On assessing the accuracy of defect free energy computations

Abstract

We develop a rigorous error analysis for coarse-graining of defect-formation free energy. For a one-dimensional constrained atomistic system, we establish the thermodynamic limit of the defect-formation free energy and obtain explicitly the rate of convergence. We then construct a sequence of coarse-grained energies with the same rate but significantly reduced computational cost. We illustrate our analytical results through explicit computations for the case of harmonic potentials and through numerical simulations.

Thermodynamic Limit of the Transition Rate of a Crystalline Defect

“…While there is a substantial literature on the scaling limit (free energy per particle), see for example [11] and references therein, we are aware of only two references that attempt to rigorously capture atomistic details of the limit N → ∞ of crystalline defects in a finite temperature setting [13,32]. While [32] considers the somewhat different setting of observables rather than formation energies there is a close connection in that those observables are localised.…”

“…Moreover, an asymptotic series in β is derived instead of focusing only on leading terms. By contrast [13] addresses the finite β regime, but severely restricts the admissible interaction laws. Both of these references are restricted to one dimension, which yields significant simplifications highlighted for example by the fact that discrete Green's functions decay exponentially.…”

Thermodynamic Limit of the Transition Rate of a Crystalline Defect

“…While there is a substantial literature on the scaling limit (free energy per particle), see e.g. [DF05] and references therein, we are aware of only two references that attempt to rigorously capture atomistic details of the limit N → ∞ of crystalline defects in a finite temperature setting [SL17,DDO18]. While [SL17] considers the somewhat different setting of observables rather than formation energies there is a close connection in that those observables are localised.…”

“…Moreover, an asymptotic series in β is derived instead of focusing only on leading terms. By contrast [DDO18] addresses the finite β regime, but severely restricts the admissible interaction laws. Both of these references are restricted to one dimension, which yields significant simplifications highlighted for example by the fact that discrete Green's functions decay exponentially.…”