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

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.

Figure 1

A schematic of the narrow escape problem in a two- and a three-dimensional domain.

Figure 2

Illustration of extremely fine and fine mesh regions.

Figure 3

Plots of (a) comparison of numerical (circles) and two-term asymptotic expression for average MFPT, and (b) relative error [see (22)] for an oblate spheroid with $N=3$.

Figure 4

Plots of (a) comparison of numerical (circles) and two-term asymptotic expression for average MFPT, and (b) relative error [see (22)] for an oblate spheroid with $N=5$.

Figure 5

Three-dimensional (transparent) plots of the numerical MFPT for the oblate spheroid at $\varepsilon =0.02$ with (a) $N=3$ and (b) $N=5$ traps. The trap parameters are given in Table 1.

Figure 6

Plots of (a) comparison of numerical (circles) and two-term asymptotic expression for average MFPT, and (b) relative error [see (22)] for a prolate spheroid with $N=3$.

Figure 7

Plots of (a) comparison of numerical (circles) and two-term asymptotic expression for average MFPT, and (b) relative error [see (22)] for a prolate spheroid with $N=5$.

Figure 8

Three-dimensional (transparent) plots of the numerically calculated MFPT (in seconds) for the prolate spheroid at $\varepsilon =0.02$ with (a) $N=3$ and (b) $N=5$ traps.

Figure 9

Biconcave disk cross-sectional view.

Figure 10

Plots of (a) comparison of numerical (circles) and two-term asymptotic expression for average MFPT, and (b) relative error [see (22)] for a biconcave disk with $N=3$.

Figure 11

Plots of (a) comparison of numerical (circles) and two-term asymptotic expression for average MFPT, and (b) relative error [see (22)] for a biconcave disk with $N=5$.

Figure 12

Three-dimensional (transparent) plots of the numerically calculated MFPT (in seconds) for the biconcave disk (blood cell) at $\varepsilon =0.02$ with (a) $N=3$ and (b) $N=5$ traps.