# $$\varGamma$$Γ-Convergence of the Heitmann–Radin Sticky Disc Energy to the Crystalline Perimeter

@article{Luca2019varGammaO,
title={\$\$\varGamma \$\$$\Gamma-Convergence of the Heitmann–Radin Sticky Disc Energy to the Crystalline Perimeter}, author={Lucia De Luca and Matteo Novaga and Marcello Ponsiglione}, journal={Journal of Nonlinear Science}, year={2019}, pages={1-27} } • Published 22 May 2018 • Mathematics • Journal of Nonlinear Science 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. 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We show that the tail energy enforces optimal packing ## References SHOWING 1-10 OF 23 REFERENCES Ground States of the 2D Sticky Disc Model: Fine Properties and N3/4 Law for the Deviation from the Asymptotic Wulff Shape We investigate ground state configurations for a general finite number N of particles of the Heitmann-Radin sticky disc pair potential model in two dimensions. Exact energy minimizers are shown to Derivation of Linearised Polycrystals from a 2D system of edge dislocations • Mathematics • 2018 In this paper we show the emergence of polycrystalline structures as a result of elastic energy minimisation. For this purpose, we introduce a variational model for two-dimensional systems of edge Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape • Physics • 2009 We investigate ground state configurations of atomic systems in two dimensions interacting via short range pair potentials. 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Classification of Particle Numbers with Unique Heitmann–Radin Minimizer • Mathematics • 2017 We show that minimizers of the Heitmann–Radin energy (Heitmann and Radin in J Stat Phys 22(3):281–287, 1980) are unique if and only if the particle number N belongs to an infinite sequence whose A Proof of Crystallization in Two Dimensions This work shows rigorously that under suitable assumptions on the potential V which are compatible with the growth behavior of the Lennard-Jones potential the ground state energy per particle converges to an explicit constant E*: where E* ∈ ℝ is the minimum of a simple function on [0,∞). Sharp$$N^{3/4}$\$N3/4 Law for the Minimizers of the Edge-Isoperimetric Problem on the Triangular Lattice
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In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking