Renewal equations for single-particle diffusion in multilayered media
File(s)
Published version
Author(s)
Bressloff, Paul C
Type
Journal Article
Abstract
Diffusion in heterogeneous media partitioned by semipermeable interfaces has a wide range of applications in the physical and life sciences, ranging from thermal conduction in composite media, gas permeation in soils, diffusion magnetic resonance imaging, drug delivery, and intercellular gap junctions. Many of these systems involve three-dimensional diffusion in an array of parallel planes with homogeneity in the lateral directions, so that they can be reduced to effective one-dimensional (1D) models. In this paper we develop a probabilistic model of single-particle diffusion in 1D multilayered media by constructing a multilayered version of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of reflected BM, each of which is restricted to a single layer. Each round of reflected BM is killed when the local time at one end of the layer exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of a boundary.) The particle then immediately resumes reflected BM in the same layer or the layer on the other side of the boundary with equal probability, and the process is iterated We proceed by constructing a last renewal equation for multilayered snapping out BM that relates the full probability density to the probability densities of partially reflected BM in each layer. We then show how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that the renewal equation and corresponding multilayer diffusion equation are equivalent. We illustrate the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries of the domain. Finally, we use the renewal approach to incorporate a generalization of snapping out BM based on the encounter-based method for surface absorption; each round of reflected BM is now killed according to a nonexponential distribution for each local time threshold. This is achieved by considering a corresponding first renewal equation that relates the full probability density to the FPT densities for killing each round of reflected BM. We show that for certain configurations, nonexponential killing leads to an effective time-dependent permeability that is normalizable but heavy-tailed.
Date Issued
2023-08-31
Date Acceptance
2023-03-15
Citation
SIAM Journal on Applied Mathematics, 2023, 83 (4), pp.1518-1545
ISSN
0036-1399
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Start Page
1518
End Page
1545
Journal / Book Title
SIAM Journal on Applied Mathematics
Volume
83
Issue
4
Copyright Statement
© 2023 Society for Industrial and Applied Mathematics
Identifier
http://dx.doi.org/10.1137/23m1545835
Publication Status
Published
Date Publish Online
2023-07-26