Approach to Equilibrium of Quantum Mechanical Gibbsian Ensembles

Abstract

In this paper we study the approach to equilibrium of an arbitrary physical system in contact with a thermal reservoir when the internal dynamics of the system is described by its Hamiltonian and the interaction with the reservoir can be described by a linear stochastic kernel which in general can be time dependent. We find a generalization of the Markov process represented by the master equation to the space of density matrices and we prove that the necessary and sufficient condition that any solution of an equation of our general structure approach equilibrium monotonically is that equilibrium is a stationary solution. We prove that the Helmholtz free energy is a minimum as a functional of the density matrix and thus is a suitable measure of deviation from canonicity. We also find another measure of deviation which is equally good and easier to use. We show, as an example, that Bloch's microscopic density matrices fulfill our criteria and thus approach equilibrium monotonically. Finally, we extend our results for the thermal reservoir to more general reservoirs.