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Validity and failure of the Boltzmann approximation of kinetic annihilation
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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. 146. ISSN 09388974

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Official URL: http://dx.doi.org/10.1007/s003320099049y
Abstract
This paper introduces a new method to show the validity of a continuum description for the deterministic dynamics of many interacting particles. Here the manyparticle 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 BoltzmannGrad scaling. A key element is a characterization of the manyparticle 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 singlebody distribution. A counterexample is given for a concentrated initial density f (0) even to shortterm 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 
Publisher:  Springer 
ISSN:  09388974 
Official Date:  February 2010 
Volume:  Vol.20 
Number:  No.1 
Number of Pages:  46 
Page Range:  pp. 146 
Identification Number:  10.1007/s003320099049y 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Open Access 
References:  [BBS83] C. Boldrighini, L.A. Bunimovich, Y.G. Sinai. On the Boltzmann equation for the Lorentz gas.J. 
URI:  http://wrap.warwick.ac.uk/id/eprint/6546 
Data sourced from Thomson Reuters' Web of Knowledge
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