  We present general and rigorous results showing that the microcanonical and
canonical ensembles are equivalent at all three levels of description
considered in statistical mechanics - namely, thermodynamics, equilibrium
macrostates, and microstate measures - whenever the microcanonical entropy is
concave as a function of the energy density in the thermodynamic limit. This is
proved for any classical many-particle systems for which thermodynamic
functions and equilibrium macrostates exist and are defined via large deviation
principles, generalizing many previous results obtained for specific classes of
systems and observables. Similar results hold for other dual ensembles, such as
the canonical and grand-canonical ensembles, in addition to path ensembles used
for describing nonequilibrium systems driven in steady states.
