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E-raamat: Modeling Anomalous Diffusion: From Statistics To Mathematics

(Lanzhou Univ, China), (Lanzhou Univ, China), (Lanzhou Univ, China), (Lanzhou Univ, China)
  • Formaat: 268 pages
  • Ilmumisaeg: 06-Jan-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811213014
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Modeling Anomalous Diffusion: From Statistics To MathematicsSuurem pilt
  • Formaat: 268 pages
  • Ilmumisaeg: 06-Jan-2020
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811213014
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This book focuses on modeling the anomalous diffusion phenomena, being ubiquitous in the natural world. Both the microscopic models (stochastic processes) and macroscopic models (partial differential equations) have been built up. The relationships between the two kinds of models are clarified, and based on these models, some statistical observables are analyzed. From statistics to mathematics, the built models show their power with their associated applications. This book is important for students to develop basic skills to be able to succeed in their future research. In addition to introducing the related models or methods, it also provides the corresponding applications and simulation results, which will attract more readers ranging from mathematicians to physicists or chemists, to name a few.

Preface v
1 Stochastic Models 1(24)
1.1 Background Knowledge in Statistics and Probability
1(5)
1.1.1 Random Variables and Distributions
1(4)
1.1.2 Moments
5(1)
1.2 Algorithm for the Generation of Random Variables
6(1)
1.3 Continuous Time Random Walk and Levy Process
6(7)
1.3.1 Continuous Time Random Walk
7(2)
1.3.2 Propagator Function
9(3)
1.3.3 Levy Process
12(1)
1.4 Levy Flight, Levy Walk, and Subordinated Processes
13(8)
1.4.1 Levy Flight
13(3)
1.4.2 Levy Walk
16(3)
1.4.3 Subordinator
19(2)
1.5 Langevin Pictures for Levy Flights
21(1)
1.6 Continuous Time Random Walk and Levy Walk with Multiple Internal States
22(3)
2 Fokker-Planck Equations 25(16)
2.1 Fractional Derivative and Integral
25(6)
2.1.1 Griinwald-Letnikov Fractional Derivative
26(1)
2.1.2 Riemann-Liouville Fractional Derivative
27(2)
2.1.3 Fractional Substantial Derivative
29(1)
2.1.4 Laplace Transform of Fractional Derivative
30(1)
2.2 Derivation of Fractional Fokker-Planck Equation
31(2)
2.3 Solution of Fractional Fokker-Planck Equation
33(8)
2.3.1 Integral Form of the Solution for Fokker-Planck Equation
33(2)
2.3.2 Solution for Force Free Fractional Diffusion
35(1)
2.3.3 Solution for Biased Fractional Wiener Process
36(1)
2.3.4 Solution Obtained by Separation of Variables
37(4)
3 Feynman-Kac Equations 41(34)
3.1 Brownian Functionals
41(1)
3.2 Fractional Feynman-Kac Equations
42(5)
3.2.1 Forward Fractional Feynman-Kac Equation
43(2)
3.2.2 Backward Fractional Feynman-Kac Equation
45(1)
3.2.3 Distribution of Occupation Times
46(1)
3.3 Tempered Fractional Feynman-Kac Equations
47(13)
3.3.1 Model and Tempered Dynamics
47(1)
3.3.2 Tempered Fractional Feynman-Kac Equations of Random Walk on a One-Dimensional Lattice
48(4)
3.3.3 Tempered Fractional Feynman-Kac Equations of Random Walk with Forces
52(2)
3.3.4 Distribution of Occupation Time in Half Space
54(2)
3.3.5 Distribution of First Passage Time
56(1)
3.3.6 Distribution of Maximal Displacement
57(1)
3.3.7 Fluctuations of Occupation Fraction
58(2)
3.4 Feynman-Kac Equations Revisited: Langevin Picture
60(15)
3.4.1 Forward Feynman-Kac Equation
60(4)
3.4.2 Backward Feynman-Kac Equation
64(3)
3.4.3 Distribution of Occupation Time in Positive Half Space
67(3)
3.4.4 Distribution of First Passage Time
70(1)
3.4.5 Area under Random Walk Curve
71(4)
4 Aging Fokker-Planck and Feynman-Kac Equations 75(42)
