The $q$-canonical ensemble as a consequence of Bayesian superstatistics

TL;DR

This paper demonstrates that the $q$-canonical ensemble arises naturally from a Bayesian superstatistics framework as the maximum Shannon-Jaynes entropy distribution under noninformative constraints, without relying on specific physical assumptions.

Contribution

It provides an information-theoretic explanation for the $q$-canonical ensemble as a maximum entropy distribution within Bayesian superstatistics, independent of system-specific physics.

Findings

$q$-canonical distributions maximize Shannon-Jaynes entropy under noninformative constraints.

The mathematical structure of superstatistics uniquely identifies the $q$-canonical form.

Results support the view that $q$-canonical ensembles are fundamentally information-theoretic in origin.

Abstract

Superstatistics is a generalization of equilibrium statistical mechanics that describes systems in nonequilibrium steady states. Among the possible superstatistical distributions, the $q$-canonical ensemble (also known as Tsallis' statistics, and in plasma physics as Kappa distributions) is probably the most widely used, however the current explanations of its origin are not completely consistent. In this work it is shown that, under a Bayesian interpretation of superstatistics, the origin of the $q$-canonical ensemble can be explained as the superstatistical distribution with maximum Shannon-Jaynes entropy under noninformative constraints. The $q$-canonical distributions are singled out by the mathematical structure of superstatistics itself, and thus no assumptions about the physics of the systems of interest, or regarding their complexity or range of interactions, are needed. These results support the thesis that the success of the $q$-canonical ensemble is information-theoretical in nature, and explainable in terms of the original maximum entropy principle by Jaynes.