Validity and failure of the Boltzmann approximation of kinetic annihilation
Matthies, Karsten and Theil, Florian. (2010) Validity and failure of the Boltzmann approximation of kinetic annihilation. Journal of Nonlinear Science, Vol.20 (No.1). pp. 1-46. ISSN 0938-8974
WRAP_Mathies_1070268-lb-130611-wrap_theil_longtime_validity_gainless.pdf - Accepted Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Official URL: http://dx.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.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Mathematics|
|Library of Congress Subject Headings (LCSH):||Field theory (Physics), Particles -- Mathematical models, Kinetic theory of matter|
|Journal or Publication Title:||Journal of Nonlinear Science|
|Official Date:||February 2010|
|Number of Pages:||46|
|Page Range:||pp. 1-46|
|Access rights to Published version:||Open Access|
[BBS83] C. Boldrighini, L.A. Bunimovich, Y.G. Sinai. On the Boltzmann equation for the Lorentz gas.J.
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