Crystallization in two dimensions and a discrete Gauss-Bonnet theorem

DOI10.1007/s00332-017-9401-6zbMath1382.82047arXiv1605.00034OpenAlexW3105442708MaRDI QIDQ1702987

Lucia De Luca, Gero Friesecke

Publication date: 1 March 2018

Published in: Journal of Nonlinear Science (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1605.00034

zbMATH Keywords

energy minimizationGauss-Bonnet theoreminteraction potentialdiscrete differential geometrycrystallization

Mathematics Subject Classification ID

Crystalline structure (74E15) Statistical mechanics of crystals (82D25) Planar graphs; geometric and topological aspects of graph theory (05C10)

Related Items (23)

Optimality of the triangular lattice for Lennard–Jones type lattice energies: a computer-assisted method ⋮ On Born's conjecture about optimal distribution of charges for an infinite ionic crystal ⋮ Typical ground states for large sets of interactions ⋮ Local variational study of 2d lattice energies and application to Lennard–Jones type interactions ⋮ Classification of particle numbers with unique Heitmann-Radin minimizer ⋮ Maximal fluctuations on periodic lattices: an approach via quantitative Wulff inequalities ⋮ Three‐dimensional lattice ground states for Riesz and Lennard‐Jones–type energies ⋮ Emergence of Wulff-crystals from atomistic systems on the FCC and HCP lattices ⋮ A proof of finite crystallization via stratification ⋮ Pathwise regularisation of singular interacting particle systems and their mean field limits ⋮ Asymptotic optimality of the triangular lattice for a class of optimal location problems ⋮ On the optimality of the rock-salt structure among lattices with charge distributions ⋮ Emergence of rigid polycrystals from atomistic systems with Heitmann-Radin sticky disk energy ⋮ Crystallization to the square lattice for a two-body potential ⋮ Optimal lattice configurations for interacting spatially extended particles ⋮ Optimal and non-optimal lattices for non-completely monotone interaction potentials ⋮ Minimizing lattice structures for Morse potential energy in two and three dimensions ⋮ Finite Crystallization and Wulff shape emergence for ionic compounds in the square lattice ⋮ Crystallization in the hexagonal lattice for ionic dimers ⋮ Vectorial crystallization problems and collective behavior ⋮ Some recent results on 2D crystallization for sticky disc models and generalizations for systems of oriented particles ⋮ \(\mathit\Gamma\)-convergence of the Heitmann-Radin sticky disc energy to the crystalline perimeter ⋮ On energy ground states among crystal lattice structures with prescribed bonds

Cites Work

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• A discrete Gauss-Bonnet type theorem
• Face-centered cubic crystallization of atomistic configurations
• Sharp \(N^{3/4}\) law for the minimizers of the edge-isoperimetric problem on the triangular lattice
• Lower bound for the interatomic distance in Lennard-Jones clusters
• Flows in complex networks: theory, algorithms, and application to lennard-Jones cluster rearrangement
• On the crystallization of 2D hexagonal lattices
• A proof of crystallization in two dimensions
• The crystallization conjecture: a review
• Gradient theory for plasticity via homogenization of discrete dislocations
• Geometric measure theory.
• Crystallization in carbon nanostructures
• Packing twelve spherical caps to maximize tangencies
• Classification of particle numbers with unique Heitmann-Radin minimizer
• Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape
• Ground states of the 2D sticky disc model: fine properties and \(N^{3/4}\) law for the deviation from the asymptotic Wulff shape
• Combinatorial curvature for planar graphs

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