Kappa and q Indices: Dependence on the Degrees of Freedom

Abstract

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.

The nonextensive parameter for the rotating astrophysical systems with power-law distributions

“…For example, in self-gravitating systems, NSM has been applied to study the negative heat capacity [2,3], the equilibrium structure and stability [4][5][6], the dark matter distributions [7][8][9] and the rotations [10][11][12][13][14], etc. In the space plasmas, NSM has been applied to study various waves and instability [15][16][17][18] and solar wind [19][20][21], and properties of the plasmas with the κ-distributions [22][23][24][25][26][27][28][29][30][31]. In particular, we have recently seen detailed studies and deep understanding of the κ-distributions in the space plasmas and their statistical properties [27][28][29][30][31].…”

“…In the space plasmas, NSM has been applied to study various waves and instability [15][16][17][18] and solar wind [19][20][21], and properties of the plasmas with the κ-distributions [22][23][24][25][26][27][28][29][30][31]. In particular, we have recently seen detailed studies and deep understanding of the κ-distributions in the space plasmas and their statistical properties [27][28][29][30][31].…”

“…Such a Hamiltonian factorization was established under the probabilistically independent postulate [36], i.e., under the assumption that the kinetic degrees of freedom are independent (uncorrelated) with the potential degrees of freedom [29][30][31]. Following the line of Ref.…”

“…Usually, the q-distribution has been considered equal to the κ-distributions in the space plasmas [19,22,[24][25][26][27][28][29][30][31].…”

Kappa Function as a Unifying Framework for Discrete Population Modeling

“…The use of the kappa function has become increasingly widespread across physics and other disciplines, with the number of papers following, remarkably, an exponential growth (Livadiotis [2015b]). The kappa function was empirically derived half a century ago as a suitable fitting-model describing observations of space and other plasma particle populations (Olbert [1968], Vasyliunas [1968]).…”