Locally and globally optimal configurations of N particles on the sphere with applications in the narrow escape and narrow capture problems.

Abstract

Determination of optimal arrangements of N particles on a sphere is a well-known problem in physics. A famous example of such is the Thomson problem of finding equilibrium configurations of electrical charges on a sphere. More recently, however, similar problems involving other potentials and nonspherical domains have arisen in biophysical systems. Many optimal configurations have previously been computed, especially for the Thomson problem; however, few results exist for potentials that correspond to more applied problems. Here we numerically compute optimal configurations corresponding to the narrow escape and narrow capture problems in biophysics. We provide comprehensive tables of global energy minima for N≤120 and local energy minima for N≤65, and we exclude all saddle points. Local minima up to N=120 are available online.