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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. doi:10.1007/s00332-009-9049-y 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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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 | ||||
Official Date: | February 2010 | ||||
Dates: |
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Volume: | Vol.20 | ||||
Number: | No.1 | ||||
Number of Pages: | 46 | ||||
Page Range: | pp. 1-46 | ||||
DOI: | 10.1007/s00332-009-9049-y | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Open Access (Creative Commons) | ||||
Date of first compliant deposit: | 3 December 2015 | ||||
Date of first compliant Open Access: | 3 December 2015 |
Data sourced from Thomson Reuters' Web of Knowledge
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