Validity and failure of the Boltzmann approximation of kinetic annihilation


Matthies, K. and Theil, F., 2010. Validity and failure of the Boltzmann approximation of kinetic annihilation. Journal of Nonlinear Science, 20 (1), pp. 1-46.

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    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.


    Item Type Articles
    CreatorsMatthies, K.and Theil, F.
    DepartmentsFaculty of Science > Mathematical Sciences
    Publisher StatementMatthies_JNS_2010_20_1_1.pdf: The original publication is available at
    ID Code17837


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