Comptes Rendus
no. 10
Partial Differential Equations
Macroscopic limit of self-driven particles with orientation interaction
[Limite macroscopique de particules autopropulsées avec interaction d'orientation]

Présenté par : Philippe G. Ciarlet

Pierre Degond1 ; Sébastien Motsch1
1 Institut de mathématiques de Toulouse, UMR 5219 (CNRS-UPS-INSA-UT1-UT2), équipe MIP, Université P. Sabatier, 31062 Toulouse cedex 09, France
Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 555-560.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.10.024
Pierre Degond 1 ; Sébastien Motsch 1

1 Institut de mathématiques de Toulouse, UMR 5219 (CNRS-UPS-INSA-UT1-UT2), équipe MIP, Université P. Sabatier, 31062 Toulouse cedex 09, France 
@article{CRMATH_2007__345_10_555_0,
     author = {Pierre Degond and S\'ebastien Motsch},
     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},
}
TY  - JOUR
AU  - Pierre Degond
AU  - Sébastien Motsch
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
DO  - 10.1016/j.crma.2007.10.024
LA  - en
ID  - CRMATH_2007__345_10_555_0
ER  - 
%0 Journal Article
%A Pierre Degond
%A Sébastien Motsch
%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
%R 10.1016/j.crma.2007.10.024
%G en
%F CRMATH_2007__345_10_555_0
Pierre Degond; Sébastien Motsch. Macroscopic limit of self-driven particles with orientation interaction. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 555-560. doi : 10.1016/j.crma.2007.10.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.024/

[1] M. Aldana; C. Huepe Phase transitions in self-driven many-particle systems and related non-equilibrium models: a network approach, J. Stat. Phys., Volume 112 (2003) no. 1/2, pp. 135-153

[2] I.D. Couzin; J. Krause; R. James; G.D. Ruxton; N.R. Franks Collective memory and spatial sorting in animal groups, J. Theor. Biol., Volume 218 (2002), pp. 1-11

[3] F. Cucker; S. Smale Emergent behavior in flocks, IEEE Trans. Automat. Control, Volume 52 (2007), pp. 852-862

[4] P. Degond, S. Motsch, Large-scale dynamics of the persistent turning walker model of fish behavior, preprint

[5] P. Degond, S. Motsch, Continuum limit of self-driven particles with orientation interaction, preprint

[6] M.R. D'Orsogna; Y.L. Chuang; A.L. Bertozzi; L.S. Chayes Self-propelled particles with soft-core interactions: pattern, stability and collapse, Lett. Phys. Rev., Volume 96 (2006), p. 104302

[7] J. Gautrais, S. Motsch, C. Jost, M. Soria, A. Campo, R. Fournier, S. Bianco, G. Théraulaz, Analyzing fish movement as a persistent turning walker, in preparation

[8] G. Grégoire; H. Chaté Onset of collective and cohesive motion, Lett. Phys. Rev., Volume 92 (2004), p. 025702

[9] V.L. Kulinskii; V.I. Ratushnaya; A.V. Zvelindovsky; D. Bedeaux Hydrodynamic model for a system of self-propelling particles with conservative kinematic constraints, Europhys. Lett., Volume 71 (2005), pp. 207-213

[10] A. Mogilner; L. Edelstein-Keshet A non-local model for a swarm, J. Math. Biol., Volume 38 (1999), pp. 534-570

[11] A. Mogilner; L. Edelstein-Keshet; L. Bent; A. Spiros Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol., Volume 47 (2003), pp. 353-389

[12] V.I. Ratushnaya; D. Bedeaux; V.L. Kulinskii; A.V. Zvelindovsky Collective behaviour of self propelling particles with kinematic constraints; the relations between the discrete and the continuous description, Physica A, Volume 381 (2007), pp. 39-46

[13] V.I. Ratushnaya; V.L. Kulinskii; A.V. Zvelindovsky; D. Bedeaux Hydrodynamic model for the system of self propelling particles with conservative kinematic constraints; two dimensional stationary solutions, Physica A, Volume 366 (2006), pp. 107-114

[14] C.M. Topaz; A.L. Bertozzi Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., Volume 65 (2004), pp. 152-174

[15] C.M. Topaz; A.L. Bertozzi; M.A. Lewis A nonlocal continuum model for biological aggregation, Bull. Math. Biol., Volume 68 (2006), pp. 1601-1623

[16] T. Vicsek; A. Szirok; B. Ben-Jacob; I. Cohen; O. Shochet Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., Volume 75 (1995) no. 6, pp. 1226-1229

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Dérivation de modèles couplés dérive-diffusion/cinétique par une méthode de décomposition en vitesse

Nicolas Crouseilles

C. R. Math (2002)