Optimal lattice configurations for interacting spatially extended particles

TL;DR

This paper analyzes the optimal arrangements of extended particles on a 2D lattice, proving the triangular lattice minimizes energy under certain conditions related to mass distribution and potential functions.

Contribution

It establishes the global minimality of the triangular lattice for extended particles with specific mass distributions and potential functions in two-dimensional settings.

Findings

Triangular lattice is globally optimal for concentrated mass distributions.

Optimality holds for completely monotone potential and density functions.

Results apply to fixed density configurations.

Abstract

We investigate lattice energies for radially symmetric, spatially extended particles interacting via a radial potential and arranged on the sites of a two-dimensional Bravais lattice. We show the global minimality of the triangular lattice among Bravais lattices of fixed density in two cases: In the first case, the distribution of mass is sufficiently concentrated around the lattice points, and the mass concentration depends on the density we have fixed. In the second case, both interacting potential and density of the distribution of mass are described by completely monotone functions in which case the optimality holds at any fixed density.