Mean exit time for diffusion on irregular domains

TL;DR

This paper introduces a perturbation method to accurately compute the mean first passage time for diffusion on irregular domains, extending classical solutions from simple geometries to more complex, realistic shapes.

Contribution

The authors develop a perturbation approach based on exact solutions for simple geometries to estimate mean exit times on irregular domains, validated on real-world maps.

Findings

Perturbation solutions closely match numerical results.

Few terms in series provide high accuracy.

Method applicable to complex natural domains.

Abstract

Many problems in physics, biology, and economics depend upon the duration of time required for a diffusing particle to cross a boundary. As such, calculations of the distribution of first passage time, and in particular the mean first passage time, is an active area of research relevant to many disciplines. Exact results for the mean first passage time for diffusion on simple geometries, such as lines, discs and spheres, are well--known. In contrast, computational methods are often used to study the first passage time for diffusion on more realistic geometries where closed--form solutions of the governing elliptic boundary value problem are not available. Here, we develop a perturbation solution to calculate the mean first passage time on irregular domains formed by perturbing the boundary of a disc or an ellipse. Classical perturbation expansion solutions are then constructed using the exact solutions available on a disc and an ellipse. We apply the perturbation solutions to compute the mean first exit time on two naturally--occurring irregular domains: a map of Tasmania, an island state of Australia, and a map of Taiwan. Comparing the perturbation solutions with numerical solutions of the elliptic boundary value problem on these irregular domains confirms that we obtain a very accurate solution with a few terms in the series only. Matlab software to implement all calculations is available on GitHub.