A consistent description of fluctuations requires negative temperatures

We review two definitions of temperature in statistical mechanics, ${{T}_{\text{B}}}$ and ${{T}_{\text{G}}}$ , corresponding to two possible definitions of entropy, ${{S}_{\text{B}}}$ and ${{S}_{\text{G}}}$ , 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 ${{S}_{\text{B}}}$ , 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 $N\sim 100$ is sufficient) and without long-range interactions. We first discuss the basic role of ${{T}_{\text{B}}}$ , even when negative, as the parameter describing fluctuations of observables in a sub-system. Then, we focus on how ${{T}_{\text{B}}}$ can be measured dynamically, i.e. averaging over a single long experimental trajectory. The same approach cannot be used in a generic system for ${{T}_{\text{G}}}$ , 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 ${{S}_{\text{B}}}$ , or equivalently at small $|{{T}_{\text{B}}}|$ , depends on the sign of ${{T}_{\text{B}}}$ .