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.

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We theoretically investigate an exciton-coupled electron transfer (XCET) process that is conversion of an exciton into a charge transfer state. This conversion happens in an exciton transfer (XT) process, and the electron moves away in an electron transfer (ET) process in multiple environments (baths). This XCET process plays an essential role in the harvesting of solar energy in biological and photovoltaic materials. We develop a practical theoretical model to study the efficiency of the XCET process that occurs either in consecutive or concerted processes under the influence of non-Markovian baths. The role of quantum coherence in the XT-ET system and the baths is investigated using reduced hierarchal equations of motion (HEOM). This model includes independent baths for each XT and ET state, in addition to a XCET bath for the conversion process. We found that, while quantum system-bath coherence is important in the XT and ET processes, coherence between the XT and ET processes must be suppressed in order to realize that an efficient irreversible XCET process through the weak off-diagonal interaction between the XT and ET bridge sites arises from an XCET bath.

The influence of Ta addition on the solidification microstructure, solute distribution and hardness after quenching and tempering treatments was investigated for a high-carbon high-speed steel type cast alloy (Fe1.9%C0.5%Mn4.9%Cr5.0%Mo5.07.2%V0.41.4%Ta). The compositions of V and Ta were systematically changed to improve the distribution of hard MC carbides in the hypoeutectic range. Electron probe micro-analysis (EPMA) and X-ray diffraction (XRD) identified an oval microstructure as MC carbides containing mainly V and Ta, and a lamellar structure as M 2 C carbides containing mainly Fe and Mo among the austenite (£) dendrites. Redistribution of alloying elements during the solidification sequence of primary £, £ + MC and M 2 C eutectic structure could be calculated from the Scheil-Gulliver equation and the initial composition. The macro-hardness of the quenched specimens gradually increased with increasing quenching temperature until a maximum was reached. This indicates that macro-hardness of the quenched specimens depends on both the amount and hardness of martensite matrix. All specimens which were tempered at 723873 K showed secondary hardening. Furthermore, hardening of the specimens was most apparent when specimens containing large amounts of retained £ were tempered at an optimum temperature. For example, the hardness of specimens with added Ta increased to around 900 HV after tempering at 823 K. These results suggest that the macro-hardness of tempered specimens is governed by the maximum amount of carbon in the £ matrix at quenching temperature, the degree of transformation from retained £ to martensite, and the precipitation and distribution of secondary carbides.