Phase-space contraction rate for classical mixed states

Physical systems with nonreciprocal 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. 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.