Approximation of Crystalline Defects at Finite Temperature

Abstract

The present paper aims at developing a theory of computation of crystalline defects at finite temperature. In a one-dimensional setting we introduce Gibbs distributions corresponding to such defects and rigorously establish their asymptotic expansion. We then give an example of using such asymptotic expansion to compare the accuracy of computations using the free boundary conditions and using an atomistic-to-continuum coupling method.

Thermodynamic Limit of the Transition Rate of a Crystalline Defect

“…Apart from being interesting in their own right, our results provide the analytical foundations for a rigorous derivation of coarse-grained models [5,19,33,36], and of numerical and multi-scale models at finite temperature [3,4,22,32,33] which JB and CO are supported by ERC Starting Grant 335120 and by EPSRC Grant EP/R043612/1. MHD was supported by ERC Starting Grant 335120 entirely lack the solid foundations that static zero-temperature multi-scale schemes enjoy [23,24,26].…”

“…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.…”

On assessing the accuracy of defect free energy computations

“…Furthermore, it does not take defects into account. The work [SL14] is in spirit much closer to ours and in particular does take defects into account. The main difference to our work is that [SL14] consider "low" temperature via an asymptotic series expansion.…”

“…A great number of numerical schemes on spatial coarse-graining of the free energy have been developed in the literature, see for instance in [DTMP05, MVH + 10] and references therein. However, a rigorous analysis on the accuracy of these schemes is still underdeveloped; we are only aware of the references [BBLP10,SL14].…”

Distribution of Cracks in a Chain of Atoms at Low Temperature

“…In particular, the R-dependence is explicitly seen to enter in exponentially small correction terms only. Harmonic approximations in case of more general pair potentials v would require to replace v (a) by more complicated terms from Hessians or WKB expansions [28,39], see also [32,Section 2.3]. (For related techniques in the context of computational approximation schemes for the simulation of atomistic materials, see [4,7,9,39].)…”