Phase space contraction rate for classical mixed states

TL;DR

This paper links geometric non-Hamiltonian dynamics with classical density matrix theory, defining phase space contraction rates for mixed states to quantify entropy flow in dissipative systems.

Contribution

It extends the classical density matrix framework to include mixed states with both statistical and mechanical components, relating contraction rates to entropy exchange.

Findings

Defines phase space contraction rate for extended mixed states.

Connects contraction rate to entropy flow in nonequilibrium steady states.

Provides a geometric interpretation of dissipative dynamics.

Abstract

Physical systems with non-reciprocal or dissipative forces evolve according to a generalization of Liouville's equation that accounts for the expansion and contraction of phase space volume. Here, we connect geometric descriptions of these non-Hamiltonian dynamics to a recently established classical density matrix theory. In this theory, the evolution of a ``maximally mixed'' classical density matrix is related to the well-known phase space contraction rate that, when ensemble averaged, is the rate of entropy exchange with the surroundings. Here, we extend the definition of mixed states to include statistical and mechanical components, describing both the deformations of local phase space regions and the evolution of ensembles within them. As a result, the equation of motion for this mixed state represents the rate of contraction for an ensemble of dissipative trajectories. Recognizing this density matrix as a covariance matrix, its contraction rate is another measure of entropy flow characterizing nonequilibrium steady states.