  In systems with long-range interactions, since energy is a non-additive
quantity, ensemble inequivalence can arise: it is possible that different
statistical ensembles lead to different equilibrium descriptions, even in the
thermodynamic limit. The microcanonical ensemble should be considered the
physically correct equilibrium distribution as long as the system is isolated.
The canonical ensemble, on the other hand, can always be defined
mathematically, but it is quite natural to wonder to which physical situations
it does correspond. We show numerically and, in some cases, analytically, that
the equilibrium properties of a generalized Hamiltonian mean-field model in
which ensemble inequivalence is present are correctly described by the
canonical distribution in (at least) two different scenarios: a) when the
system is coupled via local interactions to a large reservoir (even if the
reservoir shows, in turn, ensemble inequivalence) and b) when the mean-field
interaction between a small part of a system and the rest of it is weakened by
some kind of screening.
