Universality classes for the Fisher metric derived from relative group entropy

Abstract

We consider the Fisher metric which results from the Hessian of the relative group entropy, that we call group Fisher metric. In particular, the metrics corresponding to the Boltzmann–Gibbs, Tsallis, Kaniadakis and Abe universality classes are obtained. We prove that the scalar curvature derived from the group Fisher metric results in a multiple of the Boltzmann–Gibbs one, with the factor of proportionality given by the local properties of the group entropy. We analyze, for the Tsallis universality class, the 2D correlated model that presents a softening and strengthening of the scalar curvature, and we illustrate with the canonical ensemble of a pair of interacting harmonic oscillators as well as a quartic harmonic oscillator.