Diffusion through permeable interfaces: Fundamental equations and their application to first-passage and local time statistics

TL;DR

This paper derives a fundamental diffusion equation for systems with permeable interfaces, enabling analysis of first-passage and local time statistics in complex heterogeneous environments.

Contribution

It introduces a new microscopic derivation of the diffusion equation with permeable interfaces and explores its implications for first-passage and local time statistics.

Findings

Discovery of a dependence regime of mean first-passage time on barrier location.

General formalism applicable to multiple barriers and reactive heterogeneities.

Insights into diffusion dynamics in heterogeneous and force-influenced domains.

Abstract

The diffusion equation is the primary tool to study the movement dynamics of a free Brownian particle, but when spatial heterogeneities in the form of permeable interfaces are present, no fundamental equation has been derived. Here we obtain such an equation from a microscopic description using a lattice random walk model. The sought after Fokker-Planck description and the corresponding backward Kolmogorov equation are employed to investigate first-passage and local time statistics and gain new insights. Among them a surprising phenomenon, in the case of a semibounded domain, is the appearance of a regime of dependence and independence on the location of the permeable barrier in the mean first-passage time. The new formalism is completely general: it allows to study the dynamics in the presence of multiple permeable barriers as well as reactive heterogeneities in bounded or unbounded domains and under the influence of external forces.