Nonequilibrium thermomechanics of Gaussian phase packet crystals: Application to the quasistatic quasicontinuum method
P. Gupta, M. Ortiz and D. Kochmann 
Journal of the Mechanics and Physics of Solids, 153, 104495 (2021).

ABSTRACT
The quasicontinuum (QC) method was originally introduced to bridge across length
scales by coarse-graining an atomistic ensemble to significantly larger
continuum scales at zero temperature, thus overcoming the crucial length-scale
limitation of classical atomic-scale simulation techniques while solely relying
on atomic-scale input (in the form of interatomic potentials). An associated
challenge lies in bridging across time scales to overcome the time- scale
limitations of atomistics at finite temperature. To address the biggest
challenge, bridging across both length and time scales, only a few techniques
exist, and most of those are limited to conditions of constant temperature.
Here, we present a new general strategy for the space–time coarsening of an
atomistic ensemble, which introduces thermomechanical coupling. Specifically, we
evolve the statistics of an atomistic ensemble in phase space over time by
applying the Liouville equation to an approximation of the ensemble’s
probability distribution (which further admits a variational formulation). To
this end, we approximate a crystalline solid as a lattice of lumped correlated
Gaussian phase packets occupying atomic lattice sites, and we investigate the
resulting quasistatics and dynamics of the system. By definition, phase packets
account for the dynamics of crystalline lattices at finite temperature through
the statistical variances of atomic momenta and positions. We show that
momentum–space correlation allows for an exchange between potential and kinetic
contributions to the crystal’s Hamiltonian. Consequently, local adiabatic
heating due to atomic site motion is captured. Moreover, in the quasistatic
limit, the governing equations reduce to the minimization of thermodynamic
potentials (similar to maximum-entropy formulation previously introduced for
finite-temperature QC), and they yield the local equation of state, which we
derive for isothermal, isobaric, and isentropic conditions. Since our
formulation without interatomic correlations precludes irreversible heat
transport, we demonstrate its combination with thermal transport models to
describe realistic atomic- level processes, and we discuss opportunities for
capturing atomic-level thermal transport by including interatomic correlations
in the Gaussian phase packet formulation. Overall, our Gaussian phase packet
approach offers a promising avenue for finite-temperature non- equilibrium
quasicontinuum techniques, which may be combined with thermal transport models
and extended to other approximations of the probability distribution as well as
to exploit the variational structure.