Kappa distribution in the presence of a potential energy

Abstract

The present paper develops the theory and formulations of the kappa distributions that describe particle systems characterized by a nonzero potential energy. As yet, kappa distributions were used for the statistical description of the velocity or kinetic energy of particles but not of the potential energy. With the results provided here, it is straightforward to use the developed kappa distributions to describe any particle population of space plasmas subject to a nonnegligible potential energy. Starting from the kappa distribution of the Hamiltonian function, we develop the distributions that describe either the complete phase space or the marginal spaces of positions and velocities. The study shows, among others: (a) The kappa distributions of velocities that describe space plasmas can be vastly different from the standard formulation of the kappa distribution, because of the presence of a potential energy; the correct formulation should be given by the marginal kappa distribution of velocities by integrating the distribution of the Hamiltonian over the potential energy. (b) The long-standing problem of the divergence of the Boltzmannian exponential distribution for bounded radial potentials is solved using kappa distributions of negative kappa index. (c) Anisotropic distributions of velocities can exist in the presence of a velocity-dependent potential. (d) A variety of applications, including derivations/verifications of the following: (i) the Jeans' , the most frequent, and the maximum radii in spherical/linear gravitational potentials; (ii) the Virial theorem for power law potentials; (iii) the generalized barometric formula, (iv) the plasma density profiles in Saturnian magnetosphere, and (v) the average electron magnetic moment in Earth's magnetotail.

On the characteristics of obliquely propagating electrostatic structures in non-Maxwellian plasmas in the presence of ion pressure anisotropy

“…On the other hand, M 1 is not on P ? , as evident from Equation (32). In an analogous manner, we have plotted M 1;2 against P ?…”

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

“…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].…”

“…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 and q Indices: Dependence on the Degrees of Freedom

“…After the connection of empirical kappa distributions with the solid background of non-extensive statistical mechanics [1], it is straightforward now to generalize the distribution from the description of velocities to the description of the Hamiltonian [13]. More specifically, we are able to handle the existence of a non-negligible potential energy using the mathematical formulations and the physical theory of kappa distributions:…”

“…Single types of these distributions are usually sufficient to model the space plasma populations (e.g., see [5][6][7][8][9][10]); however, more complicated models of kappa distribution superpositions have been employed to describe rare features (e.g., anisotropy) [11,12]. A kappa distribution is primarily formulated to describe a Hamiltonian [13], i.e., the sum of the particle's kinetic and potential energies. However, the potential energy of a particle is small compared to its kinetic energy and can often be ignored; then, the system's statistical description is reduced to the kappa distribution of the particle velocities.…”