Fractional subdiffusion approach to permeable barrier crossing

Abstract

As random displacements of many biological entities such as animals and cells often display statistical characteristics that depart from diffusion, over the years various mathematical representations to describe the so-called anomalous movement statistics have been developed. In the subdiffusive regime, that is when the variance of the occupation probability grows sublinearly in time, the fractional diffusion formalism has been employed extensively, both when space is homogeneous and in the presence of a deterministic potential. But when subdiffusive motion occurs in a spatially disordered environment, the literature is still in its infancy. Here we help to fill this gap by studying the dynamics of a subdiffusive particle in the presence of a simple spatial heterogeneity, namely a thin permeable barrier. We do so by extending to subdiffusion a recent birth-death coupled equation approach to represent diffusive barrier crossing, and by deriving a generalisation of the so-called permeable boundary condition. By using the defect technique with coupled fractional diffusion equations we are able to find analytically the dynamics of the occupation probability in space and Laplace domain, and the temporal dependence at long times of the first and second moment. As an application we consider a system composed of an infinite array of periodically placed asymmetric identical permeable barriers and study its long time dynamics. When the permeability of the barriers is unequal, despite the intrinsic motion of the particle being subdiffusive, we identify a transition from sub to superdiffusive in the growth exponent of the mean square displacement due to the coupling between the effective drift of the asymmetric barriers and subdiffusive waiting times.