Macroscopic models of collective motion and self-organization
Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exposé no. 1, 27 p.

In this paper, we review recent developments on the derivation and properties of macroscopic models of collective motion and self-organization. The starting point is a model of self-propelled particles interacting with its neighbors through alignment. We successively derive a mean-field model and its hydrodynamic limit. The resulting macroscopic model is the Self-Organized Hydrodynamics (SOH). We review the available existence results and known properties of the SOH model and discuss it in view of its possible extensions to other kinds of collective motion.

@article{SLSEDP_2012-2013____A1_0,
author = {Degond, Pierre and Frouvelle, Amic and Liu, Jian-Guo and Motsch, Sebastien and Navoret, Laurent},
title = {Macroscopic models of collective motion and self-organization},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:1},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2012-2013},
doi = {10.5802/slsedp.32},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/slsedp.32/}
}
Degond, Pierre; Frouvelle, Amic; Liu, Jian-Guo; Motsch, Sebastien; Navoret, Laurent. Macroscopic models of collective motion and self-organization. Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exposé no. 1, 27 p. doi : 10.5802/slsedp.32. http://archive.numdam.org/articles/10.5802/slsedp.32/

[1] I. Aoki, A simulation study on the schooling mechanism in fish, Bulletin of the Japan Society of Scientific Fisheries, 48 (1982) 1081-1088.

[2] C. Appert-Rolland, P. Degond, S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw. Heterog. Media, 6 (2011) 351-381. | MR 2826750

[3] A. Aw, M. Rascle, Resurrection of second order models of traffic flow, SIAM J. Appl. Math., 60 (2000) 916-938 | MR 1750085 | Zbl 0957.35086

[4] P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality: an explanation of $1/f$ noise, Phys. Rev. Lett., 59 (1987) 381-384. | MR 949160 | Zbl 1230.37103

[5] A. Barbaro, P. Degond, Phase transition and diffusion among socially interacting self-propelled agents, Discrete Contin. Dyn. Syst. Ser. B, to appear.

[6] A. Baskaran, M. C. Marchetti, Nonequilibrium statistical mechanics of self-propelled hard rods, J. Stat. Mech. Theory Exp., (2010) P04019.

[7] S. Bazazi, J. Buhl, J. J. Hale, M. L. Anstey, G. A. Sword, S. J. Simpson, I. D. Couzin, Collective Motion and Cannibalism in Locust Migratory Bands, Current Biology 18 (2008) 735-739.

[8] F. Berthelin, P. Degond, M. Delitala, M. Rascle, A model for the formation and evolution of traffic jams, Arch. Rat. Mech. Anal., 187 (2008) 185-220. | MR 2366138 | Zbl 1153.90003

[9] F. Berthelin, P. Degond, V. Le Blanc, S. Moutari, J. Royer, M. Rascle, A Traffic-Flow Model with Constraints for the Modeling of Traffic Jams, Math. Models Methods Appl. Sci., 18 Suppl. (2008) 1269-1298. | MR 2438216 | Zbl 1197.35159

[10] 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.

[11] F. Bolley, J. A. Cañizo, J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2011) 339-343. | MR 2855983 | Zbl 1239.91127

[12] F. Brown, Micromagnetics, Wiley, New York, 1963.

[13] E. Carlen, R. Chatelin, P. Degond, and B Wennberg, Kinetic hierarchy and propagation of chaos in biological swarm models, Phys. D, appeared online.

[14] E. Carlen, P. Degond, and B Wennberg, Kinetic limits for pair-interaction driven master equations and biological swarm models, Math. Models Methods Appl. Sci., 23 (2013) 1339-1376.

[15] J. A. Carrillo, M. Fornasier, J. Rosado, G. Toscani, Asymptotic Flocking Dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42 (2010) 218-236. | MR 2596552 | Zbl 1223.35058

[16] H. Chaté, F. Ginelli, G. Grégoire, F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E 77 (2008) 046113 (15 p.)

[17] G. Q. Chen, C. D. Levermore, T. P. Liu, Hyperbolic conservation laws with stiff relaxation and entropy, Comm. Pure Appl. Math., 47 (1994) 787-830. | MR 1280989 | Zbl 0806.35112

[18] 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. | MR 2369988

[19] 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. | MR 2027139

[20] F. Cucker, Er. Mordecki, Flocking in noisy environments, J. Math. Pures Appl., 89 (2008) 278-296. | MR 2401690

[21] F. Cucker, S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007) 852-862. | MR 2324245

[22] A. Cziròk, E. Ben-Jacob, I. Cohen, T. Vicsek, Formation of complex bacterial colonies via self-generated vortices, Phys. Rev. E, 54 (1996) 1791-18091.

[23] P. Degond, M. Delitala, Modelling and simulation of vehicular traffic jam formation, Kinet. Relat. Models, 1 (2008) 279-293. | MR 2393278 | Zbl 1153.90356

[24] P. Degond, A. Frouvelle, J-G. Liu, Macroscopic limits and phase transition in a system of self-propelled particles, J. Nonlinear Sci., appeared online.

