Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA.
Phys Rev E. 2023 Jan;107(1-1):014110. doi: 10.1103/PhysRevE.107.014110.
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 κ_{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 κ_{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.
扩散通过半渗透界面具有广泛的应用，从分子通过生物膜的传输到使用人工膜的反渗透水净化。在单粒子水平上，可以用所谓的突然布朗运动（BM）来建模具有恒定渗透率κ_{0}的屏障的一维扩散。后者将限制在屏障左侧或右侧的连续几轮部分反射的 BM 缝合在一起。当每个回合的布朗局部时间超过由 κ_{0}参数化的指数随机变量时，每个回合都会在屏障处被杀死（吸收）。然后，以相等的概率立即在任一侧开始新的回合。最近已经表明，突然 BM 的概率密度满足更新方程，该方程将完整密度与屏障两侧部分反射 BM 的概率密度相关联。此外，可以构建更新方程的广义版本，以纳入基于遭遇的非马尔可夫吸收模型。在本文中，我们将突然 BM 的更新理论扩展到有界域和更高空间维度中的单粒子扩散。在每种情况下，我们都展示了更新方程的解如何满足具有界面处可渗透边界条件的经典扩散方程。也就是说，界面处的概率通量是连续的，并且与界面两侧的密度差成正比。我们还考虑了一个不对称界面的例子，其中每个吸收事件后的方向切换是有偏差的。最后，我们展示了如何将基于遭遇的吸收模型纳入通过球对称界面的单粒子扩散中。我们发现，即使在界面两侧应用相同的非马尔可夫吸收模型，所得渗透率也是具有记忆的不对称时变函数。此外，渗透率函数往往是重尾的。
