Advanced Search
Article Contents
Article Contents

Phase transition and diffusion among socially interacting self-propelled agents

Abstract / Introduction Related Papers Cited by
  • We consider a hydrodynamic model of swarming behavior derived from the kinetic description of a particle system combining a noisy Cucker-Smale consensus force and self-propulsion. In the large self-propulsive force limit, we provide evidence of a phase transition from disordered to ordered motion which manifests itself as a change of type of the limit model (from hyperbolic to diffusive) at the crossing of a critical noise intensity. In the hyperbolic regime, the resulting model, referred to as the `Self-Organized Hydrodynamics (SOH)', consists of a system of compressible Euler equations with a speed constraint. We show that the range of SOH models obtained by this limit is restricted. To waive this restriction, we compute the Navier-Stokes diffusive corrections to the hydrodynamic model. Adding these diffusive corrections, the limit of a large propulsive force yields unrestricted SOH models and offers an alternative to the derivation of the SOH using kinetic models with speed constraints.
    Mathematics Subject Classification: 35L60, 35K55, 35Q80, 82C05, 82C22, 82C70, 92D50.


    \begin{equation} \\ \end{equation}
  • [1]

    I. Aoki, A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982), 1081-1088.doi: 10.2331/suisan.48.1081.


    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdrakovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237.doi: 10.1073/pnas.0711437105.


    A. B. T. Barbaro, J. A. Cañizo, J. A. Carrillo and P. Degond, Phase transitions in a Cucker-Smale model with self-propulsion, in preparation.


    A. B. T. Barbaro, K. Taylor, P. F. Trethewey, L. Youseff and B. Birnir, Discrete and continuous models of the dynamics of pelagic fish: Applications to the capelin, Mathematics and Computers in Simulation, 79 (2009), 3397-3414.doi: 10.1016/j.matcom.2008.11.018.


    A. Barbaro, B. Einarsson, B. Birnir, S. Sigurdsson, H. Valdimarsson, O. K. Pálsson, S. Sveinbjornsson and Th. Sigurdsson, Modelling and simulations of the migration of pelagic fish, ICES Journal of Marine Science, 66 (2009), 826-838.doi: 10.1093/icesjms/fsp067.


    Ch. Becco, N. Vandewalle, J. Delcourt and P. Poncin, Experimental evidences of a structural and dynamical transition in fish schools, Physica A, 367 (2006), 487-793.doi: 10.1016/j.physa.2005.11.041.


    E. Bertin, M. Droz and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E, 74 (2006), 022101.doi: 10.1103/PhysRevE.74.022101.


    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.doi: 10.1088/1751-8113/42/44/445001.


    A. L. Bertozzi, J. A. Carrillo and T. Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity, 22 (2009), 683-710.doi: 10.1088/0951-7715/22/3/009.


    M. Bostan and J. A. Carrillo, Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming, Math. Models Methods Appl. Sci., 23 (2013), 2353-2393.doi: 10.1142/S0218202513500346.


    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.doi: 10.1142/S0218202511005702.


    F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339-343.doi: 10.1016/j.aml.2011.09.011.


    H. Brézis, Analyse Fonctionnelle. Thèorie et Applications, Masson, Paris, 1983.


    J. Buhl, D. J. T. Sumpter, I. D. Couzin, J. J. Hale, E. Despland, E. R. Miller and S. J. Simpson, From disorder to order in marching locusts, Science, 312 (2006), 1402-1406.doi: 10.1126/science.1125142.


    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.doi: 10.1137/090757290.


    J. A. Carrillo, A. Klar, S. Martin and S. Tiwari, Self-propelled interacting particle systems with roosting force, Math. Models Methods Appl. Sci., 20 (2010), 1533-1552.doi: 10.1142/S0218202510004684.


    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.doi: 10.1016/j.physd.2007.05.007.


    I. D. Couzin and N. R. Franks., Self-organized lane formation and optimized traffic flow in army ants, Proceedings of the Royal Society B: Biological Sciences, 270 (2003), 139-146.doi: 10.1098/rspb.2002.2210.


    I. D. Couzin and J. Krause, Self-organization and collective behavior in vertebrates, Advance in the Study of Behavior, 32 (2003), 1-74.doi: 10.1016/S0065-3454(03)01001-5.


    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.doi: 10.1006/jtbi.2002.3065.


    F. Cucker and E. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008), 278-296.doi: 10.1016/j.matpur.2007.12.002.


    F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.doi: 10.1109/TAC.2007.895842.


    P. Degond, Macroscopic limits of the Boltzmann equation: A review, in Modeling and Computational Methods for Kinetic Equations, (eds. P. Degond, L. Pareschi, and G. Russo), Birkhäuser Boston, (2004), 3-57.


    P. Degond, A. Frouvelle and J.-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonl. Sci., 23 (2013), 427-456.doi: 10.1007/s00332-012-9157-y.


