Calculus of Variations and Geometric Measure Theory
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L. De Luca - G. Friesecke

Crystallization in two dimensions and a discrete Gauss-Bonnet theorem

created by deluca on 30 Jan 2017
modified on 04 Jul 2018


Published Paper

Inserted: 30 jan 2017
Last Updated: 4 jul 2018

Journal: J. Nonlinear Sci.
Volume: 28
Number: 1
Pages: 69--90
Year: 2017
Doi: 10.1007/s00332-017-9401-6


We show that the emerging field of discrete differential geometry can be usefully brought to bear on crystallization problems. In particular, we give a simplified proof of the Heitmann-Radin crystallization theorem \cite{HR}, which concerns a system of $N$ identical atoms in two dimensions interacting via the idealized pair potential $V(r)=+\infty$ if $r<1$, $-1$ if $r=1$, $0$ if $r>1$. This is done by endowing the bond graph of a general particle configuration with a suitable notion of {\it discrete curvature}, and appealing to a {\it discrete Gauss-Bonnet theorem} \cite{Knill1} which, as its continuous cousins, relates the sumintegral of the curvature to topological invariants. This leads to an exact geometric decomposition of the Heitmann-Radin energy into (i) a combinatorial bulk term, (ii) a combinatorial perimeter, (iii) a multiple of the Euler characteristic, and (iv) a natural topological energy contribution due to defects. An analogous exact geometric decomposition is also established for soft potentials such as the Lennard-Jones potential $V(r)=r^{-6}-2r^{-12}$, where two additional contributions arise, (v) elastic energy and (vi) energy due to non-bonded interactions.


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