Comment on " Negative Heat Capacity for a Cluster of 147 Sodium Atoms "

In a recent letter [1], Schmidt et al. analyze a region where the specific heat of a Na + 147 cluster becomes negative. In Boltzmann-Gibbs (BG) statistical mechanics the specific heat can never be negative in the canonical ensemble. However, as the authors are well aware, no such property is valid in general for the microcanonical ensemble. This is indeed the case of a variety of physical situations ranging from self-gravitating systems to melting atomic clusters and fragmenting nuclei (see [1] and references therein). An important feature that all such systems have in common is thermodynamic nonextensivity, reflected, as also mentioned in [1], in the fact that the energy associated with N particles is not proportional to N. Nonextensivity occurs everytime the range of the interactions is not negligible compared to the (linear) size of the system. Two important physical examples are: – short-range interactions in small systems – long-range interactions in systems of any size [2–4]. Now, if the system is isolated, nonextensive, in an equilibrium-like state, and exhibits, for some energy range, a negative specific heat, then not one but at least two possibilities must be considered: 1) the system is in its thermodynamically stable state and then the BG microcanonical ensemble is to be used, as preconised in [5] and done in [1]. 2) the system is in some kind of dynamic metastable state, and then some other thermo-statistical approach might well be necessary. One such alternative description is nonextensive statistical mechanics [3], which has proved to be useful in a variety of complex situations (e.g., fully developed turbulence [6], electron-positron annihilation producing hadronic jets [7], motion of Hydra viridissima [8]). The metaequilibrium state to which we make reference as a second possibility has indeed been observed in magnetic-like Hamiltonians such as arrays of long-range coupled planar rotators (the so called HMF [9] and its generalization, the α − XY model [10]), among 1