Elastic constants from microscopic strain fluctuations

Abstract

Fluctuations of the instantaneous local Lagrangian strain ${\ensuremath{\epsilon}}_{\mathrm{ij}}(\mathbf{r},\mathbf{t}),$ measured with respect to a static ``reference'' lattice, are used to obtain accurate estimates of the elastic constants of model solids from atomistic computer simulations. The measured strains are systematically coarse-grained by averaging them within subsystems (of size ${L}_{b})$ of a system (of total size $L)$ in the canonical ensemble. Using a simple finite size scaling theory we predict the behavior of the fluctuations $〈{\ensuremath{\epsilon}}_{\mathrm{ij}}{\ensuremath{\epsilon}}_{\mathrm{kl}}〉$ as a function of ${L}_{b}/L$ and extract elastic constants of the system in the thermodynamic limit at nonzero temperature. Our method is simple to implement, efficient, and general enough to be able to handle a wide class of model systems, including those with singular potentials without any essential modification. We illustrate the technique by computing isothermal elastic constants of ``hard'' and ``soft'' disk triangular solids in two dimensions from Monte Carlo and molecular dynamics simulations. We compare our results with those from earlier simulations and theory.