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

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.

References in corpus (3)

• Crystallization in two dimensions and a discrete Gauss-Bonnet theorem
• Density of polyhedral partitions
• Classification of particle numbers with unique Heitmann-Radin minimizer

Cited by in corpus (3)

• Functionals defined on piecewise rigid functions: Integral representation and $\Gamma$-convergence
• Emergence of rigid Polycrystals from atomistic Systems with Heitmann-Radin sticky disk energy
• Distribution of cracks in a chain of atoms at low temperature