Tsallis statistics and fully developed turbulence

Superstatistics

“…We note that for the above five distribution functions the first-order corrections to the Boltzmann factor e −β 0 E in eqs. (6), (10), (14), (16), (20) can all be written in a universal form. When expressing the generalized Boltzmann factor in terms of the average temperature β 0 and the variance σ 2 of the various probability densities f (β), one always obtains the same result for small σ,…”

“…Several types of generalized stochastic dynamics have been recently constructed for which Tsallis statistics can be proved rigorously [4,5,6]. Physical applications include 2-d and 3-d turbulence [7]- [14], momentum spectra of hadronic particles produced in e + e − annihilation experiments [15,16], the statistics of cosmic rays [17], and many other phenomena.…”

Analysis of velocity fluctuation in turbulence based on generalized statistics

“…is the so-called scaling exponent of the velocity structure function, whose expression was derived first by the present authors [15][16][17][18]. In this paper, we will use the formula (6) in order to extract the value of the intermittency exponent µ for the best fit to the measured scaling exponents by the method of least squares.…”

“…is the multifractal spectrum [15][16][17][18], derived by the relation P (n) (α) ∝ δ 1−f (α) n [9,18], that reveals how densely each singularity, labeled by α, fills physical space.…”

“…On this line, an investigation of turbulence based on the generalized entropy, i.e., Rényi's [11] or Tsallis' [12,13], was started by the present authors [14][15][16][17][18][19][20]. After a rather preliminary investigation of the p-model [14], it has been developed further to derive the analytical expression for the scaling exponents of velocity structure function [15][16][17][18], and to determine the probability density function (PDF) of velocity fluctuations [18][19][20] by a self-consistent statistical mechanical approach.With the help of the analytical formulae derived in [15][16][17][18][19][20], we will analyze, in this paper, the PDF's of velocity fluctuations observed in the beautiful DNS (i.e., the direct numerical simulation) conducted by Gotoh et al [21]. We will deal with the data at the Taylor microscale Reynolds number R λ = 381, since at this Reynolds number Gotoh et al observed the PDF with accuracy up to order of 10 −9 ∼ 10 −10 , in contrast with any previous experiments, real or numerical.…”