Equation with Reaction Terms 7.4.1 Diffusion-Reaction Processes References 8 Relaxation Under Geometric Constraints I: Classical Processes 8.1 Introduction 8.2 The Comb Model 8.3 Quenched and Annealed Disorder Mechanisms in Comb-Models 8.4 Generalized Fractal Structure of Backbones 8.5 Comb-Model and Reaction Diffusion 8.6 Diffusion and Reaction References 9 Relaxation Under Geometric Constraints II: Quantum Processes 9.1 Introduction 9.2 Constrained Quantum Motion in δ-Potential 9.3 Time Evolution and Asymptotic Behavior 9.4 Time-Dependent Schrödinger Equation in 3D 9.4.1 3D Constrained Quantum Model 9.4.2 Green's Function and Schrödinger Equation 9.4.3 Reduced Green's Functions and Fractional Derivatives 9.5 Time-Dependent Schrödinger Equation in Non-integer Dimensions 9.5.1 Marginal Probability Density Functions 9.5.2 Constrained Diffusion in Non-integer Dimensions References Index