Subdiffusive random walk in a membrane system. The generalized method of images approach

TL;DR

This paper develops a generalized method of images to derive Green's functions for subdiffusive processes in membrane systems, comparing models of particle-membrane interactions and applying the approach to slow subdiffusion.

Contribution

It introduces a unified method to obtain Green's functions for subdiffusion in membrane systems, accommodating different membrane permeabilities and models.

Findings

Green's functions are equivalent for partially permeable membranes in both models.

Green's functions differ for one-sided fully permeable membranes between models.

The generalized method of images is applicable to various subdiffusion models.

Abstract

In this paper we study subdiffusion in a system with a thin membrane. At the beginning, the random walk of a particle is considered in a system with a discrete time and space variable and then the probability describing the evolution of the particle's position (Green's function) is transformed into a continuous system. Two models are considered differing here from each other regarding the assumptions about how the particle is stopped or reflected by the membrane when the particle attempts to pass through the membrane fails. We show that for a system in which a membrane is partially permeable with respect to both its sides the Green's functions obtained for both models within the {\it continuous time random walk formalism} are equivalent to each other and expressed by the functions presented in the paper: T. Koszto{\l}owicz, Phys. Rev. E \textbf{91}, 022102 (2015), except the values defined at the membrane's surfaces. We also show that for a system with a one--sidedly fully permeable membrane, the Green's functions obtained within both models are not equivalent to each other. We also present the generalized method of images which provides the Green's functions for the membrane system, obtained in this paper. This method, which has a simple physical interpretation, is of a general nature and, in our opinion, can be used to find the Green's functions for a system with a thin membrane in which various models of subdiffusion can be applied. As an example, we find the Green's functions for the particular case of a `slow--subdiffusion' process in a system with a thin membrane.