Crystallization in two dimensions and a discrete Gauss-Bonnet theorem. (English) Zbl 1382.82047
Summary: 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 [R. C. Heitmann and C. Radin, “The ground state for sticky disks”, J. Stat. Phys. 22, No. 3, 281–287 (1980; doi:10.1007/BF01014644)], 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 discrete curvature, and appealing to a discrete Gauss-Bonnet theorem [O. Knill, Elem. Math. 67, No. 1, 1–17 (2012; Zbl 1269.53011)] which, as its continuous cousins, relates the sum/integral 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.
MSC:
| 82D25 | Statistical mechanics of crystals |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 74E15 | Crystalline structure |
Keywords:
crystallization; interaction potential; discrete differential geometry; energy minimization; Gauss-Bonnet theoremCitations:
Zbl 1269.53011References:
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