The biparametric Fisher-R\'enyi complexity measure and its application to the multidimensional blackbody radiation

Abstract

We introduce a biparametric Fisher-R\'enyi complexity measure for general probability distributions and we discuss its properties. This notion, which is composed of two entropy-like components (the R\'enyi entropy and the biparametric Fisher information), generalizes the basic Fisher-Shannon measure and the previous complexity quantifiers of Fisher-R\'enyi type. Then, we illustrate the usefulness of this notion by carrying out a information-theoretical analysis of the spectral energy density of a $d$-dimensional blackbody at temperature $T$. It is shown that the biparametric Fisher-R\'enyi measure of this quantum system has a universal character in the sense that it does not depend on temperature nor on any physical constant (e.g., Planck constant, speed of light, Boltzmann constant), but only on the space dimensionality $d$. Moreover, it decreases when $d$ is increasing, but exhibits a non trivial behavior for a fixed $d$ and a varying parameter, which somehow brings up a non standard structure of the blackbody $d$-dimensional density distribution.