Statistical mechanics of systems with negative temperature

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 a half exhibit positive temperature, because their entropy is nondecreasing function energy. Since more than ago it has been realized may not be the case for some physical systems as incompressible fluids, nuclear magnetic chains, lasers, cold atoms optical waveguides. review these examples discuss peculiar thermodynamic properties, which have associated presence regimes, characterized temperatures. As reported in review, ambiguity inherent definition recurrently raised harsh debate about possibility considering temperature states genuine equilibrium ones. Here show are consistent with well non-equilibrium thermodynamics. In particular, thermometry, thermodynamics cyclic transformations, ensemble equivalence, fluctuation-dissipation relations, response theory even transport processes reformulated include them, thus dissipating any prejudice exceptionality, typically presumed manifestation transient metastable effects.