Biparametric complexities and the generalized Planck radiation law

TL;DR

This paper introduces biparametric complexity measures for continuous probability distributions, applies them to blackbody radiation in various dimensions, and reveals their universal, dimension-dependent behavior related to quantum effects and spectral properties.

Contribution

It defines new biparametric complexity measures and demonstrates their universal, dimension-dependent properties in blackbody radiation, linking complexity to spectral and quantum characteristics.

Findings

Complexity measures are independent of temperature and physical constants.

They depend only on spatial dimensionality, showing universal behavior.

Complexity quantifiers exhibit temperature-dependent behavior similar to Wien's law.

Abstract

Complexity theory embodies some of the hardest, most fundamental and most challenging open problems in modern science. The very term complexity is very elusive, so that the main goal of this theory is to find meaningful quantifiers for it. In fact we need various measures to take into account the multiple facets of this term. Here some biparametric Cr\'amer-Rao and Heisenberg-R\'enyi measures of complexity of continuous probability distributions are defined and discussed. Then, they are applied to the blackbody radiation at temperature T in a d-dimensional universe. It is found that these dimensionless quantities do not depend on T nor on any physical constants. So, they have an universal character in the sense that they only depend on the spatial dimensionality. To determine these complexity quantifiers we have calculated their dispersion (typical deviations) and entropy (R\'enyi entropies and the generalized Fisher information) constituents. They are found to have a temperature-dependent behavior similar to the celebrated Wien's displacement law of the dominant frequency $\nu_{max}$ at which the spectrum reaches its maximum. Moreover, they allow us to gain insights into new aspects of the d-dimensional blackbody spectrum and about the quantification of quantum effects associated with space dimensionality.