  It has been proved for a class of mean-field and long-range systems that the
concavity of the thermodynamic entropy determines whether the microcanonical
and canonical ensembles are equivalent at the level of their equilibrium
states, i.e., whether they give rise to the same equilibrium states. Here we
show that this correspondence is actually a general result of statistical
mechanics: it holds for any many-body system for which equilibrium states can
be defined and in principle calculated. The same correspondence applies for
other dual statistical ensembles, such as the canonical and grand-canonical
ensembles.
