# Validity and failure of the Boltzmann approximation of kinetic annihilation

### Reference:

Matthies, K. and Theil, F., 2010. Validity and failure of the Boltzmann approximation of kinetic annihilation. *Journal of Nonlinear Science*, 20 (1), pp. 1-46.

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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.

### Details

Item Type | Articles |

Creators | Matthies, K.and Theil, F. |

DOI | 10.1007/s00332-009-9049-y |

Departments | Faculty of Science > Mathematical Sciences |

Publisher Statement | Matthies_JNS_2010_20_1_1.pdf: The original publication is available at www.springerlink.com |

Refereed | Yes |

Status | Published |

ID Code | 17837 |

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