# Isodiametry, variance, and regular simplices from particle interactions

## TL;DR

This paper demonstrates that particles interacting via a specific potential form regular simplices at energy minima, with phase transitions inferred from variance bounds, advancing understanding of particle configurations and energy minimization.

## Contribution

It introduces a novel isodiametric variance bound that characterizes regular simplices as energy minimizers in particle interaction models.

## Key findings

- Regular simplices uniquely minimize energy for certain interaction exponents.
- Configurations with less attraction are local minimizers in the $d_
abla$ metric.
- Phase transitions are inferred from the variance bounds and configuration stability.

## Abstract

Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained -- apart from translations and rotations -- by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show these configurations are at least local energy minimizers in the relevant $d_\infty$ metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based on a simple isodiametric variance bound which characterizes regular simplices: it shows that among probability measures on ${\mathbf R}^n$ whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass $1/(n+1)$ to each vertex of a (unit-diameter) regular simplex.