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

Abstract: We present a model of diffusion in heterogeneous environment, which qualitatively reflects the transport properties of a polymeric membrane with carbon nanotubes. We derived Fokker-Planck equation from system of stochastic equations, responding diffusion regime in polymer and ballistic regime in nanotube areas. We show how probability density function is changed in presence of nanotube areas. Hereupon nonlinear time dependence of mean square displacement is observed, that indicates anomalous diffusion regime. Thus, the model explains mechanism of appearance of anomalous diffusion for diffusion-ballistic regime. Our approach does not suppose any type of distributions and using fractional differentiate apparatus.