Narrow-escape problem for the unit sphere: Homogenization limit, optimal arrangements of large numbers of traps, and the N2 conjecture

A narrow-escape problem is considered to calculate the mean first passage time (MFPT) needed for a Brownian particle to leave a unit sphere through one of its N small boundary windows (traps). A procedure is established to calculate optimal arrangements of N≫1 equal small boundary traps that minimize the asymptotic MFPT. Based on observed characteristics of such arrangements, a remarkable property is discovered, that is, the sum of squared pairwise distances between optimally arranged N traps on a unit sphere is integer, equal to N². It is observed for 2≤N≤1004 with high precision. It is conjectured that this is the case for such optimal arrangements for all N. A dilute trap limit of homogenization theory when N→∞ can be used to replace the strongly heterogeneous Dirichlet-Neumann MFPT problem with a spherically symmetric Robin problem for which an exact solution is readily found. Parameters of the Robin homogenization problem are computed that capture the first four terms of the asymptotic MFPT. Close agreement of asymptotic and homogenization MFPT values is demonstrated. The homogenization approach provides a radically faster way to estimate the MFPT since it is given by a simple formula and does not involve expensive global optimization to determine locations of N≫1 boundary traps.