Dissipation: The phase-space perspective

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

4 pages, 3 figures (4 figure files), accepted for PRL

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