Origin of generalized entropies and generalized statistical mechanics for superstatistical multifractal systems

摘要

We consider a multifractal structure as a mixture of fractal substructures and introduce a distribution function f(a), where a is a fractal dimension. Then we can introduce g(p) ~ ~μf _(-Inp)e~(-y)f(y)dy and show that the distribution functions f (a) in the form of f(α) = δ(α - 1), f (α) = δ(a - θ), f (a) = 1/α-1, f(y) = y~(α-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.