The covariant dissipation function for transient nonequilibrium states

It has recently become apparent that the dissipation function, first defined by Evans and Searles [J. Chem. Phys. 113, 3503 (2000)], is one of the most important functions in classical nonequilibrium statistical mechanics. It is the argument of the Evans-Searles fluctuation theorem, the dissipation theorem, and the relaxation theorems. It is a function of both the initial distribution and the dynamics. We pose the following question: How does the dissipation function change if we define that function with respect to the time evolving phase space distribution as one relaxes from the initial equilibrium distribution toward the nonequilibrium steady state distribution? We prove that this covariant dissipation function has a rather simple fixed relationship to the dissipation function defined with respect to the initial distribution function. We also show that there is no exact, time-local, Evans-Searles nonequilibrium steady state fluctuation relation for deterministic systems. Only an asymptotic version exists.