    P. Degond, J-G. Liu, S. Motsch and V. Panferov, Hydrodynamic models of self-organized dynamics: Derivation and existence theory, Methods Appl. Anal., 20 (2013), 89-114.doi: 10.4310/MAA.2013.v20.n2.a1.


    P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Mathematical Models and Methods in Applied Sciences, 18 (2008), 1193-1215.doi: 10.1142/S0218202508003005.


    P. Degond, L. Navoret, R. Bon and D. Sanchez, Congestion in a macroscopic model of self-driven particles modeling gregariousness, J. Stat. Phys., 138 (2010), 85-125.doi: 10.1007/s10955-009-9879-x.


    P. Degond and T. Yang, Diffusion in a continuum model of self-propelled particles with alignment interaction, Mathematical Models and Methods in Applied Sciences, 20 (2010), 1459-1490.doi: 10.1142/S0218202510004659.


    J. Deseigne, O. Dauchot and H. Chaté, Collective motion of vibrated polar disks, Phys. Rev. Lett., 105 (2010), 098001.doi: 10.1103/PhysRevLett.105.098001.


    M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. Chayes, Self-propelled particles with soft-core interactions: patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.doi: 10.1103/PhysRevLett.96.104302.


    R. Eftimie, G. de Vries and M. A. Lewis, Complex spatial group patterns result from different animal communication mechanisms, Proc. Natl. Acad. Sci. USA, 104 (2007), 6974-6979.doi: 10.1073/pnas.0611483104.


    M. Fornasier, J. Haskovec and G. Toscani, Fluid dynamic description of flocking via the Povzner-Boltzmann equation, Physica D, 240 (2011), 21-31.doi: 10.1016/j.physd.2010.08.003.


    A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Models Methods Appl. Sci., 22 (2012), 1250011-1250051.doi: 10.1142/S021820251250011X.


    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.doi: 10.1137/110823912.


    S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.doi: 10.4310/CMS.2009.v7.n2.a9.


    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.doi: 10.4310/CMS.2009.v7.n2.a2.


    S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008), 415-435.doi: 10.3934/krm.2008.1.415.


    C. K. Hemelrijk and H. Hildenbrandt, Some causes of the variable shape of flocks of birds, PLOS ONE, 6 (2011), e22479.doi: 10.1371/journal.pone.0022479.


    Y. Katz, K. Tunstrom, C. C. Ioannou, C. Huepe and I. D. Couzin, Inferring the structure and dynamics of interactions in schooling fish, Proc. Nat. Acad. Sci., 108 (2011), 18720-18725.doi: 10.1073/pnas.1107583108.


    R. Lukeman, Y.-X. Li and L. Edelstein-Keshet, Inferring individual rules from collective behavior, Proc. Nat. Acad. Sci. USA, 107 (2010), 12576-12580.doi: 10.1073/pnas.1001763107.


    A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.doi: 10.1007/s002850050158.


    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.doi: 10.1007/s00285-003-0209-7.


    S. Mishra, A. Baskaran and C. Marchetti, Fluctuations and pattern formation in self-propelled particles, Phys. Rev. E, 81 (2010), 061916.doi: 10.1103/PhysRevE.81.061916.


    S. Motsch and L. Navoret, Numerical simulations of a nonconservative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Modeling and Simulation, 9 (2011), 1253-1275.doi: 10.1137/100794067.


    S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947.doi: 10.1007/s10955-011-0285-9.


    V. I. Ratushnaya, D. Bedeaux, V. L. Kulinskii and A. V. Zvelindovsky, Collective behavior of self-propelling particles with kinematic constraints: the relation between the discrete and the continuous description, Physica A, 381 (2007), 39-46.doi: 10.1016/j.physa.2007.03.045.


    J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 68 (2007/08), 694-719. doi: 10.1137/060673254.


    N. J. Suematsu, S. Nakata, A. Awazu and H. Nishimori, Collective behavior of inanimate boats, Phys. Rev. E, 81 (2010), 056210.doi: 10.1103/PhysRevE.81.056210.


    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.doi: 10.1103/PhysRevLett.75.4326.


    J. Toner, Y. Tu and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics, 318 (2005), 170-244.doi: 10.1016/j.aop.2005.04.011.


    C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math, 65 (2004), 152-174.doi: 10.1137/S0036139903437424.


    C. M. Topaz, A. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.doi: 10.1007/s11538-006-9088-6.


    Y. Tu, J. Toner and M. Ulm, Sound waves and the absence of Galilean invariance in flocks, Phys. Rev. Lett., 80 (1998), 4819-4822.doi: 10.1103/PhysRevLett.80.4819.


    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.doi: 10.1103/PhysRevLett.75.1226.


    T. Vicsek and A. Zafeiris, Collective motion, Physics Reports, 517 (2012), 71-140.


    M. Yamao, H. Naoki and S. Ishii, Multi-cellular logistics of collective cell migration, PLoS ONE, 6 (2011), e27950.doi: 10.1371/journal.pone.0027950.

  • 加载中

Article Metrics

HTML views() PDF downloads(130) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint