Validity and failure of the Boltzmann approximation of kinetic annihilation

K Matthies, F Theil

OAI: oai:purehost.bath.ac.uk:publications/8d194c3d-89e9-41b4-a99b-3bd597df541e • DOI: https://doi.org/10.1007/s00332-009-9049-y

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 f (0)(v). Assuming that the moments of order less than three of f (0) 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 us to restrict the analysis to a finite number of interacting particles and enabling us to extract a single-body distribution. A counter-example is given for a concentrated initial density f (0) even to short-term validity.