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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. 1-46. ISSN 0938-8974

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Official URL: http://dx.doi.org/10.1007/s00332-009-9049-y

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 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
Publisher:	Springer
ISSN:	0938-8974
Date:	February 2010
Volume:	Vol.20
Number:	No.1
Number of Pages:	46
Page Range:	pp. 1-46
Identification Number:	10.1007/s00332-009-9049-y
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/6546