Distribution of Statistical Observables for Anomalous and Nonergodic Diffusions: From Statistics to Mathematics
- Length: 227 pages
- Edition: 1
- Language: English
- Publisher: CRC Pr I Llc
- Publication Date: 2022-04-12
- ISBN-10: 1032245212
- ISBN-13: 9781032245218
- Sales Rank: #0 (See Top 100 Books)
This book investigates statistical observables for anomalous and nonergodic dynamics, focusing on the dynamical behaviors of particles modelled by non-Brownian stochastic processes in the complex real-world environment.
Statistical observables are widely used for anomalous and nonergodic stochastic systems, thus serving as a key to uncover their dynamics. This study explores the cutting edge of anomalous and nonergodic diffusion from the perspectives of mathematics, computer science, statistical and biological physics, and chemistry. With this interdisciplinary approach, multiple physical applications and mathematical issues are discussed, including stochastic and deterministic modelling, analyses of (stochastic) partial differential equations (PDEs), scientific computations and stochastic analyses, etc. Through regularity analysis, numerical scheme design and numerical experiments, the book also derives the governing equations for the probability density function of statistical observables, linking stochastic processes with PDEs.
The book will appeal to both researchers of electrical engineering expert in the niche area of statistical observables and stochastic systems and scientists in a broad range of fields interested in anomalous diffusion, especially applied mathematicians and statistical physicists.
Cover Half Title Title Page Copyright Page Contents Preface CHAPTER 1: Statistical Observables 1.1. INTRODUCTION 1.1.1. Physical Models 1.1.1.1. Continuous time random walk 1.1.1.2. Langevin equation 1.1.2. Stochastic Processes 1.1.2.1. Levy process 1.1.2.2. Subordinator 1.1.2.3. Time-changed process 1.2. POSITION 1.2.1. Probability Density Function 1.2.2. Fokker-Planck Equation 1.2.2.1. Derivation from continuous time random walk 1.2.2.2. Derivation from Langevin equation 1.3. FUNCTIONAL 1.3.1. Derivation from Continuous Time Random Walk 1.3.1.1. Forward Feynman-Kac equation 1.3.1.2. Backward Feynman-Kac equation 1.3.2. Derivation from Langevin Equation 1.3.2.1. Forward Feynman-Kac equation 1.3.2.2. Backward Feynman-Kac equation 1.3.2.3. Coupled Langevin equation 1.3.3. Derivation from Ito Formula 1.4. MEAN SQUARED DISPLACEMENT 1.4.1. Green-Kubo Formula 1.4.2. Ergodic and Aging Behavior 1.5. MISCELLANEOUS ONES 1.5.1. Fractional Moments 1.5.1.1. Infinite density of rare fluctuations 1.5.1.2. Dual scaling regimes in the central part 1.5.1.3. Complementarity among different scaling regimes 1.5.1.4. Ensemble averages 1.5.2. First Passage Time and First Hitting Time 1.5.2.1. First passage time of Levy flight and Levy walk 1.5.2.2. First hitting time of Levy flight and Levy walk CHAPTER 2: Numerical Methods for the Governing Equations of PDF of Statistical Observables 2.1. NUMERICAL METHODS FOR THE TIME FRACTIONAL FOKKER-PLANCK SYSTEM WITH TWO INTERNAL STATES 2.1.1. Preliminaries 2.1.2. Equivalent Form of (2.1) and Some Useful Lemmas 2.1.3. First-Order Scheme and Error Analysis 2.1.3.1. Error estimates for the homogeneous problem 2.1.3.2. Error estimates for the inhomogeneous problem 2.1.4. Second-Order Scheme and Error Analysis 2.1.4.1. Error analysis 2.1.5. Numerical Experiments 2.1.5.1. One-dimensional cases 2.1.5.2. Two-dimensional cases 2.2. NUMERICAL METHODS FOR THE SPACE-TIME FRACTIONAL FOKKER-PLANCK SYSTEM WITH TWO INTERNAL STATES 2.2.1. Regularity of the Solution 2.2.1.1. Preliminaries 2.2.1.2. A priori estimate of the solution 2.2.2. Space Discretization and Error Analysis 2.2.3. Time Discretization and Error Analysis 2.2.4. Numerical Experiments CHAPTER 3: Numerical Methods for the Stochastic Governing Equations of PDF of Statistical Observables 3.1. ASSUMPTIONS AND GAUSSIAN PROCESSES 3.2. NUMERICAL SCHEMES FOR STOCHASTIC FRACTIONAL DIFFUSION EQUATION 3.2.1. Numerical Approximation of Stochastic Fractional Diffusion Equation with a Tempered Fractional Gaussian Noise 3.2.1.1. Regularity of the solution 3.2.1.2. Galerkin approximation for spatial discretization 3.2.1.3. Fully discrete scheme 3.2.1.4. Numerical experiments 3.3. NUMERICAL SCHEMES FOR STOCHASTIC FRACTIONAL WAVE EQUATION 3.3.1. Higher Order Approximation for Stochastic Space Fractional Wave Equation 3.3.1.1. Regularity of the solution 3.3.1.2. Galerkin approximation for spatial discretization 3.3.1.3. Fully discrete scheme 3.3.1.4. Numerical experiments 3.3.2. Galerkin Finite Element Approximation of Stochastic Fractional Wave Equations 3.3.2.1. Solution representation 3.3.2.2. Galerkin finite element approximation 3.3.2.3. Error estimates for the nonhomogeneous problem 3.3.2.4. Numerical results Bibliography Index
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