Subdiffusion in a system with a partially permeable partially absorbing wall

TL;DR

This paper models subdiffusion in a one-dimensional system with a partially permeable and absorbing wall, deriving boundary conditions involving fractional derivatives, and analyzes how wall parameters affect particle filtering efficiency.

Contribution

It introduces a new model for subdiffusion with a partially permeable and absorbing wall, deriving specific boundary conditions involving fractional derivatives.

Findings

Derived Green's functions for subdiffusion with PPAW

Established boundary conditions involving fractional derivatives

Analyzed filtering efficiency based on wall parameters

Abstract

We consider subdiffusion of a particle in a one-dimensional system with a thin partially permeable wall. Passing through the wall, the particle can be absorbed with a certain probability. We call such a wall partially permeable partially absorbing wall (PPAW). Using the diffusion model in a system with discrete time and spatial variable, probability densities (Green's functions) describing subdiffusion in the system have been derived. Knowing the Green's functions we derive boundary conditions at the wall. The boundary conditions take a specific form in which time derivatives of the fractional order controlled by the subdiffusion parameter are involved. We assume that the absorption of a particle can occur only when the particle jumps through the wall. It is not possible to temporarily retain a particle inside a thin wall. The wall can represent a thin membrane. If a system with a thick membrane inside which particles may diffuse is considered, it can be treated as a three-part with a thick membrane as the middle part. The boundary conditions at membrane surfaces can be assumed as for PPAW. The system with PPAW can be used to filter diffusing particles. The temporal evolution of the probability that the diffusing molecule has not been absorbed is considered. This function shows the efficiency of the filtering process. The impact of subdiffusion and wall parameters on this function is discussed.