Title : Scattering Theory for the Boltzmann Equation and the Arrow of Time
Abstract : We develop a scattering theory for a class of eternal solutions of the Boltzmann equation posed over all space. In three spatial dimensions, each of these solutions has thirteen conserved qualities. The Boltzmann entropy has a unique minimizer with the same thirteen conserved values. This minimizer is a local MAxwellian that is also a global soution of the Botlzmann equation - a so-called global Maxwellian. We show that each of our external solutions has a streaming asymptotic state as time goes to minus- or plus-infinity. However, it does not converge to the associated gloabl Maxwellian as time goes to infinity unless it is that global Maxwellian. The Blotzmann entropy decreases as time increases, but does not decrease to its minimum as time goes to infinity. Said another way, the final step in the traditional argument for the heat death of the universe is not valid.