Classical and quantum thermodynamics in a non-equilibrium regime: Application to thermostatic Stirling engine.

This article presents a non-equilibrium thermodynamic theory based on a thermodynamic system–bath (SB) model, extending classical thermodynamics to fully quantum regimes where the system and bath are quantum mechanically entangled. Central to this theory is the dimensionless (DL) minimum work principle, which incorporates entropy production rates into the definition of non-equilibrium thermodynamic potentials, yielding time-derivative forms of the non-equilibrium Massieu–Planck potentials (entropic potentials) and Helmholtz–Gibbs potentials (free energies). Unlike prior approaches such as stochastic thermodynamics and fluctuation theorems, this framework does not require factorized initial conditions and remains valid under non-Markovian, non-perturbative interactions. The theory is numerically validated through simulations of thermostatic quantum and classical Stirling engines using hierarchical equations of motion and quantum/classical Fokker–Planck equations, demonstrating that non-equilibrium thermodynamic processes can be analyzed via work diagrams analogous to equilibrium cases. These results provide a systematic methodology to evaluate and optimize thermodynamic potentials and entropy production in non-equilibrium quantum systems, with potential applications in designing efficient heat engines beyond traditional limits.