Kappa and q Indices: Dependence on the Degrees of Freedom

George Livadiotis

doi:10.3390/e17042062

Abstract

The kappa distributions, or their equivalent, the q-exponential distributions, are the natural generalization of the classical Boltzmann-Maxwell distributions, applied to the study of the particle populations in collisionless space plasmas. A huge step in the development of the theory of kappa distributions and their applications in space plasma physics has been achieved with the discovery that the observed kappa distributions are connected with the solid statistical background of non-extensive statistical mechanics. Now that the statistical framework has been identified, it is straightforward to improve our understanding of the nature of the kappa index (or the entropic q-index) that governs these distributions. One critical topic is the dependence of the kappa index on the degrees of freedom. In this paper, we first show how this specific dependence is naturally emerged, using the formalism of the N-particle kappa distribution of velocities. Then, the result is extended in the presence of potential energies. It is shown that the kappa index is simply related to the kinetic and potential degrees of freedom. In addition, it is shown that various problems of non-extensive statistical mechanics, such as (i) the correlation dependence on the total number of particles; and (ii) the normalization divergence for finite kappa indices, are resolved considering the kappa index dependence on the degrees of freedom.

Highlights

Thermal equilibrium is a special stationary state
Numerous independent developments in space plasma physics have revealed the peculiar statistical behavior of this category of systems: particle populations in space plasmas reside in stationary states out of thermal equilibrium
The purpose of this paper is to show and study the dependence on the degrees of freedom, of the non-equilibrium measure that governs kappa distributions, the kappa index, which is related to the entropic q-index of non-extensive statistical mechanics

Summary

Introduction

Thermal equilibrium is a special stationary state. It is the state where any flow of heat (e.g., thermal conduction, thermal radiation) is in balance. Systems at thermal equilibrium have one very special statistical feature: are their particle velocities described by stationary distribution functions, and, any of these functions is some Maxwellian distribution. Numerous independent developments in space plasma physics have revealed the peculiar statistical behavior of this category of systems: particle populations in space plasmas reside in stationary states out of thermal equilibrium. The classical Maxwell distributions are extremely rare in space plasmas. The vast majority of space plasmas reside out of thermal equilibrium, and are described by the kappa distributions (e.g., see [1,2,3,4] and references therein)

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