  Do negative absolute temperatures matter physics and specifically Statistical
Physics? We provide evidence that we can certainly answer positively to this
vexata quaestio. The great majority of models investigated by statistical
mechanics over almost one century and a half exhibit positive absolute
temperature, because their entropy is a nondecreasing function of energy. Since
more than half a century ago it has been realized that this may not be the case
for some physical systems as incompressible fluids, nuclear magnetic chains,
lasers, cold atoms and optical waveguides. We review these examples and discuss
their peculiar thermodynamic properties, which have been associated to the
presence of thermodynamic regimes, characterized by negative absolute
temperatures. As reported in this review, the ambiguity inherent the definition
of entropy has recurrently raised a harsh debate about the possibility of
considering negative temperature states as genuine thermodynamic equilibrium
ones. Here we show that negative absolute temperatures are consistent with
equilibrium as well as with non-equilibrium thermodynamics. In particular,
thermometry, thermodynamics of cyclic transformations, ensemble equivalence,
fluctuation-dissipation relations, response theory and even transport processes
can be reformulated to include them, thus dissipating any prejudice about their
exceptionality, typically presumed as a manifestation of transient metastable
effects.
