Discrete minimizers of the interaction energy in collective behavior: a brief numerical and analytic review

TL;DR

This paper reviews numerical and analytical methods for finding minimizers of interaction energy in collective behavior models, focusing on less singular potentials and their properties in two dimensions.

Contribution

It provides a comparative review of numerical algorithms and analyzes properties of minimizers for a range of pair potentials relevant to collective behavior.

Findings

GMIN algorithm effectively finds low-N minimizers

Minimizers exhibit lattice structures and Wulff shapes

Large-N limits show continuous behavior for less singular potentials

Abstract

We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are repulsive at short distances and attractive at long distances, focusing on examples which are sums of two powers. The range of powers we look at includes the well-known case of the Lennard-Jones potential, but we are also interested in less singular potentials which are relevant in collective behavior models. We report on results using the software GMIN developed by Wales and collaborators for problems in chemistry. For all cases, this algorithm gives good candidates for the minimizers for relatively low values of the particle number N. This is well-known for potentials similar to Lennard-Jones, but not for the range which is of interest in collective behavior. Standard minimization procedures have been used in the literature in this range, but they are likely to yield stationary states which are not minimizers. We illustrate numerically some properties of the minimizers in 2D, such as lattice structure, Wulff shapes, and the continuous large-N limit for locally integrable (that is, less singular) potentials.