Kappa Distributions and Isotropic Turbulence

Abstract

In this work, the two-point probability density function (PDF) for the velocity field of isotropic turbulence is modeled using the kappa distribution and the concept of superstatistics. The PDF consists of a symmetric and an anti-symmetric part, whose symmetry properties follow from the reflection symmetry of isotropic turbulence, and the associated non-trivial conditions are established. The symmetric part is modeled by the kappa distribution. The anti-symmetric part, constructed in the context of superstatistics, is a novel function whose simplest form (called “the minimal model”) is solely dictated by the symmetry conditions. We obtain that the ensemble of eddies of size up to a given length r has a temperature parameter given by the second order structure function and a kappa-index related to the second and the third order structure functions. The latter relationship depends on the inverse temperature parameter (gamma) distribution of the superstatistics and it is not specific to the minimal model. Comparison with data from direct numerical simulations (DNS) of turbulence shows that our model is applicable within the dissipation subrange of scales. Also, the derived PDF of the velocity gradient shows excellent agreement with the DNS in six orders of magnitude. Future developments, in the context of superstatistics, are also discussed.

Physical meaning of temperature in superstatistics

“…4) The ultra-strong spatial statistical dependence disagrees with the cluster decomposition principle if taken without further specifications. Hence, one may assume either that i) superstatistics applies to special systems such that all particles can be regarded as strongly spatially correlated, e.g., by Debye shielding [35], or ii) superstatistics should be scaledependent, that is, the off-equilibrium parameters depend on the spatial size of the system, and acquire their equilibrium values when the size of the system is adequately large, as it may be in the case of turbulence [7,8], see also [3,4,10]. Given assumption ii), when the largest scales are included, the average energy of particles (in the thermodynamic limit) at a certain instant of time is equal to the (constant) temperature scale of the system.…”

“…The latter remark, which was explicitly introduced in the recent works of the authors on isotropic turbulence [7,8] (see also [3,4,10] and the references therein) is a rather important conceptual addition to the theory of superstatistics, in order to free it from excessively strong assumptions regarding the nature of the systems where it is applicable. This idea will be further pursued in future publications.…”

Estimation of Turbulent Heating of Solar Wind Protons at 1 au

“…Finally, Section 4 summarizes the conclusions, highlighting the analysis outcome, that is, a method for computing the turbulent heating using simply the plasma moments and magnetic field, and for improving our understanding of the turbulence complexity. (For instance, note that several attempts were made for the description of the complexity of turbulence, based on a new time domain, e.g., Varotsos et al 2006Varotsos et al , 2014, the theory of kappa distributions, e.g., Gravanis et al 2019, and their connection with the phenomenon of wave-particle thermodynamic equilibria). The appendix demonstrates a toolbox for the derivation of the uncertainty involved in the analysis.…”

Nonextensive Statistical Mechanics: Equivalence Between Dual Entropy and Dual Probabilities

“…The q -exponential distributions are observed quite frequently in nature, and constitute a suitable generalization of the BG exponential distribution. Applications of the q -exponential distribution can be found in a wide variety of topics, among numerous others, are the following: sociology–sociometry: e.g., internet [ 6 ]; citation networks of scientific papers [ 7 ]; urban agglomeration [ 8 ]; linguistics [ 9 ]; economy [ 10 ]; biology: biochemistry [ 11 , 12 ]; ecology [ 13 , 14 ]; statistics: [ 15 , 16 , 17 , 18 ]; physics: e.g., nonlinear dynamics [ 19 , 20 ]; condensed-matter: [ 21 ]; earthquakes [ 22 , 23 , 24 , 25 , 26 ]; turbulence [ 27 , 28 ]; physical chemistry [ 29 ]; and space physics/astrophysics [ 30 , 31 , 32 ]; (a more extended bibliography of q -deformed exponential distributions can be found in [ 14 , 32 , 33 , 34 , 35 , 36 , 37 ]).…”