Constructing a statistical mechanics for Beck-Cohen superstatistics

TL;DR

This paper develops a new entropic functional and constraints that generate Beck-Cohen superstatistics, extending the statistical mechanics framework to systems with temperature fluctuations.

Contribution

It introduces an entropic functional and constraints that produce Beck-Cohen superstatistics, completing the three-stage structure of statistical mechanics for these systems.

Findings

Derived an entropic functional for superstatistics

Connected superstatistics to thermodynamic quantities

Validated with six examples of Beck-Cohen superstatistics

Abstract

The basic aspects of both Boltzmann-Gibbs (BG) and nonextensive statistical mechanics can be seen through three different stages. First, the proposal of an entropic functional ($S_{BG} =-k\sum_i p_i \ln p_i$ for the BG formalism) with the appropriate constraints ($\sum_i p_i=1$ and $\sum_i p_i E_i = U$ for the BG canonical ensemble). Second, through optimization, the equilibrium or stationary-state distribution ($p_i = e^{-\beta E_i}/Z_{BG}$ with $Z_{BG}=\sum_j e^{-\beta E_j}$ for BG). Third, the connection to thermodynamics (e.g., $F_{BG}= -\frac{1}{\beta}\ln Z_{BG}$ and $U_{BG}=-\frac{\partial}{\partial \beta} \ln Z_{BG}$). Assuming temperature fluctuations, Beck and Cohen recently proposed a generalized Boltzmann factor $B(E) = \int_0^\infty d\beta f(\beta) e^{-\beta E}$. This corresponds to the second stage above described. In this letter we solve the corresponding first stage, i.e., we present an entropic functional and its associated constraints which lead precisely to $B(E)$. We illustrate with all six admissible examples given by Beck and Cohen.