On the evaluation of Hardy's thermomechanical quantities using ensemble and time averaging

Abstract

An ensemble averaging approach was investigated for its accuracy and convergence against time averaging in computing continuum quantities such as stress, heat flux and temperature from atomistic scale quantities. For this purpose, ensemble averaging and time averaging were applied to evaluate Hardy's thermomechanical expressions (Hardy 1982 J. Chem. Phys. 76 622–8) in equilibrium conditions at two different temperatures as well as a nonequilibrium process due to shock impact on a Ni crystal modeled using molecular dynamics simulations. It was found that under equilibrium conditions, time averaging requires selection of a time interval larger than the critical time interval to obtain convergence, where the critical time interval can be estimated using the elastic properties of the material. The reason for this is because of the significant correlations among the computed thermomechanical quantities at different time instants employed in computing their time average. On the other hand, the computed thermomechanical quantities from different realizations in ensemble averaging are statistically independent, and thus convergence is always guaranteed. The computed stress, heat flux and temperature show noticeable difference in their convergence behavior while their confidence intervals increase with temperature. Contrary to equilibrium settings, time averaging is not equivalent to ensemble averaging in the case of shock wave propagation. Time averaging was shown to have poor performance in computing various thermomechanical fields by either oversmoothing the fields or failing to remove noise.