  In the study of thermodynamics for nanoscale quantum systems, a family of
quantities known as generalized free energies have been derived as necessary
and sufficient conditions that govern state transitions. These free energies
become important especially in the regime where the system of interest consists
of only a few (quantum) particles. In this work, we introduce a new family of
smoothed generalized free energies, by constructing explicit smoothing
procedures that maximize/minimize the free energies over an $ \varepsilon$-ball
of quantum states. In contrast to previously known smoothed free energies,
these quantities now allow us to make an operational statement for approximate
thermodynamic state transitions. We show that these newly defined smoothed
quantities converge to the standard free energy in the thermodynamic limit.
