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E-raamat: Functional Distribution Of Anomalous And Nonergodic Diffusion: From Stochastic Processes To Pdes

(Lanzhou Univ, China), (Nanjing Univ Of Science And Technology, China), (Lanzhou Univ, China)
  • Formaat: 260 pages
  • Ilmumisaeg: 20-Jun-2022
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811250514
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Functional Distribution Of Anomalous And Nonergodic Diffusion: From Stochastic Processes To PdesSuurem pilt
  • Formaat: 260 pages
  • Ilmumisaeg: 20-Jun-2022
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811250514
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This volume presents a pedagogical review of the functional distribution of anomalous and nonergodic diffusion and its numerical simulations, starting from the studied stochastic processes to the deterministic partial differential equations governing the probability density function of the functionals. Since the remarkable theory of Brownian motion was proposed by Einstein in 1905, it had a sustained and broad impact on diverse fields, such as physics, chemistry, biology, economics, and mathematics. The functionals of Brownian motion are later widely attractive for their extensive applications. It was Kac, who firstly realized the statistical properties of these functionals can be studied by using Feynman's path integrals.In recent decades, anomalous and nonergodic diffusions which are non-Brownian become topical issues, such as fractional Brownian motion, Lévy process, Lévy walk, among others. This volume examines the statistical properties of the non-Brownian functionals, derives the governing equations of their distributions, and shows some algorithms for solving these equations numerically.
Preface v
1 Probability Theory
1(20)
1.1 Random Variables and Probability Distributions
1(1)
1.2 Joint and Conditional Probabilities
2(2)
1.2.1 Joint Probability
2(1)
1.2.2 Conditional Probability
3(1)
1.2.3 Independence
3(1)
1.3 Moments
4(2)
1.4 Gaussian and Levy Distributions
6(6)
1.4.1 Gaussian Distribution
6(1)
1.4.2 Levy Distribution
7(3)
1.4.3 Central Limit Theorem
10(2)
1.5 Random Processes
12(3)
1.5.1 Markov Process
13(1)
1.5.2 Chapman-Kolmogorov Equation
13(1)
1.5.3 Levy Process
14(1)
1.6 Subordinators
15(6)
1.6.1 Several Subordinators
16(1)
1.6.2 Inverse Subordinator
17(2)
1.6.3 Simulations of Subordinator and Inverse Subordinator
19(2)
2 Anomalous and Nonergodic Diffusion
21(66)
2.1 Continuous Time Random Walk
21(21)
2.1.1 From Discrete to Continuous Random Walk
22(8)
2.1.2 Coupled Continuous Time Random Walk and Levy Walk
30(8)
2.1.3 Fokker-Planck Equation
38(4)
2.2 Langevin Equation
42(27)
2.2.1 Classical Langevin Equation
43(2)
2.2.2 Generalized Langevin Equation
45(7)
2.2.3 Langevin Equation Coupled with a Subordinator
52(6)
2.2.4 Langevin Equation with External Force
58(11)
2.3 Ergodic and Nonergodic Behavior
69(18)
2.3.1 Ergodicity Property in Continuous Time Random Walk Framework
70(2)
2.3.2 Ergodicity Property in Langevin Picture
72(15)
3 Functional Distributions
87(72)
3.1 Functional
87(2)
3.2 Fractional Feynman-Kac Equation
89(25)
3.2.1 Derivation in Continuous Time Random Walk Framework
89(7)
3.2.2 Derivation in Langevin Picture
96(10)
3.2.3 Multiple Internal States with Anisotropic Diffusion
106(8)
3.3 Functional Distribution Governed by Feynman-Kac Equation
114(18)
3.3.1 Occupation Time in Half-Space
114(8)
3.3.2 First-Passage Time
122(2)
3.3.3 Area under Random Walk Curve
124(6)
3.3.4 Other Functional
130(2)
3.4 Klein-Kramers Equation
132(9)
3.4.1 Force-Free Case
132(4)
3.4.2 Force Case
136(5)
3.5 First-Passage Time and Mean Exit Time
141(18)
3.5.1 First-Passage Properties for Uncoupled Langevin Equation
141(5)
3.5.2 First-Passage Properties for Coupled Langevin Equation
146(3)
3.5.3 Hitting Probability and Escape Probability
149(2)
3.5.4 Intermittent Search Strategy
151(8)
4 Algorithms for the Models Governing Functional Distribution
159(54)
4.1 Numerical Schemes for Backward Fractional Feynman-Kac Equations
160(28)
4.1.1 Basic Approximations to the Fractional Substantial Derivatives
160(8)
4.1.2 Equivalent Form of Backward Fractional Feynman-Kac Equation for Reaction and Diffusion Processes
168(1)
4.1.3 Finite Difference Method for Backward Fractional Feynman-Kac Equation
169(10)
4.1.4 Finite Element Scheme for Backward Fractional Feynman-Kac Equation
179(9)
4.2 Numerical Schemes for Forward Fractional Feynman-Kac Equation
188(25)
4.2.1 Time-Stepping Scheme for Traditional Time Fractional Diffusion Equation
189(1)
4.2.2 Time-Stepping Scheme for Time Tempered Fractional Feynman-Kac Equation
190(23)
Appendix A Fractional Calculus and Related Spaces
213(14)
A.1 Continuous Function Spaces
213(1)
A.2 Lebesgue Spaces
214(2)
A.3 Sobolev Spaces
216(1)
A.4 Definitions and Properties of Fractional Calculus
217(2)
A.5 Discretization of (Tempered) Fractional Laplacian
219(8)
Bibliography 227(16)
Index 243
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