Scale invariance and related properties of q-Gaussian systems

TL;DR

This paper explores the scale-invariance properties of q-Gaussian systems, extending Gaussian invariance principles to q-Gaussians, and discusses implications for system estimation and kinetic applications.

Contribution

It extends invariance principles from Gaussian to q-Gaussian systems using elliptic invariance, providing new insights into their structural properties.

Findings

q-Gaussian systems exhibit elliptic invariance similar to Gaussian systems.

The invariance properties help in estimating the q parameter.

Application to kinetic theory demonstrates practical relevance.

Abstract

We advance scale-invariance arguments for systems that are governed (or approximated) by a $q-$Gaussian distribution, i.e., a power law distribution with exponent $Q=1/(1-q); q \in \mathbb{R}$. The ensuing line of reasoning is then compared with that applying for Gaussian distributions, with emphasis on dimensional considerations. In particular, a Gaussian system may be part of a larger system that is not Gaussian, but, if the larger system is spherically invariant, then it is necessarily Gaussian again. We show that this result extends to q-Gaussian systems via elliptic invariance. The problem of estimating the appropriate value for $q$ is revisited. A kinetic application is also provided.