Microscopic reversibility for classical open systems

TL;DR

This paper establishes a universal symmetry in the trajectory probabilities of classical open systems under reversible Liouvillian dynamics, generalizing detailed balance and describing heat flow at a microscopic level.

Contribution

It provides a rigorous proof of microscopic reversibility for classical open systems, applicable to both conservative and dissipative dynamics with arbitrary initial states and external forces.

Findings

Derives a universal symmetry relation for trajectory probabilities.

Generalizes the detailed balance principle to non-equilibrium conditions.

Provides a microscopic expression for heat flow in open systems.

Abstract

We rigorously show that the probability to have a specific trajectory of an externally perturbed classical open system satisfies a universal symmetry for Liouvillian reversible dynamics. It connects the ratio between the probabilities of time forward and reversed trajectories to a degree of the time reversal asymmetry of the final phase space distribution. Indeed, if the final state is in equilibrium, then the forward and reversed net transition probabilities are equal, which gives a generalization of the detailed balance principle. On the other hand, when the external forcing maintains the system out of equilibrium, it expresses an asymmetry for the probabilities of the time forward and reversed trajectories. Especially, it gives a microscopic expression of the heat flowing to a system from a reservoir where the subdynamics seems like a Markovian stochastic process. Also, it turns out that the expression of the microscopic reversibility holds both for the conservative and dissipative dynamics with an arbitrary initial state and external forcing.