Abstract
In these lecture notes, prepared for the Microswimmers Summer School 2015 at Forschungszentrum Jülich, I discuss the well known Vicsek model for collective motion and its main properties. In particular, I discuss its algorithmic implementation and the basic properties of its universality class. I present results from numerical simulations and insist on the role played by symmetries and conservation laws. Analytical arguments are presented in an accessible and simplified way, but ample references are given for more advanced readings.
Article PDF
Similar content being viewed by others
References
Animal Groups in Three Dimensions, edited by J.K. Parrish, W.M. Hamner (Cambridge University Press, Cambridge, U.K., 1997)
F. Ginelli, et al., Proc. Natl. Acad. Sci. USA 112, 12729 (2015)
H.P. Zhang, et al., Proc. Natl. Acad. Sci. USA 107, 13626 (2010)
X. Trepat, et al., Nature Phys. 5 426 (2009)
V. Schaller, et al., Nature 467, 73 (2010)
Pliny, in Natural History, translated by H. Rackham (Harvard University Press, 1968), Vol. III, book 10, xxxii, p. 63
S. Ramaswamy, Annu. Rev. Condens. Matter Phys. 1, 323 (2010)
J. Palacci, S. Sacanna, A. Preska Steinberg, D.J. Pine, P.M. Chaikin, Science 339, 936 (2013)
V. Narayan, S. Ramaswamy, N. Menon, Science 317, 105 (2007)
J. Desaigne, O. Dauchot, H. Chaté, Phys. Rev. Lett. 105, 098001 (2010)
D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions, Frontiers in Physics 47, XIX, 326 (W.A. Benjamin, Inc, 1975)
T. Vicsek, et al., Phys. Rev. Lett. 75, 1226 (1995)
G. Grégoire, H. Chaté, Phys. Rev. Lett. 92, 025702 (2004)
H. Chaté, et al., Phys. Rev. E 77, 046113 (2008)
R.G. Winkler, Eur. Phys. J. Special Topics 225, 2079 (2016)
A. Peshkov, E. Bertin, F. Ginelli, H. Chaté, Eur. Phys. J. Special Topics 223, 1315 (2014)
H. Chaté, F. Ginelli, G. Grégoire, Phys. Rev. Lett. 99, 229601 (2007)
E. Bertin, M. Droz, G. Grégoire, Phys. Rev. E 74, 022101 (2006)
E. Bertin, M. Droz, G. Grégoire, J. Phys. A 42, 445001 (2009)
P. Kraikivski, R. Lipowsky, J. Kierfeld, Phys. Rev. Lett. 96, 258103 (2006)
M. Abkenar, K. Marx, T. Auth, G. Gompper, Phys. Rev. E 88, 062314 (2014)
F. Peruani, Eur. Phys. J. Special Topics 225, 3001 (2016)
A.P. Solon, H. Chaté, J. Tailleur Phys. Rev. Lett. 114, 068101 (2015)
J-B. Caussin et al., Phys. Rev. Lett. 112, 148102 (2014)
F. Ginelli, H. Chaté, Phys. Rev. Lett. 105, 168103 (2010)
M. Ballerini, et al., Proc. Natl. Acad. Sci. USA 105, 1232 (2008)
J. Gautrais, et al., PLoS Comp. Bio., 8 e1002678 (2012)
A. Peshkov, et al., Phys. Rev. Lett. 109, 098101 (2012)
N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17, 1133 (1966)
J.M. Kosterlitz, D. Thouless, J. Phys. C 6, 1181 (1973)
J. Toner, unpublished. I am deeply indebited to John Toner for having introduced me to this simple but illuminating argument
J. Toner, Y. Tu, Phys. Rev. Lett. 75, 4326 (1995)
J. Toner, Y. Tu, Phys. Rev. E 58, 4828 (1998)
J. Toner, Phys. Rev. E 86, 031918 (2012)
J. Toner, Y. Tu, S. Ramaswamy, Ann. Phys. (Berlin) 318, 170 (2005)
V.L. Pokrovskii, A.Z. Patashinskii, Fluctuation Theory of Phase Transitions, 2nd edn. (Pergamon, Oxford, 1979; Nauka, Moscow, 1982)
A. Cavagna, et al., Proc. Natl. Acad. Sci. USA 107, 11865 (2010)
S. Ramaswamy, R.A. Simha, J. Toner, Europhys. Lett. 62, 196 (2003)
C.W. Reynolds, Computer Graphics 21, 25 (1987)
G. Grégoire, H. Chaté, Y. Tu, Physica D 181, 157 (2003)
D.J.G. Pearce, et al., Proc. Natl. Acad. Sci. USA 111, 10422 (2014)
F.D.C. Farrell, M.C. Marchetti, D. Marenduzzo, J. Tailleur, Phys. Rev. Lett. 108, 248101 (2012)
F. Ginelli, F. Peruani, M. Baer, H. Chaté, Phys. Rev. Lett. 104, 184502 (2010)
H. Chaté, F. Ginelli, R. Montagne, Phys. Rev. Lett. 96, 180602 (2006)
J. Toner, Phys. Rev. Lett. 108, 088102 (2012)
Y.-h. Tu, M. Ulm, J. Toner, Phys. Rev. Lett. 80, 4819 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ginelli, F. The Physics of the Vicsek model. Eur. Phys. J. Spec. Top. 225, 2099–2117 (2016). https://doi.org/10.1140/epjst/e2016-60066-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjst/e2016-60066-8