An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems

The mean first passage time (MFPT) is calculated for a Brownian particle in a bounded twoor three-dimensional domain that contains N small non-overlapping absorbing windows on its boundary. The reciprocal of the MFPT of such narrow escape problems has wide applications in cellular biology where it may be used as an effective first order rate constant to describe, for example, the nuclear export of messenger RNA molecules through nuclear pores. In the asymptotic limit where the absorbing patches have small measure, the method of matched asymptotic expansions is used to calculate the MFPT in an arbitrary two dimensional domain with smooth boundary and for the unit sphere in three dimensions. In both cases the asymptotic results for the MFPT depend on the surface Neumann Green’s function of the corresponding domain and its associated regular part. In the two dimensional case, the known analytical formulae for the surface Neumann Green’s function for the unit disk and the unit square provide explicit asymptotic approximations to the MFPT for these special domains. For arbitrary two-dimensional domains, the asymptotic MFPT is evaluated by developing a novel boundary integral method to numerically calculate the required surface Neumann Green’s function. For the unit sphere, a three-term asymptotic expansion for the MFPT is derived for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial arrangement of the absorbing windows on the boundary of the sphere. The MFPT is minimized for particular trap configurations that minimize a certain discrete variational problem. This variational problem is closely related to the well-known optimization problem of determining the minimum energy configuration for N repelling Coulomb charges on the unit sphere. Finally, our three-term asymptotic expansion for the averaged MFPT is shown to be in very close agreement with full numerical results.