Statistical mechanics of Hamiltonian systems with bounded kinetic terms: an insight into negative temperature

Some physical systems are characterized by regimes in which entropy is a decreasing function of the internal energy, meaning that they can achieve “negative absolute temperature”. Such states have been experimentally realized in various contexts, from two-dimensional hydrodynamics to nuclear spins and Bose condensates, and many important theoretical results are available. The usage of negative values of the temperature, however, is not universally accepted, and during the last years a stimulating debate about the possibility to include this formalism in the framework of Statistical Mechanics has been attracting the attention of many authors. Motivated by the open questions on this topic, in the present Thesis we study a class of Hamiltonian systems characterized by bounded kinetic terms; these models achieve negative temperature in their high-energy regimes, therefore they provide a preferential tool for the analytical and numerical investigation of such states. First we characterize the equilibrium properties of these models. We discuss the possibility to achieve thermalization between systems at negative temperature, the consequent validity of a Zeroth Principle of Thermodynamics also for these states, and the non-trivial case of long-range interactions, inducing inequivalence between statistical ensembles. Then, aiming at a consistent generalization of Einstein rela- tion and Langevin formalism to cases with negative temperature, we address the problem of Brownian motion of slow particles characterized by bounded kinetic terms, coupled to suitable thermal baths. Classical topics of out-of-equilibrium Statistical Mechanics, such as response theory and Fourier transport, are also con- sidered for this particular class of Hamiltonian systems. The aim of this project is to show that negative temperatures give a consistent description of the statistical properties of the chosen class of mechanical models; moreover, the introduction of this formalism is shown to be a necessary condition for the extension of usual results of Statistical Mechanics to these systems. This systematic study is expected to provide a useful analogy for more realistic phys- ical models, and a deeper understanding of the fundamental aspects of negative temperature.