Anomalous diffusion in porous fractal media

Abstract

We suggest a model of a diffusive process inside a fractal sponge structure, which is a generalization of the diffusion processes on a comb and fractal mesh structure. The sponge model is considered as the direct product of Cantor sets. It is shown that the corresponding one-dimensional diffusion process is governed by a generalized Fokker-Planck equation with a power-law memory kernel and a position-dependent diffusion coefficient. That is, the fractal structure of the medium induces memory effects and heterogeneity in the transport system. The considered model may be of interest to describe anomalous heat transport in porous fractal media.