Generalized entropies and the transformation group of superstatistics

TL;DR

This paper explores the connection between superstatistics, generalized entropies, and the transformation group, revealing duality in entropy construction methods and the impact on thermodynamic properties.

Contribution

It introduces a duality between two entropy construction methods in superstatistics and characterizes the transformation group as the Euclidean group in one dimension.

Findings

Two methods to construct entropy from superstatistical distributions are identified.

The transformation group of superstatistics distributions is the Euclidean group in one dimension.

Thermodynamic properties differ depending on the entropy construction approach.

Abstract

Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures $\beta$, so that the probability distribution is $p(\epsilon_i) \propto \int_{0}^{\infty} f(\beta) e^{-\beta \epsilon_i}d\beta$, where the `kernel' $f(\beta)$ is nonnegative and normalized ($\int f(\beta)d \beta =1$). We discuss the relation between this distribution and the generalized entropic form $S=\sum_i s(p_i)$. The first three Shannon-Khinchin axioms are assumed to hold. It then turns out that for a given distribution there are two different ways to construct the entropy. One approach uses escort probabilities and the other does not; the question of which to use must be decided empirically. The two approaches are related by a duality. The thermodynamic properties of the system can be quite different for the two approaches. In that connection we present the transformation laws for the superstatistical distributions under macroscopic state changes. The transformation group is the Euclidean group in one dimension.