Anisotropic Kappa Distributions. I. Formulation Based on Particle Correlations

G. Livadiotis,G. Nicolaou,F. Allegrini

doi:10.3847/1538-4365/abd4ed

Abstract

Abstract We develop the theoretical basis for the connection of the variety of anisotropic distributions with the statistical correlations among particles’ velocity components. By examining the most common anisotropic distribution function, we derive the correlation coefficient among particle energies, show how this correlation is connected to the effective dimensionality of the velocity distribution, and derive the connection between anisotropy and adiabatic polytropic index. Having established the importance of the correlation among particles in the formulation of anisotropic kappa distributions, we generalize these distributions within the framework of nonextensive statistical mechanics and based on the types of homogeneous or heterogeneous correlations among the particles’ velocity components. The formulation of the developed generalized distributions mediates the main two types of anisotropic kappa distributions that consider either (a) equal correlations, or (b) zero correlations, among different velocity components. Finally, the developed anisotropic kappa distributions are expressed in terms of the energy and pitch angle in arbitrary reference frames.

Highlights

The temperature is a well-defined quantity in the theoretical framework of kappa distributions and nonextensive statistical mechanics (Tsallis 2009; Livadiotis & McComas 2009; 2010)
We examined the typical cases, where the velocity components of each particle are either (a) homogeneously correlated, that is, the correlations from all components are characterized by the same kappa index, or (b) uncorrelated, that is, the correlations between different components are characterized by kappa index equal to infinity
We develop the general case, where the velocity components can be heterogeneously correlated with an arbitrary heterogeneity, namely, the correlations between different components can be characterized by any kappa index

Summary

Introduction

The temperature is a well-defined quantity in the theoretical framework of kappa distributions and nonextensive statistical mechanics (Tsallis 2009; Livadiotis & McComas 2009; 2010). The purpose of this analysis is to (i) determine the correlation coefficient and the involved effective dimensionality of anisotropic kappa distributions characterized with homogeneous or heterogeneous correlations among their velocity components; (ii) indicate the connection of the adiabatic polytropic index with temperature anisotropy; (iii) characterize and study the types of homogeneous/heterogeneous correlations among the particles velocity components; (iv) formulate the correlation relationship that characterizes the partition of 2D joint kappa distribution into the two marginal 1D kappa distributions, as emerges from nonextensive statistical mechanics; (v) generalize the formulae of anisotropic kappa distributions, based on the various types of homogeneous/heterogeneous correlations; (vi) describe and examine the anisotropic kappa distributions in (a) the comoving reference frame with respect to the velocity components, (b) arbitrary S/C frame with respect to to the triplet of energy, pitch angle, and azimuth, and (c) the more complicate form of azimuth independent distributions with respect to energy and pitch angle. Using these distributions is vital for understanding several important properties of kappa distributions, which is impossible to be done with one-particle distributions (e.g., Gravanis et al 2020)

Derivation of correlation coefficient

II uII4

Anisotropic distributions with heterogeneously correlated velocity components

Anisotropic distributions with nontrivial heterogeneity

Summary of anisotropic distributions formulae

Formalism of anisotropic kappa distributions in arbitrary reference frames

Conclusions