We propose a theory which establishes a consistency between thermodynamics and quantum pure states. In the theory, thermodynamical entropy is defined in terms of parameter that characterizes quantum pure states.
We study an ensemble of quantum pure states, the thermalization resilient ensemble (TRE), providing the statistical characterization of the thermal equilibrium of isolated quantum systems. Following a previous work where the ensemble was defined based on the invariance of the average populations upon thermal contact of identical systems, here we introduce a general methodology to generate quantum states according to the TRE statistic. The sampling is employed to characterize the ensemble distribution of thermodynamic functions like the entropy, internal energy, and temperature. The possibility of defining the temperature also for isolated quantum systems with a limited number of degrees of freedom is a distinctive feature of the TRE statistic which has no counterpart in other quantum statistical ensembles. The results are illustrated by explicit calculations for spin model systems.
A way to construct Boltzmann entropy, i.e., the entropy as a function of a microscopic pure state, for quantum field systems is proposed. Operators that shift the field in wave-vector space are used in the construction. By employing an assumption that some terms emerging due to the shift are negligible in the thermodynamic limit, it is shown that, for almost all states in the ensemble of pure states corresponding to a thermodynamic state, the value of the proposed Boltzmann entropy coincides with that of the thermodynamic entropy for the thermodynamic state. For general self-interacting fields, the Boltzmann entropy evolves with time under Hamiltonian dynamics, so that it is capable of characterizing the thermalization of isolated quantum field systems.
We investigate a thermally isolated quantum many-body system with an external control represented by a time-dependent parameter. We formulate a path integral in terms of thermal pure states and derive an effective action for trajectories in a thermodynamic state space, where the entropy appears with its conjugate variable. In particular, for quasi-static operations, the symmetry for the uniform translation of the conjugate variable emerges in the path integral. This leads to the entropy as a Noether invariant.