Memory formalism in the passive diffusion across highly heterogeneous systems

In this note, we present an extension of the Fick's second law by introducing a memory formalism based on derivatives of fractional order to take into account the passive diffusion process across two different membranous systems, i.e., a biological membrane, where its structural complexity suggests the introduction of a space-dependent diffusion constant, and a heterogeneous highly porous system at a macroscopic level, composed of an impenetrant matrix and pores across which the solute diffuse. This approach has been employed to describe some recent experimental results concerning the transdermal diffusion of drugs through human stratum corneum and the flux of water observed across a sand layer under a constant hydrostatic pressure difference. Despite the two examples we report concern with deeply different heterogeneous systems at a different scale length, a reasonable good agreement is obtained in both the two cases.