Thermodynamic quantum Fokker–Planck equations and their application to thermostatic Stirling engine.

The article focuses on the development and numerical implementation of thermodynamic quantum Fokker–Planck equations (T-QFPE) for simulating quantum and classical thermodynamic processes in anharmonic Brownian systems coupled to multiple Ohmic heat baths at varying temperatures. The T-QFPE extend low-temperature quantum Fokker–Planck equations by incorporating time-dependent Matsubara frequencies to model thermostatic processes, enabling the evaluation of thermodynamic intensive and extensive variables and potentials. Numerical validation against analytical solutions of the Brownian oscillator demonstrates the accuracy of the approach and reveals limitations of Markovian quantum master equations in fully quantum regimes due to bath entanglement effects. As a demonstration, the authors simulate a thermostatic Stirling engine under quasi-static conditions, showing that in the quantum case, work done by external fields decreases with increasing system–bath coupling strength, unlike the classical case. The provided C++ codes utilize Open Multi-Processing and CUDA for efficient computation and are supplied as supplementary material to facilitate further research in quantum thermodynamics.