A random walk simulation of fractional diffusion

Abstract

Conventional diffusion, described by an equation whose fundamental solution is a Gaussian and characterized by a second moment linear in time, can be simulated by an ensemble of random walks. Anomalous diffusion, characterized by a second moment non-linear in time, may be described by a diffusion equation with a fractional time derivative and can be simulated by a continuous time random walk whose waiting time density distribution is related to the generalized Mittag–Leffler function. An alternative approach is presented for the simulation of fractional diffusion based on the fact that Gaussians provide an orthornormal basis for the expansion of any function. The fundamental solution of the fractional diffusion equation is represented by a truncated linear combination of Gaussians and this results in simulations of fractional diffusion by a set of random walks.