  Ensemble inequivalence has been observed in several systems. In particular it
has been recently shown that negative specific heat can arise in the
microcanonical ensemble in the thermodynamic limit for systems with long-range
interactions. We display a connection between such behaviour and a mean-field
like structure of the partition function. Since short-range models cannot
display this kind of behaviour, this strongly suggests that such systems are
necessarily non-mean field in the sense indicated here. We further show that a
broad class of systems with non-integrable interactions are indeed of
mean-field type in the sense specified, so that they are expected to display
ensemble inequivalence as well as the peculiar behaviour described above in the
microcanonical ensemble.
