  We present a scheme to evaluate thermodynamic variables for a system coupled
to a heat bath under a time-dependent external force using the quasi-static
Helmholtz energy from the numerically "exact" hierarchical equations of motion
(HEOM). We computed the entropy produced by a spin system strongly coupled to a
non-Markovian heat bath for various temperatures. We showed that when changes
to the external perturbation occurred sufficiently slowly, the system always
reached thermal equilibrium. Thus, we calculated the Boltzmann entropy and the
von Neumann entropy for an isothermal process, as well as various thermodynamic
variables, such as changes of internal energies, heat, and work, for a system
in quasi-static equilibrium based on the HEOM. We found that, although the
characteristic features of the system entropies in the Boltzmann and von
Neumann cases as a function of the system--bath coupling strength are similar,
those for the total entropy production are completely different. The total
entropy production in the Boltzmann case is always positive, whereas that in
the von Neumann case becomes negative if we chose a thermal equilibrium state
of the total system (an unfactorized thermal equilibrium state) as the initial
state. This is because the total entropy production in the von Neumann case
does not properly take into account the contribution of the entropy from the
system--bath interaction. Thus, the Boltzmann entropy must be used to
investigate entropy production in the fully quantum regime. Finally, we
examined the applicability of the Jarzynski equality.
