Renewal equations for single-particle diffusion through a semipermeable interface

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 reflected 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. In each case we show how the solution of the renewal equation satisfies the classical diffusion equation with a permeable boundary condition at the interface. That is, the probability flux across the interface is continuous and proportional to the difference in densities on either side of the interface. 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.

Figure 1

Diffusion through a closed semipermeable membrane in ${\mathbb{R}}^d$.

Figure 2

Brownian motion in the interval $[-L^{\prime },L]$ with a semipermeable membrane at $x=0$ and reflecting boundary conditions at $x=-L^{\prime },L$.

Figure 3

Asymmetric semipermeable barrier at $x=0$ with a reflecting boundary at $x=L^{\prime }=-1$ and an absorbing boundary at $x=L=2$. Plots of MFPT $\mathbb{E}\left[{\mathcal{T}}_L\right]$ as function of the initial position $x_0$ for various $\alpha$ and ${\kappa }_0$. We also set $D=1$.

Figure 4

Decomposition of a higher-dimensional snapping out BM into two partially reflected BMs corresponding to (a) ${\mathbf{X}}_t\in {\mathcal{M}}^c$ and (b) ${\mathbf{X}}_t\in \mathcal{M}$, respectively.

Figure 5

Cylinder construction across the semipermeable membrane. See text for details.

Figure 6

Diffusion through a spherically symmetric semipermeable interface in ${\mathbb{R}}^d$. (a) For $d=2$ the interface is a circle of radius $R$ (b) For $d=1$ there exist two semipermeable barriers at $r=\pm R$ and a totally reflecting barrier at $r=0^{\pm }$. The solution is reflection symmetric about $r=0$.