Analytic description of anomalous diffusion in heterogeneous environments: Fokker-Planck equation without fractional derivatives.

Abstract

This article presents a one-dimensional model of diffusion in a heterogeneous environment, which qualitatively reflects the transport properties of a polymeric membrane with carbon nanotube areas. We derive the Fokker-Planck equation from a system of stochastic equations using a diffusion regime in polymers and a ballistic diffusion regime in nanotube areas. We demonstrate how the probability density function changes in the presence of nanotubes. The mean-square displacement of nonlinear time dependence is observed, indicating anomalous diffusion. This model explains the mechanism of anomalous diffusion in a ballistic-diffusion regime. Our approach does not suppose any type of distribution and does not use a fractional differentiate apparatus.