Exact equality between dissipation and irreversibility

We show, through a reformulation of the Crooks theorem and the Jarzynski equality, that the average dissipation for a system perturbed to go from one equilibrium state to another one, is exactly given by $\langle W \rangle_{diss} = \langle W \rangle -\Delta F =kT D(\rho\|\widetilde{\rho})= kT \langle \ln (\rho/\widetilde{\rho})\rangle$, where $\rho$ and $\widetilde{\rho}$ are the phase space density of the system measured at the same but otherwise arbitrary intermediate point in time, for the forward and backward process. $D(\rho\|\widetilde{\rho})$ is the relative entropy of $\rho$ versus $\widetilde{\rho}$.