Physical systems with non-reciprocal or dissipative forces evolve according to a generalization ofLiouville’s equation that accounts for the expansion and contraction of phase space volume. Here, weconnect geometric descriptions of these non-Hamiltonian dynamics to a recently established classicaldensity matrix theory. In this theory, the evolution of a “maximally mixed” classical density matrixis related to the well-known phase space contraction rate that, when ensemble averaged, is therate of entropy exchange with the surroundings. Here, we extend the definition of mixed statesto include statistical and mechanical components, describing both the deformations of local phasespace regions and the evolution of ensembles within them. As a result, the equation of motionfor 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 ofentropy flow characterizing nonequilibrium steady states