Dissipation: The phase-space perspective

TL;DR

This paper refines the work theorem to precisely relate average dissipation in Hamiltonian systems to phase space densities and relative entropy, providing more accurate inequalities than the second law and implications for irreversible computation.

Contribution

It introduces a phase-space perspective that refines the work theorem, linking dissipation to phase space densities and relative entropy for arbitrary far-from-equilibrium transitions.

Findings

Exact expression for average dissipation in terms of phase space densities.

More accurate inequalities than the second law including Landauer's principle.

Dissipation related to relative entropy of phase space distributions.

Abstract

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 $<W_{diss} > = < W > -\Delta F =kT D(\rho\|\widetilde{\rho})= kT < \ln (\rho/\widetilde{\rho})>$, where $\rho$ and $\widetilde{\rho}$ 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. $D(\rho\|\widetilde{\rho})$ is the relative entropy of $\rho$ versus $\widetilde{\rho}$. 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.