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Surface energies in a two-dimensional mass-spring model for crystals. (English) Zbl 1269.82065
Summary: We study an atomistic pair potential-energy \(E^{(n)}(y\)) that describes the elastic behavior of two-dimensional crystals with \(n\) atoms, where \(y \in {\mathbb R}^{2\times n}\) characterizes the particle positions. The main focus is on the asymptotic analysis of the ground state energy as \(n\) tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of \(E^{(n)}\) admits an asymptotic expansion involving fractional powers of \(n\): \[ \min_y E^{(n)}(y) = n E_{\mathrm{bulk}}+ \sqrt{n} E_{\mathrm{surface}} +o(\sqrt{n}), \qquad n \to \infty. \] The bulk energy density \(E_{\mathrm{bulk}}\) is given by an explicit expression involving the interaction potentials. The surface energy \(E_{\mathrm{surface}}\) can be expressed as a surface integral, where the integrand depends only on the surface normal and the interaction potentials. The evaluation of the integrand involves solving a discrete algebraic Riccati equation. Numerical simulations suggest that the integrand is a continuous but nowhere differentiable function of the surface normal.

MSC:
82D25 Statistical mechanical studies of crystals
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
74Q05 Homogenization in equilibrium problems of solid mechanics
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