General solution of a Boltzmann equation, and the formation of Maxwellian tails

Abstract

A classical Boltzmann equation is studied. The equation describes the evolution towards the Maxwellian equilibrium state of a homogeneous, isotropic gas where the collision cross section is inversely proportional to the relative velocity of the colliding particles. After Tjon & Wu (1979), the problem is transformed into a mathematically equivalent one, itself a model Boltzmann equation in two dimensions. Working in the context of the latter equation, a formal derivation of the general solution is presented. First a countable ensemble of particular solutions, called pure solutions , is constructed. From these, via a non-linear combination mechanism, the general solution is obtained in a form appropriate for direct numerical computation. The validity of the solution depends upon its containment in a well defined Hilbert space H~ Given that the initial condition lies within H~ it is proved that at least for a small finite time interval it remains in H~.