$\Gamma$-convergence of the Heitmann-Radin sticky disc energy to the crystalline perimeter

TL;DR

This paper studies the asymptotic behavior of low-energy configurations of sticky disc models, showing convergence to crystalline structures and deriving a limit perimeter functional that captures anisotropic properties.

Contribution

It establishes a $ ext{Gamma}$-convergence result for the Heitmann-Radin energy to a crystalline perimeter, revealing the emergence of polycrystalline and single crystal structures.

Findings

Empirical measures converge to sets of finite perimeter.

Microscopic lattice orientation converges to a locally constant function.

The $ ext{Gamma}$-limit is an anisotropic perimeter for single crystals.

Abstract

We consider low energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle, i.e., the so called kissing number. For configurations whose energy scales like the perimeter, we prove a compactness result which shows the emergence of polycrystalline structures: The empirical measure converges to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function. Whenever the limit configuration is a single crystal, i.e., it has constant orientation, we show that the $\Gamma$-limit is the anisotropic perimeter, corresponding to the Finsler metric determined by the orientation of the single crystal.