Generalised asymptotic classes for additive and non-additive entropies

Abstract

In the field of non-extensive statistical mechanics it is common to focus more attention on the family of parameter-dependent entropies rather than on those strictly depending only on the probability in which case there is no need to adjust a specific parameter. In particular, there exist two non-parametric entropy measures, , that resemble the Boltzmann-Gibbs (BG) entropy, SB, in the thermodynamic limit, whereas a difference between them arises inasmuch as the statistical system possesses a small number of microstates. The difference, although slight, accounts for meaningful physical consequences such as effective forces and inner interactions among constituents. Yet, in this letter we are to report some of the analytical attributes associated to entropies via the formulation introduced by Hanel and Thurner. These two functionals allow to construct a generalised classification of entropy measures in terms of their defining equivalence classes, which are determined by a pair of scaling exponents (c, d). As a result, it has been identified that and SB belong to the same asymptotic, equivalence class. The latter is an interesting fact since it does not occur for non-logarithmic, parameter-dependent entropies. Following this scheme, we also briefly discuss the features of the Sharma-Mittal, Rényi and Tsallis entropies in the asymptotic limit.