# Classifying minimum energy states for interacting particles: Regular simplices

arXiv: https://arxiv.org/abs/2109.07091 · HTML: https://mlsanity.com/paper/2109.07091/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Sep 2021.

## Built on by (most cited first, 1 of 1)

- [Contrastive Self-Supervised Learning As Neural Manifold Packing](https://mlsanity.com/paper/2506.13717/) (2025)

## Abstract (arXiv)

Densities of particles on Rn which interact pairwise through an attractive-repulsive power-law potential Wal,bt(x) = |x|^al/al-|x|^bt/bt have often been used to explain patterns produced by biological and physical systems. In the mildly repulsive regime al> bt ≥ 2 with n ≥ 2, we show there exists a decreasing homeomorphism alDeltan from [2,4] to itself such that: distributing the particles uniformly over the vertices of a regular unit diameter n-simplex minimizes the potential energy if and only if al≥ alDen(bt). Moreover this minimum is uniquely attained up to rigid motions when al > alDen(bt). We estimate alDen(bt) above and below, and identify its limit as the dimension grows large. These results are derived from a new northeast comparison principle in the space of exponents. At the endpoint (al,bt)=(4,2) of this transition curve, we characterize all minimizers by showing they lie on a sphere and share all first and second moments with the spherical shell. Suitably modified versions of these statements are also established (i) for Wα,β and corresponding energies in the case where n=1, and (ii) for the attractive-repulsive potentials D_al(x) = |x|^al(allog |x|-1) that arise in the limit bt nearrow al.