In this work we derive a mathematical model for an open system that exchanges particles and momentum with a reservoir from their joint Hamiltonian dynamics. The complexity of this many-particle problem is addressed by introducing a countable set of n-particle phase space distribution functions just for the open subsystem, while accounting for the reservoir only in terms of statistical expectations. From the Liouville equation for the full system we derive a set of coupled Liouville-type equations for the n-particle distributions by marginalization with respect to reservoir states. The resulting equation hierarchy describes the external momentum forcing of the open system by the reservoir across its boundaries, and it covers the effects of particle exchanges, which induce probability transfers between the n- and (n+1)-particle distributions. Similarities and differences with the Bergmann-Lebowitz model of open systems (P.G.Bergmann, J.L. Lebowitz, Phys.Rev., 99:578--587 (1955)) are discussed in the context of the implementation of these guiding principles in a computational scheme for molecular simulations.

中文翻译：

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开放系统 n 粒子分布函数的 Liouville 型方程

在这项工作中，我们推导出了一个开放系统的数学模型，该系统从它们的联合哈密顿动力学中与储层交换粒子和动量。这个多粒子问题的复杂性通过为开放子系统引入一组可数的 n 粒子相空间分布函数来解决，同时仅根据统计预期考虑储层。从整个系统的 Liouville 方程，我们通过相对于储层状态的边缘化导出一组耦合的 Liouville 型方程，用于 n 粒子分布。由此产生的方程层次描述了储层越过其边界对开放系统施加的外部动量，它涵盖了粒子交换的影响，这导致了 n 和 (n+1) 粒子分布之间的概率转移。