Bayesian approach to superstatistics

TL;DR

This paper refines the superstatistics framework by emphasizing the importance of including the density of states and applying Bayesian analysis to interpret the probability distribution of fluctuating intensive variables.

Contribution

It introduces a corrected theoretical foundation for superstatistics by incorporating the density of states and offers a Bayesian interpretation of the distribution of the inverse temperature.

Findings

Highlighting the necessity of including the density of states in superstatistics

Providing a Bayesian interpretation for the probability density f(b)

Clarifying the fundamental basis of superstatistics theory

Abstract

The superstatistics approach recently introduced by Beck [C. Beck and E.G.D. Cohen, Physica A 322, 267 (2003)] is a formalism that aims to deal in a unifying way with a large variety of complex nonequilibrium systems, for which spatio-temporal fluctuations of one intensive variable (the "temperature" 1/b) are supposed to exist. The intuitive explanation provided by Beck for superstatistics is based on the ansatz that the system under consideration, during its evolution, travels within its phase space which is partitioned into cells. Within each cell, the system is described by ordinary Maxwell-Boltzmann statistical mechanics, i.e., its statistical distribution is the canonical one, but b varies from cell to cell, with its own probability density f(b). In this work we first address that the explicit inclusion of the density of states in this description is essential for its correctness. The correction is not relevant for developments of the theory, but points to the fact that its correct starting point, as well its meaning, must be found at a more basic level: the pure probability product rule involving the intensive variable b and its conjugate extensive one. The question therefore arises how to assign a meaning to these probabilities for each specific problem. We will see that it is easily answered through Bayesian analysis. This way, we are able to provide an interpretation for f(b), that was not fully elucidated till now.