Divide and Conquer: A Distributed Approach to Five Point Energy Minimization

TL;DR

This paper rigorously verifies a phase transition in 5-point energy minimization, identifying the specific s-values where different configurations are optimal, and connects to classical problems like Thomson's 5 electron problem.

Contribution

It provides a rigorous proof of the phase transition in 5-point energy minimization and precisely characterizes the minimizers for different s-values.

Findings

Triangular bi-pyramid minimizes energy for s in (0,S)

Square-based pyramid minimizes energy for s in (S,15+512/25]

Connection to Thomson's 5 electron problem at s=1

Abstract

This work rigorously verifies the phase transition in 5-point energy minimization first observed by Melnyk-Knop-Smith in 1977. More precisely, we prove that there is a constant S = [15+24/512,15+25/512] such that the triangular bi-pyramid is the energy minimizer with respect to the s-power law potential for all s in (0,S) and some pyramid with square base is the unique minimizer for all s in (S,15+512/25]. Taking s=1 gives another solution to Thomson's 5 electron problem from 1904.