Why do Solutions of the Maxwell-Boltzmann Equation Tend to be Gaussians? A Simple Answer

Abstract

The Maxwell-Boltzmann functional equation has recently attraction renewed interest since besides its importance in Boltzmann's kinetic theory of gases it also characterizes maximizers of certain bilinear estimates for solutions of the free Schrödinger equation. In this note we give a short and simple proof that, under some mild growth restrictions, any measurable complex-valued solution of the Maxwell-Boltzmann equation is a Gaussian. This covers most, if not all, of the applications.