Hierarchical equations of motion for multiple baths (HEOM-MB) and their application to Carnot cycle

TL;DR

This paper introduces a computational method based on hierarchical equations of motion for simulating thermodynamic processes in spin systems coupled to multiple baths, including Carnot cycle analysis, highlighting non-Markovian effects and thermodynamic variables.

Contribution

The work presents a novel HEOM-based code for thermodynamic simulations of spin systems with multiple baths, enabling detailed analysis of non-Markovian effects and thermodynamic cycles.

Findings

HEOM-MB accurately reproduces equilibrium distributions and correlation functions.

Lindblad master equation is inadequate for thermodynamic descriptions.

Non-Markovian effects significantly influence thermodynamic processes.

Abstract

We have developed a computer code for the thermodynamic hierarchical equations of motion 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 extensive and intensive thermodynamic variables are calculated as physical observables, and Gibbs and Helmholtz energies are evaluated as intensive and extensive work. The energy contribution of the system--bath interaction is evaluated separately from the subsystem using the hierarchical elements of the HEOM. The accuracy of the calculated results for the equilibrium distribution and the two-body correlation functions are assessed by contrasting the results with those obtained from the time-convolution-less Redfield equation. It is shown that the Lindblad master equation is inappropriate for thermodynamic description of a spin--boson system. Non-Markovian effects in thermostatic processes are investigated by sequentially turning on and off the baths at different temperatures with different switching times and system--bath coupling. In addition, the Carnot cycle is simulated under quasi-static conditions. To analyze the work done for the subsystem in the cycle, thermodynamic work diagrams are plotted as functions of intensive and extensive variables. The C++ source codes are provided as supplementary material.