Special Issue on Mathematical Models and Methods for Complex SystemsNo Access

HYDRODYNAMICS OF THE KURAMOTO–VICSEK MODEL OF ROTATING SELF-PROPELLED PARTICLES

PIERRE DEGOND

Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, F-31062 Toulouse, France

CNRS, Institut de Mathématiques de Toulouse UMR 5219, F-31062 Toulouse, France

Search for more papers by this author

GIACOMO DIMARCO

Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, F-31062 Toulouse, France

CNRS, Institut de Mathématiques de Toulouse UMR 5219, F-31062 Toulouse, France

Search for more papers by this author

, and

THI BICH NGOC MAC

Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, F-31062 Toulouse, France

CNRS, Institut de Mathématiques de Toulouse UMR 5219, F-31062 Toulouse, France

Search for more papers by this author

https://doi.org/10.1142/S0218202513400095Cited by:34 (Source: Crossref)

Abstract

We consider an Individual-Based Model for self-rotating particles interacting through local alignment and investigate its macroscopic limit. This model describes self-propelled particles moving in the plane and trying to synchronize their rotation motion with their neighbors. It combines the Kuramoto model of synchronization and the Vicsek model of swarm formation. We study the mean-field kinetic and hydrodynamic limits of this system within two different scalings. In the small angular velocity regime, the resulting model is a slight modification of the "Self-Organized Hydrodynamic" model which has been previously introduced by the first author. In the large angular velocity case, a new type of hydrodynamic model is obtained. A preliminary study of the linearized stability is proposed.

Keywords:

