Asymptotic analysis of narrow escape problems in nonspherical three-dimensional domains.

Abstract

Narrow escape problems consider the calculation of the mean first passage time (MFPT) for a particle undergoing Brownian motion in a domain with a boundary that is everywhere reflecting except for at finitely many small holes. Asymptotic methods for solving these problems involve finding approximations for the MFPT and average MFPT that increase in accuracy with decreasing hole sizes. While relatively much is known for the two-dimensional case, the results available for general three-dimensional domains are rather limited. This paper addresses the problem of finding the average MFPT for a class of three-dimensional domains bounded by the level surface of an orthogonal coordinate system. In particular, this class includes spheroids and other solids of revolution. The primary result presented is a two-term asymptotic expansion for the average MFPT of such domains containing an arbitrary number of holes. Steps are taken towards finding higher-order asymptotic expansions for both the average MFPT and the MFPT in these domains. The results for the average MFPT are compared to full numerical calculations performed with the comsol Multiphysics finite element solver for three distinct domains: prolate and oblate spheroids and biconcave disks. This comparison shows good agreement with the proposed two-term expansion of the average MFPT in the three domains.