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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.

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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