Validity and failure of the Boltzmann approximation of kinetic annihilation

K Matthies, F Theil

Research output: Contribution to journalArticle

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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.
LanguageEnglish
Pages1-46
Number of pages46
JournalJournal of Nonlinear Science
Volume20
Issue number1
Early online date25 Jul 2009
DOIs
StatusPublished - 1 Feb 2010

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Boltzmann equation
Annihilation
Ludwig Boltzmann
Kinetics
Approximation
Collision
Poisson Point Process
Hard Spheres
Boltzmann Equation
Counterexample
Continuum
Scaling
Moment
Configuration
Line
Arbitrary
Term

Cite this

Validity and failure of the Boltzmann approximation of kinetic annihilation. / Matthies, K; Theil, F.

In: Journal of Nonlinear Science, Vol. 20, No. 1, 01.02.2010, p. 1-46.

Research output: Contribution to journalArticle

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