  In nonadditive systems, like small systems or like long-range interacting
systems even in the thermodynamic limit, ensemble inequivalence can be related
to the occurrence of negative response functions, this in turn being connected
with anomalous concavity properties of the thermodynamic potentials associated
to the various ensembles. We show how the type and number of negative response
functions depend on which of the quantities E, V and N (energy, volume and
number of particles) are constrained in the ensemble. In particular, we
consider the unconstrained ensemble in which E, V and N fluctuate, physically
meaningful only for nonadditive systems. In fact, its partition function is
associated to the replica energy, a thermodynamic function that identically
vanishes when additivity holds, but that contains relevant information in
nonadditive systems.
