Origin of generalized entropies and generalized statistical mechanics for superstatistical multifractal systems Available to Purchase

We consider a multifractal structure as a mixture of fractal substructures and introduce a distribution function f (α), where α is a fractal dimension. Then we can introduce $g(p)∼ ∫ −lnpμe−yf(y)dy$ and show that the distribution functions f (α) in the form of $f(α) = δ(α−1)$⁠, $f(α) = δ(α−θ)$⁠, $f(α) = 1α−1$⁠, $f(y) = yα−1$ lead to the Boltzmann – Gibbs, Shafee, Tsallis and Anteneodo – Plastino entropies conformably. Here δ(x) is the Dirac delta function. Therefore the Shafee entropy corresponds to a fractal structure, the Tsallis entropy describes a multifractal structure with a homogeneous distribution of fractal substructures and the Anteneodo – Plastino entropy appears in case of a power law distribution f (y). We consider the Fokker – Planck equation for a fractal substructure and determine its stationary solution. To determine the distribution function of a multifractal structure we solve the two-dimensional Fokker – Planck equation and obtain its stationary solution. Then applying the Bayes theorem we obtain a distribution function for the entire system in the form of q-exponential function. We compare the results of the distribution functions obtained due to the superstatistical approach with the ones obtained according to the maximum entropy principle.