First Steps in Random Walks: From Tools to Applications
From Tools to Applications
1st Edition
Published on 18. August 2011
161 pages
978-0-19-155295-3 (ISBN)
Description
Random walks proved to be a useful model of many complex transport processes at the micro and macroscopical level in physics and chemistry, economics, biology and other disciplines. The book discusses the main variants of random walks and gives the most important mathematical tools for their theoretical description.
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08/2011
Oxford University Press
€108.90
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Content
- Cover
- Contents
- Abbreviations
- 1 Characteristic functions
- 1.1 First example: A random walk on a one-dimensional lattice
- 1.2 More general considerations
- 1.3 The moments
- 1.4 Random walk as a process with independent increments
- 1.5 A pedestrian approach to the Central Limit Theorem
- 1.6 When the Central Limit Theorem breaks down
- 1.7 Random walks in higher dimensions
- 2 Generating functions and applications
- 2.1 Definitions and properties
- 2.2 Tauberian theorems
- 2.3 Application to random walks: The first-passage and return probabilities
- 2.4 Mean number of distinct visited sites
- 2.5 Sparre Andersen theorem
- 3 Continuous-time random walks
- 3.1 Waiting-time distributions
- 3.2 Transforming steps into time
- 3.3 Moments of displacement in CTRW
- 3.4 Power-law waiting-time distributions
- 3.5 Mean number of steps, MSD, and probability of being at the origin
- 3.6 Other characteristic properties of heavy-tailed CTRW
- 4 CTRW and aging phenomena
- 4.1 When the process ages
- 4.2 Forward waiting time
- 4.3 PDF of the walker's positions
- 4.4 Moving time averages
- 4.5 Response to the time-dependent field
- 5 Master equations
- 5.1 A heuristic derivation of the generalized master equation
- 5.2 A note on time-dependent transition probabilities
- 5.3 Relation between the solutions to the generalized and the customary master equations
- 5.4 Generalized Fokker-anck and diffusion equations
- 6 Fractional diffusion and Fokker-Planck equations for subdiffusion
- 6.1 Riemann-Liouville and Weyl derivatives
- 6.2 Grünwald-Letnikov representation
- 6.3 Fractional diffusion equation
- 6.4 Eigenfunction expansion
- 6.5 Subordination and the forms of the PDFs
- 7 Lévy flights
- 7.1 General Lévy distributions
- 7.2 Space-fractional diffsion equation for Lévy flights
- 7.3 Leapover
- 7.4 Simulation of Lévy distributions
- 8 Coupled CTRW and Lévy walks
- 8.1 Space-time coupled CTRWs
- 8.2 Lévy walks
- 8.3 Lévy walk interrupted by rests
- 9 Simple reactions: A + B : B
- 9.1 Configurational averaging
- 9.2 A target problem
- 9.3 Trapping problem
- 9.4 Asymptotics of trapping kinetics in one dimension
- 9.5 Trapping in higher dimensions
- 10 Random walks on percolation structures
- 10.1 Some facts about percolation
- 10.2 Fractals
- 10.3 Random walks on fractals
- 10.4 Calculating the spectral dimension
- 10.5 Using the spectral dimension
- 10.6 The role of finite clusters
- Index
- A
- C
- D
- E
- F
- G
- I
- K
- L
- M
- P
- R
- S
- T
- W
