  We develop a thermodynamic theory in the non-equilibrium regime by extending
a theory based on the dimensionless (DL) minimum work principle previously
developed for a thermodynamic system-bath model [S.Koyanagi and
Y.Tanimura,J.Chem.Phys.160,(2024)]. Our results are described by
non-equilibrium thermodynamic potentials expressed in time-derivative form in
terms of extensive and intensive variables. This is made possible through the
incorporation of waste heat, which is equivalent to the loss work consumed by
the bath, into the definitions of thermodynamic potentials in a non-equilibrium
regime. These potentials can be evaluated from the DL
non-equilibrium-to-equilibrium minimum work principle, which is derived from
the principle of DL minimum work and is equivalent to the second law of
thermodynamics. We thus obtain the non-equilibrium Massieu-Planck potentials as
entropic potentials and the non-equilibrium Helmholtz-Gibbs potentials as free
energies. Our results are verified numerically by simulating a Stirling engine
consisting of two isothermal and two thermostatic processes using the quantum
Fokker-Planck equations and the classical Kramers equation derived from the
thermodynamic system-bath model. We then show that any thermodynamic process
can be analyzed using a non-equilibrium work diagram analogous to the
equilibrium one for given time-dependent intensive variables. The results can
be used to develop efficient heat machines in non-equilibrium regimes.
