[070] Generalized diffusion: A microscopic approach.

James F. Lutsko and Jean P. Boon, "Generalized diffusion: A microscopic approach.", Phys. Rev. E, 77, 51103 (2008) http://jimlutsko.github.io/files/Lutsko_PRE_2008_1.pdf

Abstract

The Fokker-Planck equation for the probability f(r,t) to find a random walker at position r at time t is derived for the case that the probability to make jumps depends nonlinearly on f(r,t). The result is a generalized form of the classical Fokker-Planck equation where the effects of drift, due to a violation of detailed balance, and of external fields are also considered. It is shown that in the absence of drift and external fields a scaling solution, describing anomalous diffusion, is possible only if the nonlinearity in the jump probability is of the power law type [ f(r,t)], in which case the generalized Fokker-Planck equation reduces to the porous media equation. Monte Carlo simulations are shown to confirm the theoretical results.