Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk

Details Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk

2007-09-25

arxiv.org/abs/0709.3990v1

Springer Lecture Notes in Physics, No 621, Berlin 2003, pp. 148-166

Physics

Condensed Matter

Statistical Mechanics

24 pages, 3 figures, 10 eps files

Scientific paper

A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time of peculiar self-similar stochastic processes: an integral representation of these solutions is here presented. A more general approach to anomalous diffusion is known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.

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