Numerically “exact” simulations of entropy production in the fully quantum regime: Boltzmann entropy vs von Neumann entropy

Abstract

We present a scheme to evaluate thermodynamic variables for a system coupled to a heat bath under a time-dependent external force using the quasi-static Helmholtz energy from the numerically "exact" hierarchical equations of motion (HEOM). We computed the entropy produced by a spin system strongly coupled to a non-Markovian heat bath for various temperatures. We showed that when changes to the external perturbation occurred sufficiently slowly, the system always reached thermal equilibrium. Thus, we calculated the Boltzmann entropy and the von Neumann entropy for an isothermal process, as well as various thermodynamic variables, such as changes in internal energies, heat, and work, for a system in quasi-static equilibrium based on the HEOM. We found that although the characteristic features of the system entropies in the Boltzmann and von Neumann cases as a function of the system-bath coupling strength are similar, those for the total entropy production are completely different. The total entropy production in the Boltzmann case is always positive, whereas that in the von Neumann case becomes negative if we chose a thermal equilibrium state of the total system (an unfactorized thermal equilibrium state) as the initial state. This is because the total entropy production in the von Neumann case does not properly take into account the contribution of the entropy from the system-bath interaction. Thus, the Boltzmann entropy must be used to investigate entropy production in the fully quantum regime. Finally, we examined the applicability of the Jarzynski equality.

Open quantum dynamics theory for a complex subenvironment system with a quantum thermostat: Application to a spin heat bath

“…In terms of the vectors, |Φ(t) = e Λut |Φ and |Ψ(t) = e Λ † l t |Φ , we derive the modified HSEOM, in which the computation time is linear with t, as Eqs. ( 6) and (7). Figure 12 displays the real part of the three-body correlation function D 777 (t, 10) for (a) a TLS + spin lattice + QT ( /ω 0 = 1, ζ = 0.01, β ω 0 → ∞, blue), (b) a TLS + spin lattice ( /ω 0 = 1, ζ = 0, red), and (c) an isolated spin lattice ( /ω 0 = 0, ζ = 0, green).…”

“…[2][3][4][5][6] The heat bath is considered to be an unlimited heat source with infinite heat capacity. 7,8 However, the thermal noise from complex or frustrated materials is typically non-Gaussian, and its temporal and spatial correlations are non-uniform, although the ensemble average of the noise has a Gaussian profile when the strength of each source of noise is comparable, due to the central limit theorem. 9 The noise correlation is characterized by a simple function, for example, a stretched exponential function.…”

Autobiography of Yoshitaka Tanimura

“…The distinctive feature of the HEOM approach is that it can rigorously evaluate changes in heat and entropy not only for the system but also for the bath and the SB interaction (117). We thus showed that quantum thermodynamics can be described in the framework of statistical mechanics, even in the nonequilibrium case, by regarding nonequilibrium work as a change in quasi-static free energy (131,133).…”

Hierarchical Equations-of-Motion Method for Momentum System–Bath Coupling

“…These equations could be of interest for exploring quantum thermodynamics in the case of strong system–bath coupling. 77 …”