Constructing a statistical mechanics for Beck-Cohen superstatistics

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 (SBG=-k∑ᵢpᵢlnpᵢ for the BG formalism) with the appropriate constraints (∑ᵢpᵢ=1 and ∑ᵢpᵢEᵢ=U for the BG canonical ensemble). Second, through optimization, the equilibrium or stationary-state distribution (pᵢ=e^-βEᵢ/ZBG with ZBG=∑ⱼe^-βEⱼ for BG). Third, the connection to thermodynamics (e.g., FBG=-(1/β)lnZBG and UBG=-(∂/∂β)lnZBG). Assuming temperature fluctuations, Beck and Cohen recently proposed a generalized Boltzmann factor B(E)=∫₀^∞dβf(β)e^-βE. This corresponds to the second stage described above. In this paper, 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.