Classical non-equilibrium statistical mechanics and an "open system dynamics" perspective on quantum-classical analogy

TL;DR

This paper explores the dynamics of classical systems coupled to external environments using Green's functions, compares it with quantum master equations, and discusses implications for quantum-classical analogy.

Contribution

It develops a time-local equation of motion for open classical systems and compares it with quantum counterparts, highlighting differences relevant to quantum-classical analogy.

Findings

Derived a classical equation of motion with a unique term not present in quantum analogues

Identified a potential obstacle to quantum-classical analogy in open systems

Proposed methods to reconcile classical and quantum dynamics in open system frameworks

Abstract

It is well known that the statistics of closed classical systems evolves according to the Liouville theorem. Here we study the dynamics of the marginal statistics of classical systems coupled to external degrees of freedom, by developing a time-local equation of motion using Green's functions and a series expansion method. We also compare this equation of motion with its supposed quantum counterpart, namely the quantum master equation, which we hope could shed some light on quantum-classical analogy (QCA) from the perspective of "open system dynamics". We notice an apparent exception to QCA in this case, as the first-order classical equation of motion derived herein contains a term that does not appear to have a quantum analogue. We also propose possible ways of getting around this tension, which may help re-establish QCA (in first perturbative order). We do not draw a definitive conclusion about QCA in the context of open system dynamics but hope to provide a starting point for investigations along this line.