Sakamoto Souichi, Tanimura Yoshitaka
Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan.
J Chem Phys. 2020 Dec 21;153(23):234107. doi: 10.1063/5.0033664.
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.
我们提出了一种方案，利用数值“精确”的运动层次方程（HEOM）中的准静态亥姆霍兹能量，来评估在随时间变化的外力作用下与热浴耦合的系统的热力学变量。我们计算了在不同温度下，与非马尔可夫热浴强耦合的自旋系统产生的熵。我们表明，当外部扰动变化足够缓慢时，系统总是能达到热平衡。因此，我们基于HEOM计算了等温过程的玻尔兹曼熵和冯·诺依曼熵，以及处于准静态平衡的系统的各种热力学变量，如内能、热量和功的变化。我们发现，尽管玻尔兹曼和冯·诺依曼情况下系统熵随系统 - 热浴耦合强度的特征相似，但总熵产生的特征却完全不同。玻尔兹曼情况下的总熵产生始终为正，而在冯·诺依曼情况下，如果我们选择整个系统的热平衡态（非因式分解的热平衡态）作为初始态，总熵产生会变为负。这是因为冯·诺依曼情况下的总熵产生没有恰当地考虑系统 - 热浴相互作用的熵贡献。因此，在完全量子领域研究熵产生时必须使用玻尔兹曼熵。最后，我们检验了雅津斯基等式的适用性。
