Mean exit time for diffusion on irregular 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.

31pages, 12 figures

References in corpus (9)

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• Escape of a Sticky Particle
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• Approximate analytical solution for transient heat and mass transfer across an irregular interface
• First-passage times to a fractal boundary: local persistence exponent and its log-periodic oscillations
• Mean exit time in irregularly-shaped annular and composite disc domains
• Diffusion in a fluid flow generated by a source at the apex of a wedge
• Enhancing search efficiency through diffusive echo