Computing the Non-linear Anomalous Diffusion Equation from First Principles

We investigate asymptotically the occurrence of anomalous diffusion and its associated family of statistical evolution equations. Starting from a non-Markovian process a la Langevin we show that the mean probability distribution of the displacement of a particle follows a generalized non-linear Fokker-Planck equation. Thus we show that the anomalous behavior can be linked to a fast fluctuation process with memory from a microscopic dynamics level, and slow fluctuations of the dissipative variable. The general results can be applied to a wide range of physical systems that present a departure from the Brownian regime. (C) 2007 Elsevier B.V. All rights reserved.