The use of the Pearson differential equation to test energetic distributions in space physics as Kappa distributions; implication for Tsallis nonextensive entropy: II

Abstract

This paper presents a mathematical demonstration that particular observed energetic particle distributions of space physics in the particle velocity magnitude, v∈[0,∞), are not rigorously Kappa distributions. The method is based on the distribution function of a test particle of mass m in a heat bath of particles of mass M. The distribution function is given by a Fokker-Planck equation. The particles interact via Coulomb collisions and a second diffusion coefficient that represents the effects of wave-particle interactions. For the particular wave-particle diffusion coefficient that varies inversely with the particle velocity, the steady distribution for m/M→0 is a Kappa distribution which is the solution of a Pearson ordinary differential equation. The analysis of the observed distributions versus v∈(0,∞) employed in this paper is based on the Pearson differential equation and applied to several published distributions. The chosen distributions are representative and shown not to be Kappa distributions. Thus, the Tsallis nonextensive entropy which yields uniquely the Kappa distribution does not explain the occurrence of the myriad of nonequilibrium distributions in space physics.