Evolution of mesoscopic interactions and scattering solutions of the Boltzmann equation

Nima Moini

Published 2020-10-13Version 1

The purpose of this writing is to to study the evolution of moving particles subject to a general mesoscopic interaction. The primary examples of mesoscopic evolutions are the different variants of the Boltzmann equation which tie this work to active fields of research in mathematics, including kinetic theory, statistical mechanics and fluid dynamics.Here, we introduce the concept of uncertainty for the evolution of mesoscopic interactions and show evidence for the universal dispersion of particles independent of a specific structure of interactions. The results of this paper are valid for any mesoscopic evolution, including the Boltzmann equation, and show similarities to the theory of linear and nonlinear Schrodinger's equation. Furthermore, we will develop the notion of scattering in generality for the evolution of mesoscopic interactions and use these insights to show the existence and uniqueness of scattering solutions to the Boltzmann equation for a class of initial values with a special property.