Work relations for time-dependent states

TL;DR

This paper generalizes the relation between free energy and thermodynamic work for time-dependent states, extending it to systems with momentum degrees of freedom and connecting it to steady state thermodynamics.

Contribution

It re-derives a relation between generalized free energy and work, and extends it to include systems with momentum degrees of freedom.

Findings

Re-derivation of the free energy-work relation for over-damped systems.

Extension of the relation to systems with momentum degrees of freedom.

Connection to an extended Clausius relation in steady state thermodynamics.

Abstract

For time-dependent states generated by an external operation, a generalized free energy may be introduced by the relative entropy with respect to an equilibrium state realized after sufficient relaxation from the time-dependent states. Recently, by studying over-damped systems, Sivak and Crooks presented a formula that relates the generalized free energy with measurable thermodynamic works. We re-derive this relation with emphasizing a connection to an extended Clausius relation proposed in the framework of steady state thermodynamics. As a natural consequence, we generalize this relation to be valid for systems with momentum degrees of freedom, where the Shannon entropy in the generalized free energy is replaced by a symmetric one.