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It is well known that the statistics of closed classical systems evolvesaccording to the Liouville theorem. Here we study the dynamics of the marginalstatistics of classical systems coupled to external degrees of freedom, bydeveloping a time-local equation of motion using Green's functions and a seriesexpansion method. We also compare this equation of motion with its supposedquantum counterpart, namely the quantum master equation, which we hope couldshed some light on quantum-classical analogy (QCA) from the perspective of"open system dynamics". We notice an apparent exception to QCA in this case, asthe first-order classical equation of motion derived herein contains a termthat does not appear to have a quantum analogue. We also propose possible waysof getting around this tension, which may help re-establish QCA (in firstperturbative order). We do not draw a definitive conclusion about QCA in thecontext of open system dynamics but hope to provide a starting point forinvestigations along this line.

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