AMSC: 35Q80, 35L60, 82C22, 82C70, 92D50

References

J. A. Acebrónet al., Rev. Mod. Phys. 77, 137 (2005). Crossref, Web of Science, Google Scholar
M. Aldanaet al., Phys. Rev. Lett. 98, 095702 (2007), DOI: 10.1103/PhysRevLett.98.095702. Crossref, Web of Science, Google Scholar
I. Aoki, Bull. Jpn. Soc. Sci. Fisheries 48, 1081 (1982), DOI: 10.2331/suisan.48.1081. Crossref, Web of Science, Google Scholar
A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discr. Contin. Dynam. Syst. Ser. B, to appear . Google Scholar
A. Baskaran and M. C. Marchetti, Phys. Rev. E 77, 011920 (2008), DOI: 10.1103/PhysRevE.77.011920. Crossref, Web of Science, Google Scholar
A. Baskaran and M. C. Marchetti, J. Stat. Mech. Theory Exp. P04019 (2010). Google Scholar
N. Bellomo and J. Soler, Math. Models Methods Appl. Sci. 22, 1140006 (2012), DOI: 10.1142/S0218202511400069. Link, Web of Science, Google Scholar
E. Bertin, M. Droz and G. Grégoire, J. Phys. A: Math. Theor. 42, 445001 (2009), DOI: 10.1088/1751-8113/42/44/445001. Crossref, Web of Science, Google Scholar
L. Bertini, G. Giacomin and K. Pakdaman, J. Statist. Phys. 138, 270 (2010), DOI: 10.1007/s10955-009-9908-9. Crossref, Web of Science, Google Scholar
L. Bertini, G. Giacomin and C. Poquet, Synchronization and random long time dynamics for mean-field plane rotators, preprint , arXiv:1209.4537 . Google Scholar
F. Bolley, J. A. Cañizo and J. A. Carrillo, Appl. Math. Lett. 25, 339 (2011), DOI: 10.1016/j.aml.2011.09.011. Crossref, Web of Science, Google Scholar
F. Bolley, J. A. Cañizo and J. A. Carrillo, Math. Models Methods Appl. Sci. 21, 2179 (2011), DOI: 10.1142/S0218202511005702. Link, Web of Science, Google Scholar
J. A. Carrilloet al., SIAM J. Math. Anal. 42, 218 (2010), DOI: 10.1137/090757290. Crossref, Web of Science, Google Scholar
H. Chatéet al., Phys. Rev. E 77, 046113 (2008). Crossref, Web of Science, Google Scholar
Z. Chen and H. T. Zhang, Automatica 47, 1929 (2011), DOI: 10.1016/j.automatica.2011.03.012. Crossref, Web of Science, Google Scholar
A. A. Chepizhko and V. L. Kulinskii, Physica A 389, 5347 (2010), DOI: 10.1016/j.physa.2010.08.016. Crossref, Web of Science, Google Scholar
Y.-L. Chuanget al., Physica D 232, 33 (2007), DOI: 10.1016/j.physd.2007.05.007. Crossref, Web of Science, Google Scholar
I. D. Couzinet al., J. Theor. Biol. 218, 1 (2002), DOI: 10.1006/jtbi.2002.3065. Crossref, Web of Science, Google Scholar
F. Cucker and S. Smale, IEEE Trans. Automatic Control 52, 852 (2007), DOI: 10.1109/TAC.2007.895842. Crossref, Web of Science, Google Scholar
A. Cziròket al., Phys. Rev. E 54, 1791 (1996). Crossref, Web of Science, Google Scholar
A. Cziròk and T. Vicsek, Physica A 281, 17 (2000). Crossref, Web of Science, Google Scholar
P. Degond, A. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., appeared online . Google Scholar
P. Degond , A. Frouvelle and J.-G. Liu , A note on phase transitions for the Smoluchowski equation with dipolar potential , Hyp2012 —The 14th Int. Conf. on Hyperbolic Problems ( 2012 ) , arXiv:1212.3920 . Google Scholar
P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics , arXiv:1304.2929 . Google Scholar
P. Degond, A. Frouvelle, J.-G. Liu, S. Motsch and L. Navoret, Macroscopic models of collective motion and self-organization , arXiv:1304.6040 . Google Scholar
P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., to appear . Google Scholar
P. Degond and S. Motsch, Math. Models Methods Appl. Sci. 18, 1193 (2008), DOI: 10.1142/S0218202508003005. Link, Web of Science, Google Scholar
P. Degond and S. Motsch, J. Statist. Phys. 131, 989 (2008), DOI: 10.1007/s10955-008-9529-8. Crossref, Web of Science, Google Scholar
P. Degond and S. Motsch, J. Statist. Phys. 143, 685 (2011), DOI: 10.1007/s10955-011-0201-3. Crossref, Web of Science, Google Scholar
J. Dunstanet al., Phys. Fluids 24, 011901 (2012), DOI: 10.1063/1.3676245. Crossref, Web of Science, Google Scholar
W. M. Durham, E. Climent and R. Stocker, Phys. Rev. Lett. 106, 238102 (2011), DOI: 10.1103/PhysRevLett.106.238102. Crossref, Web of Science, Google Scholar
