Equivalence of ensembles in quantum lattice systems: States

The general analysis of the equivalence of ensembles in quantum lattice systems, which was undertaken in paper I of this series, is continued.The properties of equilibrium states are considered in a variational sense. It is then shown that there exists a canonical as well as a microcanonical variational formulation of equilibrium both of which are equivalent to the grandcanonical formulation.Equilibrium states are constructed both in the canonical and in the microcanonical formalism by means of suitable limiting procedures.It is shown, in particular, that the invariant equilibrium states for a given energy and density are those for which the maximum of the mean entropy is reached. The mean entropy thus obtained coincides with the microcanonical entropy.