  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.
