Energy minimisation of repelling particles on a toric grid

TL;DR

This paper investigates the optimal arrangements of repelling particles on a toric grid, proving a checkerboard distribution minimizes energy in many cases, supported by numerical validation and mathematical proofs.

Contribution

It proves the checkerboard distribution conjecture for specific torus sizes and repelling forces, advancing understanding of energy minimization on toric grids.

Findings

Checkerboard pattern minimizes energy for certain torus sizes.

Numerical experiments support the conjecture.

Mathematical proofs confirm the pattern in special cases.

Abstract

We explore the minimum energy configurations of repelling particles distributed over n possible locations forming a toric grid. We conjecture that the most energy-efficient way to distribute n/2 particles over this space is to place them in a checkerboard pattern. Numerical experiments validate this conjecture for reasonable choices of the repelling force. In the present paper, we prove this conjecture in a large number of special cases---most notably, when the sizes of the torus are either two or multiples of four in all dimensions and the repelling force is a completely monotonic function of the Lee distance between the particles.