Generalized entropies and the transformation group of superstatistics

<jats:p> Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures β, so that the probability distribution is <jats:inline-formula> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="pnas.1103539108eq1.gif" /> </jats:inline-formula> , where the “kernel” <jats:italic>f</jats:italic> ( <jats:italic>β</jats:italic> ) is nonnegative and normalized [∫ <jats:italic>f</jats:italic> ( <jats:italic>β</jats:italic> ) <jats:italic>dβ</jats:italic>  = 1]. We discuss the relation between this distribution and the generalized entropic form <jats:inline-formula> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="pnas.1103539108eq2.gif" /> </jats:inline-formula> . 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. </jats:p>