Pseudospectral solutions of the Fokker-Planck equation for Pearson diffusion that yields a Kappa distribution; the associated SUSY Schrödinger equation

Abstract

Numerous satellite measurements of the energy distributions of electrons and ions in the solar system and beyond can be fitted with the Kappa distribution. The origin of these Kappa distributions has attracted considerable attention. In particular, it has been used to support the Tsallis entropy formalism that purports to explain the origin of these distributions. In this paper, we compare the Kappa distribution of space plasmas modelled with a particular Fokker–Planck equation for a two component system with the linear Fokker–Planck equation that has been used to study the Student t-distribution. Pseudospectral solutions of the Fokker–Planck equation for the Student-t distribution based on the steady solution are not convergent owing to its asymptotic power law behaviour. We transform the Fokker–Planck equation to the isospectral Schrödinger equation which yields a Rosen-Morse II potential. The eigenfunctions and eigenvalues of this Schrödinger equation are obtained with a nonclassical pseudospectral method in comparison with a Fourier method and a method based on Hermite polynomials. The nonclassical pseudospectral method converges faster than these other basis sets and agrees with the known eigenvalues.