Generalized Diffusion

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 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 only possible if the nonlinearity in the jump probability is of the power law type ($\sim f^{\eta }(r,t)$), in which case the generalized Fokker-Planck equation reduces to the well-known Porous Media equation. Monte-Carlo simulations are shown to confirm the theoretical results.

29 pages, 8 figures

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• A microscopic approach to nonlinear Reaction-Diffusion: the case of morphogen gradient formation
• Molecular theory of anomalous diffusion
• Questioning the validity of non-extensive thermodynamics for classical Hamiltonian systems
• Mass transport subject to time-dependent flow with non-uniform sorption in porous media
• Exclusion processes: short range correlations induced by adhesion and contact interactions
• A nonlinear random walk approach to concentration-dependent contaminant transport in porous media
• Reaction-Diffusion Processes with Nonlinear Diffusion
• Anomalous diffusion of volcanic earthquakes
• Asymmetric discrete random walk and drift-diffusion with unequal jump times, lengths, and probabilities