Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape

TL;DR

This paper studies the ground state atomic configurations in two dimensions with short range pair potentials, showing convergence to a macroscopic cluster with a Wulff shape for certain potentials, via Gamma-convergence analysis.

Contribution

It establishes the convergence of low-energy configurations to a macroscopic cluster with a Wulff shape using Gamma-convergence, specifically for the Heitmann-Radin potential.

Findings

Ground states form a macroscopic cluster with finite surface area.

The cluster's shape converges to a Wulff shape for the Heitmann-Radin potential.

Energy converges to a macroscopic anisotropic surface energy.

Abstract

We investigate ground state configurations of atomic systems in two dimensions interacting via short range pair potentials. As the number of particles tends to infinity, we show that low-energy configurations converge to a macroscopic cluster of finite surface area and constant density, the latter being given by the density of atoms per unit volume in the triangular lattice. In the special case of the Heitmann-Radin sticky disc potential and exact ground states, we show that the macroscopic cluster has a (unique) Wulff shape. This is done by showing that the atomistic energy, after subtracting off a bulk part and re-scaling, Gamma-converges to a macroscopic anisotropic surface energy.