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The purpose of this paper is to study the evolution of moving interactingparticles on the mesoscopic scale. We will introduce an uncertainty principleand a new priori bound for the evolution of particles subject to a generalmesoscopic interaction and use this setting to demonstrate evidence for theuniversal dispersion of particles in the absence of the conventional methodslike velocity averaging lemmas or Strichartz-type estimates. We will develop ascattering theory in generality and use its frame work to show the existenceand uniqueness of a class scattering solutions to the Boltzmann equation withasymptotic stability and completeness in the $L^{\infty}$ setting.

Evolution of mesoscopic interactions and scattering solutions of the Boltzmann equation