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

Abstract

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) ∼ µ − ln p e −y f (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) = 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 twodimensional 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.

Temperature based Restricted Boltzmann Machines

“…RBMs originate from the concept of Boltzmann distribution 40 , a well known concept in physical science where temperature is a key factor of the distribution. In fact, in statistical mechanics 41 42 43 and mathematics, a Boltzmann distribution is a probability distribution of particles in a system over various possible states. Particles in this context refer to gaseous atoms or molecules, and the system of particles is assumed to have reached thermodynamic equilibrium 44 45 .…”

Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks

“…Here 𝑟 is a spatial coordinate, 𝑡 is time and the functions 𝑔 0 (𝑟, 𝑡, 𝑟 ′ , 𝑡′) and 𝑔 1 (𝑟, 𝑡, 𝑟 ′ , 𝑡′) describe the influence of long-range space interactions and temporal memory on the dynamics of the order parameter. Integration is performed over the region 𝑅 in the two-dimensional space 𝑅 2 to which (𝑟, 𝑡) belong [8]. The dynamic equation follows from the stationary principle 𝛿𝐹[𝜂, 𝑢] = 0…”

“…The value 𝑞 = 1 corresponds to the degree distribution in the form of the Gaussian distribution. At 𝑞 ≠ 1 and for large enough 𝑘 the network is characterized by the power-law degree distribution [8]. The presented algorithm shows that order and disorder are inherent in small-world systems.…”

Generalized entropies, density of states, and non-extensivity

“…A variety of generalized entropic functionals have been introduced to phenomenologically extend statistical mechanics to specific non-ergodic or strongly interacting systems, both within and outside the realm of physics including spin-like systems [9][10][11], cosmic ray energy spectra [12], multifractals [13], networks [14], quantum information [15][16][17], special relativity [18], anomalous diffusive processes [19][20][21][22][23], superstatistics [24,25], time series analysis [26][27][28] and artificial neural networks [29].…”