Boundary conditions at a thin membrane for normal diffusion equation which generate subdiffusion

TL;DR

This paper investigates how specific boundary conditions at a thin membrane can alter the normal diffusion process to exhibit subdiffusive behavior, by analyzing moments of particle distribution and modeling particle trapping.

Contribution

It introduces a novel class of boundary conditions involving integral operators with time-dependent kernels that can induce subdiffusion in a normal diffusion framework.

Findings

Boundary conditions affect the moments of particle distribution over time.

Appropriate boundary conditions can generate subdiffusive moments.

The model links membrane trapping to subdiffusive behavior.

Abstract

We consider a particle transport process in a one-dimensional system with a thin membrane, described by a normal diffusion equation. We consider two boundary conditions at the membrane that are linear combinations of integral operators, with time dependent kernels, which act on the functions and their spatial derivatives define on both membrane surfaces. We show how boundary conditions at the membrane change the temporal evolution of the first and second moments of particle position distribution (the Green's function) which is a solution to normal diffusion equation. As these moments define the kind of diffusion, an appropriate choice of boundary conditions generates the moments characteristic for subdiffusion. The interpretation of the process is based on a particle random walk model in which the subdiffusion effect is caused by anomalously long stays of the particle in the membrane.