Generalized Langevin equation for solids. II. Stochastic boundary conditions for nonequilibrium molecular dynamics simulations

The generalized Langevin equation (GLE) [L. Kantorovich, Phys. Rev. B78, 094304 (2008)], which describes dynamics of a finite and possibly highly anharmonic subsystem surrounded by an extended harmonic solid, is simplified here assuming short-range interactions between atoms. We show that in this case quite naturally the GLE can be worked into a form which corresponds to considering central atoms of the finite region as governed by usual Newtonian mechanics, while the boundary atoms are treated as Langevin atoms, i.e., they experience friction and random forces [the so-called stochastic boundary conditions (SBCs)]. We show that the random forces constitute a stationary Gaussian random process with the dispersion directly related to the friction coefficient in a usual way. Next, we rigorously demonstrate that, even though not all atoms are stochastic within the SBC model, the system should still arrive at canonical distribution at long times. Since the SBC model follows directly from the general GLE description and can perform as a correct $NVT$ thermostat, we propose that the SBC is a method of choice if one wants to do nonequilibrium molecular dynamics simulations correctly. Our derivation of SBC is physically more acceptable than given previously when the central region atoms were assumed to be much heavier than those in the surrounding lattice (the zero-frequency approximation).