Thermodynamic hierarchical equations of motion and their application to Carnot engine

Abstract

We developed a computer code for the thermodynamic hierarchical equations of motion (T-HEOM) derived from a spin subsystem coupled to multiple Drude baths at different temperatures, which are connected to or disconnected from the subsystem as a function of time. The code can simulate the reduced dynamics of the subsystem under isothermal, isentropic, thermostatic, and entropic conditions. The thermodynamic extensive and intensive variables were calculated as physical observables, and the Gibbs and Helmholtz energies were evaluated as intensive and extensive work. The contribution of energies from the system-bath interaction was evaluated separately from the subsystem using the hierarchical elements of T-HEOM. The accuracy of the calculated results for the equilibrium distribution and two-body correlation functions of the subsystem was verified by comparison with the results obtained from the time-convolution-less Redfield equation. Non-Markovian effects in thermostatic processes were investigated by sequentially turning on and off the baths of different temperatures with different switching times and system-bath coupling. As a demonstration, a comparison was made by simulating the case where the temperature of one bath was varied over time and the case where similar temperature changes were achieved by turning on and off the baths at different temperatures. In addition, the Carnot engine was simulated under quasi-static conditions. To analyze the work done for the subsystem in the cycle, thermodynamic work diagrams were plotted as functions of intensive and extensive variables. The C++ source codes are provided as supplementary material.