The Transformation-Groupoid Structure of the q-Gaussian Family

TL;DR

This paper develops a comprehensive mathematical framework for transforming q-Gaussian distributions, showing that these transformations form a groupoid, which enhances understanding of their relationships in complex systems.

Contribution

It derives a general map mma_{qq'} that transforms q-Gaussians within a broad parameter range, completing the theory of their interrelations.

Findings

The mma_{qq'} map enables transformation between any two q-Gaussians with q, q' in [-, 3).

The set of q-Gaussian transformations forms a transformation groupoid.

The results unify different q-Gaussian distributions under a common mathematical structure.

Abstract

The q-Gaussian function emerges naturally in various applications of statistical mechanics of non-ergodic and complex systems. In particular it was shown that in the theory of binary processes with correlations, the q-Gaussian can appear as a limiting distribution. Further, there exist several problems and situations where, depending on procedural or algorithmic details of data-processing, q-Gaussian distributions may yield distinct values of q, where one value is larger, the other smaller than one. To relate such pairs of q-Gaussians it would be convenient to map such distributions onto one another, ideally in a way, that any value of q can be mapped uniquely to any other value q'. So far a (duality) map from q -> q'=(7-5q)/(5-3q) was found, mapping q from the interval q\in [-\infty, 1] -> q'\in [1, 5/3]. Here we complete the theory of transformations of q-Gaussians by deriving a general map \gamma_{qq'}, that transforms normalizable q-Gaussian distributions onto one another for which q and q' are in the range of [1,3). By combining this with the previous result, a mapping from any value of q \in [-\infty,3) is possible to any other value q'\in [-\infty,3). We show that the action of \gamma_{qq'} on the set of q-Gaussian distributions is a transformation groupoid.