# Locally and Globally Optimal Configurations of $N$ Particles on the Sphere with Applications in the Narrow Escape and Narrow Capture Problems

## TL;DR

This paper numerically determines optimal particle arrangements on a sphere for biophysical problems, providing extensive data on global and local minima up to N=120, extending beyond classical physics problems.

## Contribution

It introduces numerical methods to find optimal configurations for narrow escape and capture problems, filling gaps in existing results for applied potentials.

## Key findings

- Global energy minima tables for N≤120
- Local minima data for N≤65
- Exclusion of saddle points from results

## Abstract

Determination of \emph{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 non-spherical 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 \emph{narrow escape} and \emph{narrow capture} problems in biophysics. We provide comprehensive tables of global energy minima for $N\leq120$ and local energy minima for $N\leq65$, and we exclude all saddle points. Local minima up to $N=120$ are available online.