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Systems at "negative" absolute temperatures (NAT), i.e. systems whose entropy decreases when energy increases, can be found in several research fields: important examples are two-dimensional vortices, nuclear spins and cold atoms. Far from being mere curiosities, they show rather interesting statistical properties, and it is not completely clear, a priori, whether the usual results of classical Statistical Mechanics can be extended to them: a stimulating debate on these topics is still ongoing.
Here we are interested in a class of Hamiltonians with bounded kinetic terms, which can achieve NAT and can be studied through analytical computations and numerical simulations. Our aim is to get a better insight into the properties of NAT states: in particular, results such as the Zero Principle, the (generalized) Maxwell-Boltzmann distribution for the momenta, the (in)equivalence of statistical ensembles and the Fluctuation-Dissipation relation are reviewed in this framework. In addition, we show that a generalized Langevin equation for the considered class of systems can be found using different approaches, and we check its validity through numerical simulations.