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

created by ponsiglio on 21 May 2018
modified by novaga on 16 Aug 2019

[BibTeX]

Published Paper

Inserted: 21 may 2018
Last Updated: 16 aug 2019

Journal: J. Nonlinear Sci.
Volume: 29
Number: 4
Pages: 1273-1299
Year: 2019

ArXiv: 1805.08472 PDF

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.