Hydrodynamics of the Kuramoto-Vicsek model of rotating self-propelled particles

Mathematical Models and Methods in Applied Sciences

Volumn 24, Issue 2, 2014, Pages 277-325

  Degond, Pierre ^a,b   Dimarco, Giacomo ^a,b   Mac, Thi Bich Ngoc ^a,b  

^a INSA   (France)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Alignment; Dispersion relation; Fokker Planck equation; Generalized collision invariant; Macroscopic limit; Order parameter; Von Mises Fisher distribution

Indexed keywords

COLLISION INVARIANTS; DISPERSION RELATIONS; MACROSCOPIC LIMIT; ORDER PARAMETER; VON MISES-FISHER DISTRIBUTION;

ALIGNMENT; DISPERSIONS; HYDRODYNAMICS; SYNCHRONIZATION;

ANGULAR VELOCITY;

EID: 84890541198     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202513400095     Document Type: Article

References (63)

1. J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77 (2005) 137-185.
2. M. Aldana, V. Dossetti, C. Huepe, V. M. Kenkre and H. Larralde, Phase transitions in systems of self-propelled agents and related network models, Phys. Rev. Lett. 98 (2007) 095702.
3. I. Aoki, A simulation study on the schooling mechanism in fish, Bull. Jpn. Soc. Sci. Fisheries 48 (1982) 1081-1088.
4. A. Barbaro and P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discr. Contin. Dynam. Syst. Ser. B, to appear.
5. A. Baskaran and M. C. Marchetti, Hydrodynamics of self-propelled hard rods, Phys. Rev. E 77 (2008) 011920.
6. A. Baskaran and M. C. Marchetti, Nonequilibrium statistical mechanics of selfpropelled hard rods, J. Stat. Mech. Theory Exp. (2010) P04019.
7. N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci. 22 (2012) 1140006.
8. E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor. 42 (2009) 445001.
9. L. Bertini, G. Giacomin and K. Pakdaman, Dynamical aspects of mean field plane rotators and the Kuramoto model, J. Statist. Phys. 138 (2010) 270-290.
10. L. Bertini, G. Giacomin and C. Poquet, Synchronization and random long time dynamics for mean-field plane rotators, preprint arXiv:1209.4537.
11. F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett. 25 (2011) 339-343.
12. F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci. 21 (2011) 2179-2210.
13. J. A. Carrillo, M. Fornasier, J. Rosado and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal. 42 (2010) 218-236.
14. H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E 77 (2008) 046113.
15. Z. Chen and H. T. Zhang, No-beacon collective circular motion of jointly connected multi-agents, Automatica 47 (2011) 1929-1937.
16. A. A. Chepizhko and V. L. Kulinskii, On the relation between Vicsek and Kuramoto models of spontaneous synchronization, Physica A 389 (2010) 5347-5352.
17. Y.-L. Chuang, M. R. D'Orsogna, D. Marthaler, A. L. Bertozzi and L. S. Chayes, State transitions and the continuum limit for a 2D interacting, self-propelled particle system, Physica D 232 (2007) 33-47.
18. I. D. Couzin, J. Krause, R. James, G. D. Ruxton and N. R. Franks, Collective memory and spatial sorting in animal groups, J. Theor. Biol. 218 (2002) 1-11.
19. F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automatic Control 52 (2007) 852-862.
20. A. Cziròk, E. Ben-Jacob, I. Cohen and T. Vicsek, Formation of complex bacterial colonies via self-generated vortices, Phys. Rev. E 54 (1996) 1791-18091.
21. A. Cziròk and T. Vicsek, Collective behavior of interacting self-propelled particles, Physica A 281 (2000) 17-29.
22. 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.
23. P. Degond, A. Frouvelle and J.-G. Liu, A note on phase transitions for the Smoluchowski equation with dipolar potential, Proc. of Hyp2012 - The 14th Int. Conf. on Hyperbolic Problems, Padova, Italy, 2012, arXiv:1212.3920.
24. P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, arXiv:1304.2929.
25. P. Degond, A. Frouvelle, J.-G. Liu, S. Motsch and L. Navoret, Macroscopic models of collective motion and self-organization, arXiv:1304.6040.
26. P. Degond, J.-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of selforganized dynamics: Derivation and existence theory, Methods Appl. Anal., to appear.
27. P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci. 18 (2008) 1193-1215.
28. P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Statist. Phys. 131 (2008) 989-1021.
29. P. Degond and S. Motsch, A macroscopic model for a system of swarming agents using curvature control, J. Statist. Phys. 143 (2011) 685-714.
