Scattering Theory for the Boltzmann Equation and the Arrow of Time

David Levermore
University of Maryland
Department of Mathematics

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 quantities. The Boltzmann entropy has a unique minimizer with the same thirteen conserved values. This minimizer is a local Maxwellian that is also a global solution of the Boltzmann equation --- a so-called global Maxwellian. We show that each of our eternal solutions has a streaming asymptotic state as time goes to minus or plus infinity. However it does not converge to the associated global Maxwellian as time goes to infinity unless it is that global Maxwellian. Hence, the final step in the traditional
argument for the heat death of the universe breaks down.

(joint work with Claude Bardos, Irene Gamba, and Francois Golse)

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