W. M. Durham, J. O. Kessler and R. Stocker, Science 323, 1067 (2009), DOI: 10.1126/science.1167334. Crossref, Web of Science, Google Scholar
M. Fornasier, J. Haskovec and G. Toscani, Physica D 240, 21 (2011), DOI: 10.1016/j.physd.2010.08.003. Crossref, Web of Science, Google Scholar
A. Frouvelle, Math. Models Methods Appl. Sci. 22, 1250011 (2012), DOI: 10.1142/S021820251250011X. Link, Web of Science, Google Scholar
A. Frouvelle and J.-G. Liu, SIAM J. Math. Anal. 44, 791 (2012), DOI: 10.1137/110823912. Crossref, Web of Science, Google Scholar
J. Gautraiset al., PLoS Comput. Biol. 8, e1002678 (2012), DOI: 10.1371/journal.pcbi.1002678. Crossref, Web of Science, Google Scholar
J. Gautraiset al., J. Math. Biol. 58, 429 (2009), DOI: 10.1007/s00285-008-0198-7. Crossref, Web of Science, Google Scholar
G. Giacomin, E. Luçon and C. Poquet, Coherence stability and effect of random natural frequencies in populations of coupled oscillators , arXiv:1111.3581 . Google Scholar
G. Giacominet al., SIAM J. Math. Anal. 44, 4165 (2012), DOI: 10.1137/110846452. Crossref, Web of Science, Google Scholar
G. Grégoire and H. Chaté, Phys. Rev. Lett. 92, 025702 (2004). Crossref, Web of Science, Google Scholar
S.-Y. Haet al., Quart. Appl. Math. 69, 91 (2011). Crossref, Web of Science, Google Scholar
S.-Y. Ha and J.-G. Liu, Commun. Math. Sci. 7, 297 (2009), DOI: 10.4310/CMS.2009.v7.n2.a2. Crossref, Web of Science, Google Scholar
S.-Y. Ha and E. Tadmor, Kinetic Relat. Models 1, 415 (2008). Crossref, Web of Science, Google Scholar
S. Henkes, Y. Fily and M. C. Marchetti, Phys. Rev. E 84, 040301 (2011), DOI: 10.1103/PhysRevE.84.040301. Crossref, Web of Science, Google Scholar
E. P. Hsu , Stochastic Analysis on Manifolds , Graduate Series in Mathematics ( Amer. Math. Soc. , 2002 ) . Crossref, Google Scholar
G. B. Jeffrey, Proc. Roy. Soc. London Ser A. 102, 161 (1922). Crossref, Google Scholar
Y. Kuramoto , Chemical Oscillations, Waves, and Turbulence ( Springer , 1984 ) . Crossref, Google Scholar
E. Laugaet al., Biophys. J. 90, 400 (2006), DOI: 10.1529/biophysj.105.069401. Crossref, Web of Science, Google Scholar
E. Lucon, Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model , arXiv:1204.2176 . Google Scholar
A. Mogilneret al., J. Math. Biol. 47, 353 (2003), DOI: 10.1007/s00285-003-0209-7. Crossref, Web of Science, Google Scholar
S. Motsch and L. Navoret, Multiscale Model. Simulat. 9, 1253 (2011), DOI: 10.1137/100794067. Crossref, Web of Science, Google Scholar
S. Motsch and E. Tadmor, J. Statist. Phys. 144, 923 (2011), DOI: 10.1007/s10955-011-0285-9. Crossref, Web of Science, Google Scholar
D. A. Paleyet al., IEEE Control Syst. Magn. 27, 89 (2007), DOI: 10.1109/MCS.2007.384123. Crossref, Web of Science, Google Scholar
F. Peruani, A. Deutsch and M. Bär, Phys. Rev. E 74, 030904 (2006), DOI: 10.1103/PhysRevE.74.030904. Crossref, Web of Science, Google Scholar
F. Peruani, E. M. Nicola and L. G. Morelli, New J. Phys. 12, 093029 (2010), DOI: 10.1088/1367-2630/12/9/093029. Crossref, Web of Science, Google Scholar
V. I. Ratushnayaet al., Physica A 381, 39 (2007), DOI: 10.1016/j.physa.2007.03.045. Crossref, Web of Science, Google Scholar
I. H. Riedel, K. Kruse and J. Howard, Science 309, 300 (2005), DOI: 10.1126/science.1110329. Crossref, Web of Science, Google Scholar
R. Sepulchre, D. A. Paley and N. Ehrich Leonard, IEEE Trans. Automat. Control 53, 706 (2008), DOI: 10.1109/TAC.2008.919857. Crossref, Web of Science, Google Scholar
J. Toner and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995), DOI: 10.1103/PhysRevLett.75.4326. Crossref, Web of Science, Google Scholar
J. Toner, Y. Tu and S. Ramaswamy, Ann. Phys. 318, 170 (2005), DOI: 10.1016/j.aop.2005.04.011. Crossref, Web of Science, Google Scholar
Y. Tu, J. Toner and M. Ulm, Phys. Rev. Lett. 80, 4819 (1998), DOI: 10.1103/PhysRevLett.80.4819. Crossref, Web of Science, Google Scholar
T. Vicseket al., Phys. Rev. Lett. 75, 1226 (1995), DOI: 10.1103/PhysRevLett.75.1226. Crossref, Web of Science, Google Scholar
T. Vicsek and A. Zafeiris, Phys. Rep. 517, 71 (2012), DOI: 10.1016/j.physrep.2012.03.004. Crossref, Web of Science, Google Scholar