A consistent description of fluctuations requires negative temperatures

Abstract

We review two definitions of temperature in statistical mechanics, TB and TG, corresponding to two possible definitions of entropy, SB and SG, known as surface and volume entropy respectively. We restrict our attention to a class of systems with bounded energy and such that the second derivative of SB with respect to energy is always negative: the second request is quite natural and holds in systems of obvious relevance, i.e. with a number N of degrees of freedom sufficiently large (examples are shown where N ∼ 100 is sufficient) and without long-range interactions. We first discuss the basic role of TB, even when negative, as the parameter describing fluctuations of observables in a sub-system. Then, we focus on how TB can be measured dynamically, i.e. averaging over a single long experimental trajectory. On the contrary, the same approach cannot be used in a generic system for TG, since the equipartition theorem may be spoiled 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 SB, or equivalently at small |TB|, depends on the sign of TB.

On the dispute between Boltzmann and Gibbs entropy

“…These papers have engendered a glowing debate between supporters of the Gibbs entropy [23,24,25,26,22,20,28,29,21,34] and those considering correct the Boltzmann entropy [31,32,30,41,42,43,44,45,34,46].…”

Derivation of a Langevin equation in a system with multiple scales: The case of negative temperatures

“…[10], where cold atoms in an optical lattice display both positive and negative temperatures. It has also been studied theoretically, for instance see [11,21]. For the microscopic dynamics, the spins are started from a completely random configuration.…”

Langevin equation in systems with also negative temperatures

“…In the model for negative temperatures discussed in Section 3 [18], one has K(P ) = 1 − cos(P ) and therefore Γ(P ) = −D P β sin(P ), which let β have any possible sign. It is interesting to notice that the "drift" term Γ(P ) acts consistently with the simple idea deduced from the form of f P (P ): the drift term should counteract the spreading action of the noise term in order to concentrate the distribution in its maximum.…”

“…of Eqs. (18) are computed as functions of both j and ∆t, and the limit ∆t → 0 can be inferred. Let us note that the mathematical limit ∆t → 0 in the definitions of drift and diffusion coefficients must be interpreted in a proper physical way.…”