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 $⟨{W}_{mathrm{diss}}⟩=⟨W⟩ensuremath{-}ensuremath{Delta}F=kTD(ensuremath{rho}ensuremath{parallel}stackrel{texttildelow{}}{ensuremath{rho}})=kT⟨mathrm{ln}(ensuremath{rho}/stackrel{texttildelow{}}{ensuremath{rho}})⟩$, where $ensuremath{rho}$ and $stackrel{texttildelow{}}{ensuremath{rho}}$ 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. $D(ensuremath{rho}ensuremath{parallel}stackrel{texttildelow{}}{ensuremath{rho}})$ is the relative entropy of $ensuremath{rho}$ versus $stackrel{texttildelow{}}{ensuremath{rho}}$. 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.