Estimating the Number of Stable Configurations for the Generalized Thomson Problem

TL;DR

This paper investigates the number of stable point configurations on a sphere that minimize generalized Coulomb energy, revealing a vast number of such configurations as N approaches 200.

Contribution

The authors develop methods to identify, count, and estimate the total number of stable configurations in the generalized Thomson problem.

Findings

At N near 200, there are at least tens of thousands of stable configurations.

The paper provides new techniques for counting and estimating stable configurations.

Stable configurations are abundant and complex for larger N.

Abstract

Given a natural number N, one may ask what configuration of N points on the two-sphere minimizes the discrete generalized Coulomb energy. If one applies a gradient-based numerical optimization to this problem, one encounters many configurations that are stable but not globally minimal. This led the authors of this manuscript to the question, how many stable configurations are there? In this manuscript we report methods for identifying and counting observed stable configurations, and estimating the actual number of stable configurations. These estimates indicate that for N approaching two hundred, there are at least tens of thousands of stable configurations.