Entropy production and phase space volume contraction

We inquire whether the connection between entropy production and phase space volume contraction rate reported recently for a class of thermostatted systems is an intrinsic property of a wide class of dynamical systems, or the result of the particular algorithm devised for thermostatting a system of interacting particles obeying, in the presence of nonequilibrium constraints, a time-reversible, dissipative dynamics. A nonequilibrium thermodynamics based on the balance equation for information entropy is developed for dissipative dynamical systems subjected, in addition, to a stochastic forcing. This latter accounts for the thermodynamic fluctuations accompanying the reduced description of the thermostat by a dissipative perturbation, for the interaction between the system and the external reservoirs or for perturbations of external origin. Entropy flux and entropy productionlike terms depending on the characteristics of the dynamics in phase space, particularly the rate of phase space volume contraction, are identified. Their connections with irreversible thermodynamics are explored. In particular, for thermostatted systems we find, without invoking an ad hoc conservation law between the system and the reservoir, that information entropy production is related to the opposite of the rate of phase space volume contraction to the second order in the distance from equilibrium.