Dissipation: The phase-space perspective
We show, through a refinement of the work theorem, that the average dissipation, upon perturbing a Hamiltonian system arbitrarily far out of equilibrium in a transition between two canonical equilibrium states, is exactly given by $ = < W > -ΔF =kT D(ρ\|\widetildeρ)= kT < \ln (ρ/\widetildeρ)>$, where $ρ$ and $\widetildeρ$ are the phase space density of the system measured at the same intermediate but otherwise arbitrary point in time, for the forward and backward process. $D(ρ\|\widetildeρ)$ is the relative entropy of $ρ$ versus $\widetildeρ$. This result also implies general inequalities, which are significantly more accurate than the second law and include, as a special case, the celebrated Landauer principle on the dissipation involved in irreversible computations.