Weapons and Materials Research Directorate, U.S. Army CCDC Army Research Laboratory, Aberdeen Proving Ground, Maryland 21005, USA.
J Chem Phys. 2019 Sep 14;151(10):104109. doi: 10.1063/1.5096655.
Properly simulating nonequilibrium phenomena such as thermal transport and shock wave propagation in complex condensed matter systems require the conservation of system's internal energy. This precludes the application of the coarse-grained (CG) generalized Langevin equation (GLE) dynamics due to the presence of dissipative interactions. Attempts to address this issue have been pursued both phenomenologically and from entropy-based first principles 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 two 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. We consider two choices of such observables: the total internal energy and a set of internal energies of the CG particles. The derivation is performed using the Mori-Zwanzig projection operator method in the Heisenberg picture for time evolution of thermodynamic expectations and the recently introduced interpretation of the Zwanzig projection operator [S. Izvekov, J. Chem. Phys. 146(12), 124109 (2017)] which allows an exact calculation of the memory and projected terms. We begin with equilibrium conditions and show that the GLE dynamics for the internal energy observables is purely dissipative. Our extension of the GLE dynamics to quasiequilibrium conditions (necessary to observe heat transport) is based on the generalized canonical ensemble approach and transport equation using the nonequilibrium statistical operator (NSO) method. We derive closed microscopic expressions for conductive heat transfer coefficients in the limit of neglecting dissipation in heat transfer and in the lowest order of deviation from equilibrium. After employing the Markov approximation, we compare the equations of motion to the published DPD-E equations. Our equations contain additional energy transfer terms not reported in the previous works. Additionally, we show that, despite neglecting dissipative processes in heat transport, the heat transfer coefficients and random force are related in a way reminiscent of the fluctuation-dissipation relation. The formalism presented here is sufficiently general for the rigorous formulation of the GLE dynamics for arbitrary microscopic phase space observables as well as sampling different microscopic ensembles in CG simulations.
正确模拟复杂凝聚态系统中的非平衡现象，如热输运和冲击波传播，需要系统内部能量的守恒。这排除了粗粒化（CG）广义朗之万方程（GLE）动力学的应用，因为存在耗散相互作用。已经从唯象和基于熵的第一原理出发，尝试解决这个问题，用于耗散粒子动力学（DPD，CG GLE 动力学的马尔可夫变体），通过引入 DPD 的能量守恒扩展（DPD-E）。我们在这里通过将 CG 运动方程扩展到包括微观系统的某些内能可观测量的 GLE，为 CG GLE 动力学的两个能量守恒变体提供了严格的微观推导。我们考虑了两种这样的可观测量的选择：总内能和 CG 粒子的一组内能。推导是使用海森堡图像中的 Mori-Zwanzig 投影算子方法在热力学期望的时间演化和最近引入的 Zwanzig 投影算子的解释[ S. Izvekov，J. Chem. Phys. 146(12), 124109 (2017)] 进行的，这允许对记忆和投影项进行精确计算。我们从平衡条件开始，并表明内能可观测量的 GLE 动力学是纯粹耗散的。我们将 GLE 动力学扩展到准平衡条件（观察热传输所必需的）是基于广义正则系综方法和使用非平衡统计算子（NSO）方法的输运方程。我们在忽略热传输中的耗散和从平衡的最低阶偏差的极限下，推导出传导热导率的微观封闭表达式。在采用马尔可夫近似之后，我们将运动方程与已发表的 DPD-E 方程进行了比较。我们的方程包含了以前的工作中没有报道的额外的能量传递项。此外，我们表明，尽管在热传输中忽略了耗散过程，但热导率和随机力之间的关系与波动耗散关系类似。这里提出的形式主义对于任意微观相空间可观测量的 GLE 动力学的严格表述以及在 CG 模拟中采样不同的微观系综都是足够通用的。
