Temperature-dependent pure state for the thermodynamic system and its heat reservoir

Based on the quantum Liouville and Lindblad equations, a temperature-dependent pure state was constructed for the composite isolated system consisting of a thermodynamic system and its heat reservoir. The results demonstrate that the expectation value of this pure state coincides with the statistical average of the thermodynamic system’s mixed state. Furthermore, the existence of this pure state is consistent with the eigenstate thermalization hypothesis. Additionally, the “fictitious system” introduced in thermo field dynamics can be naturally interpreted as the heat reservoir of the system, and this interpretation allows us to re-examine the relationship between the physical system and the so-called “fictitious system.”