Non-Maxwellian Velocity Distribution Functions for Coulombic Systems Out of Equilibrium

The velocity distribution function (VDF) of ions in the solar wind, as observed by spacecraft at 1 au and elsewhere in the heliosphere, exhibits a consistent trend: at low energies in the solar wind frame, the distribution is largely Maxwellian-the core; at higher but still modest energies in the solar wind frame, the distribution follows a power law (f proportional to nu(-gamma), where f is the VDF, nu is the speed in the solar wind frame, and gamma is an arbitrary spectral index parameter)-the tail-with a spectral index of gamma approximate to 5 being extremely common. Several theories have been proposed to explain this common index. Among these theories is that the tail is a natural consequence of an ensemble of particles obeying Coulomb's law. In this study, we derive a general analytical formula for the distribution of electric fields, and find that it always exhibits a power-law tail with a spectral index of exactly 9/2, or 4.5, due to the spatial power-law index of Coulomb's law. We then show how the VDF is a convolution of the distribution of electric fields with a preexisting VDF, and that for small values of time after being created, the ion VDF always exhibits a gamma = 9/2 power law, wherein the probability of the tail relative to the core depends on particle density, n, and inversely on the preexisting VDF thermal speed, nu(th). Finally, we compare our results with previous works, and find good agreement but with important distinctions.