Converting between the Cartesian tensor and spherical harmonic expansion of solutions to the Boltzmann equation

Abstract

The Vlasov–Fokker–Planck (VFP) equation, a variant of the Boltzmann equation, is frequently used to model the dynamics of laboratory and astrophysical plasmas. A common approach in solving the VFP equation in numerical and analytic studies is to expand the momentum part of the distribution function $f$ (i.e. the particle density in phase-space) in terms of known functions. The Cartesian tensor and spherical harmonic expansion have been widely used, leading to the question of how to convert between the different coefficients of the two expansions. This problem is also familiar in multipole expansions of an electrostatic (or gravitational) potential. The coefficients of the Cartesian tensor expansion of the potential are called (Cartesian) multipole moments and the ones of the spherical harmonic expansion are called spherical multipole moments. In this paper, we investigate the relation between the two kinds of multipole moments and provide a general formalism to convert between them. We subsequently apply this formalism to the coefficients of the expansions of the distribution function $f$ . A free, open-source command-line tool which implements this formalism is provided.