Time Evolution of a System of Particles Obeying Coulomb's Law and the High-speed Rollover of Suprathermal Tails in the Solar Wind

In this article, we show through a series of rigorous mathematical steps, starting with Liouville's equation and solving it for the case of two protons using the method of characteristics, that a velocity distribution function, f(u), is formed that exhibits a power law, f proportional to u ( gamma ), where gamma is -9/2, and an exponential-type rollover at large speeds, providing a potential explanation for the observations of suprathermal ions in the solar wind. The solution is valid for all times and all background conditions, though the power law does at some point merge with the core of the distribution. When it does so, a final distribution is found that agrees with overall energy conservation. Analytical approximations to the solution are discussed, along with explanations for the various phases that the solution undergoes.