4.1 Aging CTRW
75(2)
4.2 Aging Renewal Theory
77(9)
4.3 ACTRW with Tempered Power Law Waiting Time
86(4)
4.3.1 MSD
86(2)
4.3.2 Propagator Function p(x, ta, t)
88(2)
4.4 Strong Relation between Fluctuation and Response
90(4)
4.5 Fokker-Planck Equations for Tempered ACTRW
94(2)
4.6 Derivations of Aging Feynman-Kac Equation
96(9)
4.6.1 Forward Feynman-Kac Equation with Discrete Step Length PDF
97(3)
4.6.2 Forward Feynman-Kac Equation with Continuous Step Length PDF
100(2)
4.6.3 Backward Feynman-Kac Equation with Discrete Step Length PDF
102(2)
4.6.4 Backward Feynman-Kac Equation with Continuous Step Length PDF
104(1)
4.7 Application
105(12)
4.7.1 Occupation Time in Half Space for ACTRW
105(3)
4.7.2 Fluctuation of Occupation Fraction
108(2)
4.7.3 Distribution of First Passage Time
110(7)
5 Fokker-Planck and Feynman-Kac Equations with Multiple Internal States 117(34)
5.1 Model and Notations
118(2)
5.2 Fractional Fokker-Planck Equations for CTRW with Multiple Internal States
120(6)
5.3 Equations Governing Distribution of Functionals of Paths and Internal States of Process
126(6)
5.4 Some Applications of Feynman-Kac Equations and Gov- erning Equations of Functionals of Internal States
132(8)
5.5 Levy Walk with Multiple Internal States
140(1)
5.6 More Applications for CTRW and Levy Walk with Multiple Internal States
141(10)
6 Fractional Reaction Diffusion Equations and Corresponding Feynman-Kac Equations 151(24)
6.1 Fractional Reaction Diffusion Equations
151(6)
6.1.1 Reaction-Anomalous Diffusion Equations
152(3)
6.1.2 Non-Markovian Transport with Nonlinear Reactions
155(2)
6.2 Feynman-Kac Equations for Reaction and Diffusion Processes
157(18)
6.2.1 Forward Feynman-Kac Equations for Nonlinear Reaction Rate r(p(x, t))
158(5)
6.2.2 Forward Feynman-Kac Equations for Nonlinear Reaction Rate r(t)
163(1)
6.2.3 Forward Feynman-Kac Equations for Nonlinear Reaction Rate r(x)
163(4)
6.2.4 Derivation of Backward Feynman-Kac Equations
167(2)
6.2.5 Distribution of Occupation Time in Half Space and its Fluctuations
169(2)
6.2.6 Distribution of First Passage Time
171(1)
6.2.7 Distribution of Occupation Time in Half Interval
172(3)
7 Renewal Theory for Fractional Poisson Process: Typical versus Rare 175(42)
7.1 Introduction
175(2)
7.2 Model
177(3)
7.3 Number of Renewals between 0 and t
180(6)
7.3.1 Number of Renewals between 0 and t with 0 < α < 1
180(2)
7.3.2 Number of Renewals between 0 and t with 1 < α < 2
182(4)
7.4 Forward Recurrence Time
186(9)
7.4.1 Forward Recurrence Time with 0 < α < 1
187(5)
7.4.2 Forward Recurrence Time with 1 < α < 2
192(3)
7.5 Backward Recurrence Time
195(6)
7.5.1 Backward Recurrence Time with 0 < α < 1
195(5)
7.5.2 Backward Recurrence Time with 1 < α < 2
200(1)
7.6 Time Interval Straddling t
201(4)
7.6.1 Time Interval Straddling t with 0 < α < 1
202(2)
7.6.2 Time Interval Straddling Time t with 1 < α < 2
204(1)
7.7 Occupation Time
205(6)
7.7.1 Occupation Time with 0 < α < 1
206(4)
7.7.2 Occupation Time with 1 < α < 2
210(1)
7.8 Some Properties of Stable Distribution
211(1)
7.9 Discussion
212(5)
8 Governing Equation for Average First Passage Time and Transitions among Anomalous Diffusions 217(26)
8.1 Governing Equation for Average First Passage Time
218(3)
8.2 Transition among Anomalous Diffusions: CTRW Description
221(6)
8.3 Non-Negativity of Solution: Subordinated Approach, and Stochastic Representation
227(2)
8.4 MSD, Fractional Moments, and Multi-Scale
229(5)
8.5 Fractional Fokker-Planck Equation with Prabhakar Derivative
234(6)
8.5.1 Relaxation of Modes
236(1)
8.5.2 Harmonic External Potential
236(4)
8.6 A Brief Introduction of Three Parameter Mittag-Leffler Functions
240(3)
Bibliography 243(10)
Index 253
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