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 ∝ v ^- , where f is the VDF, v is the speed in the solar wind frame, and γ is an arbitrary spectral index parameter)-the tail-with a spectral index of γ ≈ 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 (Randol & Christian 2014, 2016). 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 pre-existing VDF, and that for small values of time after being created, the ion VDF always exhibits a γ = 9/2 power law, wherein the probability of the tail relative to the core depends on particle density, n, and inversely on the pre-existing VDF thermal speed, v . Finally, we compare our results with previous works, and find good agreement but with important distinctions.