Renewal equations for single-particle diffusion through a semipermeable interface

TL;DR

This paper extends renewal equations for single-particle diffusion through semipermeable interfaces to bounded domains and higher dimensions, incorporating non-Markovian absorption models and asymmetric interfaces, revealing complex, memory-dependent permeability behaviors.

Contribution

It introduces a generalized renewal theory for snapping out Brownian motion applicable to more complex geometries and non-Markovian absorption, including asymmetric and spherical interfaces.

Findings

Permeability becomes asymmetric and time-dependent with memory effects.

Heavy-tailed permeability functions emerge even with symmetric absorption models.

Extensions enable modeling in higher dimensions and bounded domains.

Abstract

Diffusion through semipermeable interfaces has a wide range of applications, ranging from molecular transport through biological membranes to reverse osmosis for water purification using artificial membranes. At the single-particle level, one-dimensional diffusion through a barrier with constant permeability $\kappa_0$ can be modeled in terms of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting BMs that are restricted to either the left or right of the barrier. Each round is killed (absorbed) at the barrier when its Brownian local time exceeds an exponential random variable parameterized by $\kappa_0$. A new round is then immediately started in either direction with equal probability. It has recently been shown that the probability density for snapping out BM satisfies a renewal equation that relates the full density to the probability densities of partially reflected BM on either side of the barrier. Moreover, generalized versions of the renewal equation can be constructed that incorporate non-Markovian, encounter-based models of absorption. In this paper we extend the renewal theory of snapping out BM to single-particle diffusion in bounded domains and higher spatial dimensions. We also consider an example of an asymmetric interface in which the directional switching after each absorption event is biased. Finally, we show how to incorporate an encounter-based model of absorption for single-particle diffusion through a spherically symmetric interface. We find that, even when the same non-Markovian model of absorption applies on either side of the interface, the resulting permeability is an asymmetric time-dependent function with memory. Moreover, the permeability functions tend to be heavy-tailed.