# Liouville-type equations for the n-particle distribution functions of an open system

## TL;DR

This paper develops a mathematical framework for open systems exchanging particles and momentum with a reservoir, deriving Liouville-type equations for n-particle distributions that incorporate reservoir effects and particle exchanges.

## Contribution

It introduces a set of coupled Liouville-type equations for open systems' n-particle distributions derived from Hamiltonian dynamics, extending previous models.

## Key findings

- Derived hierarchy of equations for open system dynamics.
- Accounted for particle exchange and external momentum forcing.
- Compared with Bergmann-Lebowitz model for open systems.

## Abstract

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.