First Steps in Random Walks From Tools to Applications

by Klafter, J.; Sokolov, I. M.

• ISBN13: 9780199234868
• ISBN10: 0199234868
• Format: Hardcover
• Copyright: 2011-11-01
• Publisher: Oxford University Press

Summary

The name "random walk" for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of "Nature". The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier such a problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays the theory of random walks has proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub- and super-diffusive transport processes as well. This book discusses the main variants of random walks and gives the most important mathematical tools for their theoretical description.

Author Biography

Joseph Klafter is Professor and Heinemann Chair of Physical Chemistry at Tel Aviv University. Igor Sokolov is Professor and Chair for Statistical Physics and Nonlinear Dynamics at Humboldt University, Berlin.

Table of Contents

Abbreviations	p. vii
Characteristic functions	p. 1
First example: A random walk on a one-dimensional lattice	p. 2
More general considerations	p. 4
The moments	p. 6
Random walk as a process with independent increments	p. 6
A pedestrian approach to the Central Limit Theorem	p. 7
When the Central Limit Theorem breaks down	p. 10
Random walks in higher dimensions	p. 13
Generating functions and applications	p. 17
Definitions and properties	p. 17
Tauberian theorems	p. 19
Application to random walks: The first-passage and return probabilities	p. 21
Mean number of distinct visited sites	p. 27
Sparre Andersen theorem	p. 29
Continuous-time random walks	p. 36
Waiting-time distributions	p. 36
Transforming steps into time	p. 39
Moments of displacement in CTRW	p. 42
Power-law waiting-time distributions	p. 43
Mean number of steps, MSD, and probability of being at the origin	p. 49
Other characteristic properties of heavy-tailed CTRW	p. 51
CTRW and aging phenomena	p. 54
When the process ages	p. 54
Forward waiting time	p. 55
PDF of the walker's positions	p. 60
Moving time averages	p. 62
Response to the time-dependent field	p. 65
Master equations	p. 68
A heuristic derivation of the generalized master equation	p. 70
A note on time-dependent transition probabilities	p. 74
Relation between the solutions to the generalized and the customary master equations	p. 75
Generalized Fokker-Planck and diffusion equations	p. 77
Fractional diffusion and Fokker-Planck equations for subdiffusion	p. 80
Riemann-Liouville and Weyl derivatives	p. 80
Grunwald-Letnikov representation	p. 83
Fractional diffusion equation	p. 84
Eigenfunction expansion	p. 89
Subordination and the forms of the PDFs	p. 92
Lévy flights	p. 97
General Lévy distributions	p. 98
Space-fractional diffusion equation for Lévy nights	p. 102
Leapover	p. 104
Simulation of Lévy distributions	p. 106
Coupled CTRW and Lévy walks	p. 110
Space-time coupled CTRWs	p. 110
Lévy walks	p. 114
Lévy walk interrupted by rests	p. 119
Simple reactions: A + B → B	p. 123
Configurational averaging	p. 123
A target problem	p. 125
Trapping problem	p. 127
Asymptotics of trapping kinetics in one dimension	p. 129
Trapping in higher dimensions	p. 132
Random walks on percolation structures	p. 135
Some facts about percolation	p. 137
Fractals	p. 139
Random walks on fractals	p. 141
Calculating the spectral dimension	p. 143
Using the spectral dimension	p. 145
The role of finite clusters	p. 147
Index	p. 151
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