Microscopic derivation of coarse-grained, energy-conserving generalized Langevin dynamics

Properly simulating non-equilibrium phenomena such as thermal transport in condensed matter systems requires the conservation of system’s internal energy. This precludes application of the coarse-grained (CG) generalized Langevin equation (GLE) dynamics. Attempts to address this issue have been pursued phenomenologically for dissipative particle dynamics (DPD, a Markovian variant of the CG GLE dynamics) by introducing an energy conserving extension of DPD (DPD-E). We present here a rigorous microscopic derivation of energy conserving variants of the CG GLE dynamics by extending the CG equations of motion to include the GLE for certain internal energy observables of the microscopic system. The derivation is performed using the Mori-Zwanzig projection operator method with the recently introduced interpretation of the Zwanzig projection operator.¹ Our extension of the GLE dynamics to quasi-equilibrium conditions (necessary to observe heat transport), are based on the generalized canonical ensemble approach and transport equation method. We derive closed microscopic expressions for conductive heat transfer coefficients. After employing the Markov approximation, we compare the equations of motion to the published DPD-E equations.
¹ S. Izvekov, J. Chem. Phys. 151 (10), 104109 (2019).