Open Quantum Dynamics Theory for Non-Equilibrium Work: Hierarchical Equations of Motion Approach

A system--bath (SB) model is considered to examine the Jarzynski equality in the fully quantum regime. In our previous paper [J. Chem. Phys. 153 (2020) 234107], we carried out "exact" numerical experiments using hierarchical equations of motion (HEOM) in which we demonstrated that the SB model describes behavior that is consistent with the first and second laws of thermodynamics and that the dynamics of the total system are time irreversible. The distinctive quantity in the Jarzynski equality is the "work characteristic function (WCF)", $\langle \exp[-\beta W] \rangle$, where $W$ is the work performed on the system and $\beta$ is the inverse temperature. In the present investigation, we consider the definitions based on the partition function (PF) and on the path, and numerically evaluate the WCF using the HEOM to determine a method for extending the Jarzynski equality to the fully quantum regime. We show that using the PF-based definition of the WCF, we obtain a result that is entirely inconsistent with the Jarzynski equality, while if we use the path-based definition, we obtain a result that approximates the Jarzynski equality, but may not be consistent with it.