Some formal properties on superstatistics and superposition of statistical factors

TL;DR

This paper explores the formal mathematical properties of superstatistics, linking it to thermodynamics and information theory, and clarifies how variance influences the parameters in superstatistical models.

Contribution

It presents a formal analysis of superstatistics, highlighting its connection to thermodynamic structures and the role of variance in its parameters.

Findings

Superstatistics can be viewed as a counterpart of the canonical partition function.

Variance of the fluctuating quantity appears in correction terms for any superstatistics.

Parameters in the statistical factor are related to the variance of the fluctuating quantity.

Abstract

By focusing on the interchangeable role in a generating function (i.e., $\beta \leftrightarrow E$ in the Laplace transform), the superstatistics proposed by Beck and Cohen can be viewed as a counterpart of the canonical partition function. Some formal properties of this superstatistics are presented in connection with thermodynamic structures and information aspects. For any combination of the local equilibrium statistical factor and the form of fluctuating field, which are ingredients of making a generic superstatistics, a variance of the fluctuating quantity appears in the correction term. This fact enables us to relate parameters contained in the statistical factor with the variance in {\it any} situation.