Generalized diffusion: A microscopic approach

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 [ approximately f;{eta}(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.