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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 ($S_{BG} =-k\sum_i p_i \ln p_i$ 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 ($p_i = e^{-βE_i}/Z_{BG}$ with $Z_{BG}=\sum_j e^{-βE_j}$ for BG). Third, the connection to thermodynamics (e.g., $F_{BG}= -\frac{1}β\ln Z_{BG}$ and $U_{BG}=-\frac{\partial}{\partial β} \ln Z_{BG}$). Assuming temperature fluctuations, Beck and Cohen recently proposed a generalized Boltzmann factor $B(E) = \int_0^\infty dβf(β) e^{-βE}$. 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.