An Asymptotic Analysis of the Mean First Passage Time for Narrow Escape Problems: Part II: The Sphere

The mean first passage time (MFPT) is calculated for a Brownian particle in a spherical domain in R that contains N small non-overlapping absorbing windows, or traps, on its boundary. For the unit sphere, the method of matched asymptotic expansions is used to derive an explicit three-term asymptotic expansion for the MFPT for the case of N small locally circular absorbing windows. The third term in this expansion, not previously calculated, depends explicitly on the spatial configuration of the absorbing windows on the boundary of the sphere. The three-term asymptotic expansion for the average MFPT is shown to be in very close agreement with full numerical results. The average MFPT is shown to be minimized for 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 point charges on the unit sphere. Numerical results, based on global optimization methods, are given for both the optimum discrete energy and the arrangements of the centers {x1, . . . , xN} of N circular traps on the boundary of the sphere. A scaling law for the optimum discrete energy, valid for N 1, is also derived.