Nonequilibrium representative ensembles for isolated quantum systems

TL;DR

This paper develops a framework for describing the dynamics of isolated quantum systems out of equilibrium using representative ensembles, highlighting the role of a grand Hamiltonian and the concept of quasi-stationary states.

Contribution

It introduces a theorem relating field operator commutators to variational derivatives and establishes the equivalence of variational and Heisenberg equations, advancing the understanding of nonequilibrium quantum dynamics.

Findings

Finite quantum systems do not fully equilibrate but approach quasi-stationary states.

Microcanonical ensemble is a special case of representative ensembles.

Principle of minimal information defines quasi-stationary ensembles.

Abstract

An isolated quantum system is considered, prepared in a nonequilibrium initial state. In order to uniquely define the system dynamics, one has to construct a representative statistical ensemble. From the principle of least action it follows that the role of the evolution generator is played by a grand Hamiltonian, but not merely by its energy part. A theorem is proved expressing the commutators of field operators with operator products through variational derivatives of these products. A consequence of this theorem is the equivalence of the variational equations for field operators with the Heisenberg equations for the latter. A finite quantum system cannot equilibrate in the strict sense. But it can tend to a quasi-stationary state characterized by ergodic averages and the appropriate representative ensemble depending on initial conditions. Microcanonical ensemble, arising in the eigenstate thermalization, is just a particular case of representative ensembles. Quasi-stationary representative ensembles are defined by the principle of minimal information. The latter also implies the minimization of an effective thermodynamic potential.