A consistent description of fluctuations requires negative temperatures. (10th December 2015) - Cerino, Luca; Puglisi, Andrea; Vulpiani, Angelo
Abstract: We review two definitions of temperature in statistical mechanics, and, corresponding to two possible definitions of entropy, and, known as surface and volume entropy respectively. We limit our attention to a class of systems with bounded energy, and such that the second derivative of, with respect to energy, is always negative. The second condition holds in systems where the number N of degrees of freedom is sufficiently large (examples are shown where is sufficient) and without long-range interactions. We first discuss the basic role of, even when negative, as the parameter describing fluctuations of observables in a sub-system. Then, we focus on how can be measured dynamically, i.e. averaging over a single long experimental trajectory. The same approach cannot be used in a generic system for, since the equipartition theorem may be impaired by boundary effects due to the limited energy. These general results are substantiated by the numerical study of a Hamiltonian model of interacting rotators with bounded kinetic energy. The numerical results confirm that the kind of configurational order realized in the regions at small, or equivalently at small, depends on the sign of .