# Classification of particle numbers with unique Heitmann-Radin minimizer

## TL;DR

This paper characterizes when minimizers of the Heitmann-Radin energy are unique, identifying a specific sequence of particle numbers for which this holds, using discrete differential geometry techniques.

## Contribution

It provides a complete classification of particle numbers with unique Heitmann-Radin minimizers, extending understanding of energy minimization in discrete systems.

## Key findings

- Unique minimizers occur for a specific infinite sequence of particle numbers.
- The sequence of particle numbers with unique minimizers is explicitly characterized.
- The proof employs advanced discrete differential geometry methods.

## Abstract

We show that minimizers of the Heitmann-Radin energy (R. C. Heitmann, C. Radin, J. Stat. Phys. 22, 281-287, 1980) are unique if and only if the particle number N belongs to an infinite sequence whose first thirty-five elements are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120 (see the paper for a closed-form description of this sequence). The proof relies on the discrete differential geometry techniques introduced in (L. De Luca, G. Friesecke, preprint).