30. J. Dunstan, G. Miño, E. Clement and R. Soto, A two-sphere model for bacteria swimming near solid surfaces, Phys. Fluids 24 (2012) 011901.
31. W. M. Durham, E. Climent and R. Stocker, Gyrotaxis in a steady vortical flow, Phys. Rev. Lett. 106 (2011) 238102.
32. W. M. Durham, J. O. Kessler and R. Stocker, Disruption of vertical motility by shear triggers formation of thin phytoplankton layers, Science 323 (2009) 1067-1070.
33. M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via the Povzner-Boltzmann equation, Physica D 240 (2011) 21-31.
34. A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci. 22 (2012) 1250011.
35. A. Frouvelle and J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math. Anal. 44 (2012) 791-826.
36. J. Gautrais, F. Ginelli, R. Fournier, S. Blanco, M. Soria, H. Chaté and G. Theraulaz, Deciphering interactions in moving animal groups, PLoS Comput. Biol. 8 (2012) e1002678.
37. J. Gautrais, C. Jost, M. Soria, A. Campo, S. Motsch, R. Fournier, S. Blanco and G. Theraulaz, Analyzing fish movement as a persistent turning walker, J. Math. Biol. 58 (2009) 429-445.
38. G. Giacomin, E. Lu̧con and C. Poquet, Coherence stability and effect of random natural frequencies in populations of coupled oscillators, arXiv:1111.3581.
39. G. Giacomin, K. Pakdmana, X. Pellegrin and C. Poquet, Transitions in active rotator systems: Invariant hyperbolic manifold approach, SIAM J. Math. Anal. 44 (2012) 4165-4194.
40. G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett. 92 (2004) 025702.
41. S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math. 69 (2011) 91-103.
42. S.-Y. Ha and J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 7 (2009) 297-325.
43. S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic Relat. Models 1 (2008) 415-435.
44. S. Henkes, Y. Fily and M. C. Marchetti, Active jamming: Self-propelled soft particles at high density, Phys. Rev. E 84 (2011) 040301.
45. E. P. Hsu, Stochastic Analysis on Manifolds, Graduate Series in Mathematics (Amer. Math. Soc. 2002).
46. G. B. Jeffrey, The motion of ellipsoidal particles immersed in a viscous fluid, Proc. Roy. Soc. London Ser A. 102 (1922) 161-179.
47. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, 1984).
48. E. Lauga, W. R. DiLuzio, G. M. Whitesides and H. A. Stone, Swimming in circles: Motion of bacteria near solid boundaries, Biophys. J. 90 (2006) 400-412.
49. E. Lu̧con, Large time asymptotics for the fluctuation SPDE in the Kuramoto synchronization model, arXiv:1204.2176.
50. A. Mogilner, L. Edelstein-Keshet, L. Bent and A. Spiros, Mutual interactions, potentials, and individual distance in a social aggregation, J. Math. Biol. 47 (2003) 353-389.
51. S. Motsch and L. Navoret, Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Model. Simulat. 9 (2011) 1253-1275.
52. S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Statist. Phys. 144 (2011) 923-947.
53. D. A. Paley, N. Ehrich Leonard, R. Sepulchre, D. Grünbaum and J. J. Parrish, Oscillator models and collective motion, IEEE Control Syst. Magn. 27 (2007) 89-105.
54. F. Peruani, A. Deutsch and M. Bär, Nonequilibrium clustering of self-propelled rods, Phys. Rev. E 74 (2006) 030904.
55. F. Peruani, E. M. Nicola and L. G. Morelli, Mobility induces global synchronization of oscillators in periodic extended systems, New J. Phys. 12 (2010) 093029.
56. V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii and A. V. Zvelindovsky, Collective behavior of self propelling particles with kinematic constraints: The relations between the discrete and the continuous description, Physica A 381 (2007) 39-46.
57. I. H. Riedel, K. Kruse and J. Howard, A self-organized vortex array of hydrodynamically entrained sperm cells, Science 309 (2005) 300-303.
58. R. Sepulchre, D. A. Paley and N. Ehrich Leonard, Stabilization of planar collective motion with limited communication, IEEE Trans. Automat. Control 53 (2008) 706-719.
59. J. Toner and Y. Tu, Flocks, Long-range order in a two-dimensional dynamical XY model: How birds fly together, Phys. Rev. Lett. 75 (1995) 4326-4329.
60. J. Toner, Y. Tu and S. Ramaswamy, Hydrodynamics and phases of flocks, Ann. Phys. 318 (2005) 170-244.
61. Y. Tu, J. Toner and M. Ulm, Sound waves and the absence of Galilean invariance in flocks, Phys. Rev. Lett. 80 (1998) 4819-4822.
62. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (1995) 1226-1229.
63. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517 (2012) 71-140.