Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

This paper introduces a new method to show the validity of a continuum description for the deterministic dynamics of many interacting particles. Here the many particle evolution is analyzed for a hard sphere flow with the addition that after a collision the collided particles are removed from the system. We consider random initial configurations which are drawn from a Poisson point process with spatially homogeneous velocity density f0(v). Assuming that the moments of order less than three of f0 are finite and no mass is concentrated on lines, the homogeneous Boltzmann equation without gain term is derived for arbitrary long times in the Boltzmann-Grad scaling. A key element is a characterization of the many particle flow by a hierarchy of trees which encode the possible collisions. The occurring trees are shown to have favorable properties with a high probability, allowing to restrict the analysis to a finite number of interacting particles, enabling us to extract a single-body d...

• Type: Journal
• Year: 2010
• Where: JNS
• Authors: Karsten Matthies, Florian Theil
• Keywords: Concentrated Initial Density, Homogeneous Boltzmann Equation, Homogeneous Velocity Density, JNS 2010