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 $⟨{W}diss⟩=⟨W⟩{-}{Δ}F=kTD({ρ}{∥}{{}}{{ρ}})=kT⟨ln({ρ}/{{}}{{ρ}})⟩$, where ${ρ}$ and ${{}}{{ρ}}$ 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. $D({ρ}{∥}{{}}{{ρ}})$ is the relative entropy of ${ρ}$ versus ${{}}{{ρ}}$. 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.