[Limite macroscopique de particules autopropulsées avec interaction d'orientation]
L'algorithme discret de Couzin–Vicsek (CVA) a été proposé pour modéliser l'interaction d'individus au sein de sociétés animales comme les bancs de poissons. Dans cette Note, nous proposons une version cinétique (champ-moyen) de l'algorithme CVA et en donnons la limite macroscopique formelle. Le modèle macroscopique final comprend une équation de conservation pour la densité des individus et une équation non-conservative pour le vecteur directeur de la vitesse moyenne. Ce résultat est basé sur l'introduction d'un concept non-conventionnel d'invariant collisionnel de l'opérateur de collision.
The discrete Couzin–Vicsek algorithm (CVA) has been proposed to model the interactions of individuals among animal societies such as schools of fish. In this Note, we propose a kinetic (mean-field) version of the CVA model and provide its formal macroscopic limit. The final macroscopic model involves a conservation equation for the density of the individuals and a non-conservative equation for the director of the mean velocity. The result is based on the introduction of a non-conventional concept of a collisional invariant of the collision operator.
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@article{CRMATH_2007__345_10_555_0,
author = {Degond, Pierre and Motsch, S\'ebastien},
title = {Macroscopic limit of self-driven particles with orientation interaction},
journal = {Comptes Rendus. Math\'ematique},
pages = {555--560},
publisher = {Elsevier},
volume = {345},
number = {10},
year = {2007},
doi = {10.1016/j.crma.2007.10.024},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2007.10.024/}
}
TY - JOUR AU - Degond, Pierre AU - Motsch, Sébastien TI - Macroscopic limit of self-driven particles with orientation interaction JO - Comptes Rendus. Mathématique PY - 2007 SP - 555 EP - 560 VL - 345 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2007.10.024/ DO - 10.1016/j.crma.2007.10.024 LA - en ID - CRMATH_2007__345_10_555_0 ER -
%0 Journal Article %A Degond, Pierre %A Motsch, Sébastien %T Macroscopic limit of self-driven particles with orientation interaction %J Comptes Rendus. Mathématique %D 2007 %P 555-560 %V 345 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2007.10.024/ %R 10.1016/j.crma.2007.10.024 %G en %F CRMATH_2007__345_10_555_0
Degond, Pierre; Motsch, Sébastien. Macroscopic limit of self-driven particles with orientation interaction. Comptes Rendus. Mathématique, Tome 345 (2007) no. 10, pp. 555-560. doi : 10.1016/j.crma.2007.10.024. http://www.numdam.org/articles/10.1016/j.crma.2007.10.024/
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