  Thermodynamic probes can be used to deduce microscopic internal dynamics of
nanoscale quantum systems. Several direct entropy measurement protocols based
on charge transport measurements have been proposed and experimentally applied
to single-electron devices. To date, these methods have relied on
(quasi-)equilibrium conditions between the nanoscale quantum system and its
environment, which constitutes only a small subset of the experimental
conditions available. In this paper, we establish a thermodynamic analysis
method based on stochastic thermodynamics, that is valid far from equilibrium
conditions, is applicable to a broad range of single-electron devices and
allows us to find the difference in entropy between the charge states of the
nanodevice, as well as a characteristic of any selection rules governing
electron transfers. We apply this non-equilibrium entropy measurement protocol
to a single-molecule device in which the internal dynamics can be described by
a two-site Hubbard model.