[25] P. Degond, A. Frouvelle, J.-G. Liu, A note on phase transitions for the Smoluchowski equation with dipolar potential, submitted.

[26] P. Degond, A. Frouvelle, J-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, preprint.

[27] P. Degond, J. Hua, Self-Organized Hydrodynamics with congestion and path formation in crowds, J. Comput. Phys., 237 (2013) 299-319.

[28] P. Degond, J. Hua, L. Navoret, Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., 230 (2011) 8057-8088. | MR 2835410

[29] P. Degond, J-G. Liu, Hydrodynamics of self-alignment interactions with precession and derivation of the Landau-Lifschitz-Gilbert equation, Math. Models Methods Appl. Sci., 22 Suppl. 1 (2012) 1140001 (18 pages). | MR 2974181

[30] P. Degond, J-G. Liu, S. Motsch, V. Panferov, Hydrodynamic models of self-organized dynamics: derivation and existence theory, Methods Appl. Anal., to appear.

[31] P. Degond, J.-G. Liu, C. Ringhofer, A Nash equilibrium macroscopic closure for kinetic models coupled with Mean-Field Games, submitted. arXiv:1212.6130.

[32] P. Degond, S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 Suppl. (2008) 1193-1215. | MR 2438213 | Zbl 1157.35492

[33] P. Degond, S. Motsch, Large scale dynamics of the Persistent Turning Walker model of fish behavior, J. Stat. Phys., 131 (2008) 989-1021. | MR 2407377 | Zbl 1214.82075

[34] P. Degond, S. Motsch, A macroscopic model for a system of swarming agents using curvature control, J. Stat. Phys., 143 (2011) 685-714 | MR 2800660 | Zbl 1222.82071

[35] P. Degond, L. Navoret, R. Bon, D. Sanchez, Congestion in a macroscopic model of self-driven particles modeling gregariousness, J. Stat. Phys., 138 (2010) 85-125. | Zbl 1187.82086

[36] P. Degond, T. Yang, Diffusion in a continuum model of self-propelled particles with alignment interaction, Math. Models Methods Appl. Sci., 20 Suppl. (2010) 1459-1490.

[37] M. L. Domeier, P. L. Colin, Tropical reef fish spawning aggregations: defined and reviewed, Bulletin of Marine Science, 60 (1997) 698-726.

[38] A. Frouvelle, A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters, Math. Mod. Meth. Appl. Sci., 22 (2012) 1250011 (40 p.). | MR 2924786 | Zbl 1241.35200

[39] A. Frouvelle, J.-G. Liu, Dynamics in a kinetic model of oriented particles with phase transition, SIAM J. Math. Anal., 44 (2012) 791-826. | MR 2914250 | Zbl 1248.35097

[40] J. Gautrais, C. Jost, M. Soria, A. Campo, S. Motsch, R. Fournier, S. Blanco, G. Theraulaz, Analyzing fish movement as a persistent turning walker, J. Math. Biol., 58 (2009) 429-445. | MR 2470196 | Zbl 1153.92038

[41] J. Gautrais, F. Ginelli, R. Fournier, S. Blanco, M. Soria, H. Chaté, G. Theraulaz, Deciphering interactions in moving animal groups. Plos Comput. Biol., 8 (2012) e1002678. | MR 2993806

[42] S. -Y. Ha, J.-G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7 (2009) 297-325. | MR 2536440 | Zbl 1177.92003

[43] S.-Y. Ha, E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinetic and Related Models, 1 (2008) 415-435. | MR 2425606

[44] E. P. Hsu, Stochastic Analysis on Manifolds, Graduate Series in Mathematics, American Mathematical Society, 2002. | MR 1882015 | Zbl 0994.58019

[45] A. Khuong, G. Theraulaz, C. Jost, A. Perna, J. Gautrais, A computational model of ant nest morphogenesis, in “Advances in Artificial Life, ECAL 2011 - Synthesis and Simulation of Living Systems”, MIT Press, 2011, pp. 404-411.

[46] P.L. LeFloch. Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form, Comm. Partial Differential Equations, 13 (1988) 669-727. | MR 934378 | Zbl 0683.35049

[47] 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. | MR 2024502 | Zbl 1054.92053

[48] J. Monod, Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology, Alfred A. Knopf, New York, 1971.

[49] S. Motsch, L. Navoret, Numerical simulations of a non-conservative hyperbolic system with geometric constraints describing swarming behavior, Multiscale Model. Simul., 9 (2011) 1253-1275. | MR 2846932 | Zbl 1251.35172

[50] S. Motsch, E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011) 923-947. | MR 2836613 | Zbl 1230.82037

[51] V. V. Rusanov, Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR 1 (1961) 267-279. | MR 147083

[52] J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM J. Appl. Math., 58 (2007) 694-719. | MR 2375291

[53] 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.

[54] J. Toner, Y. Tu and S. Ramaswamy, Hydrodynamics and phases of flocks, Annals of Physics, 318 (2005) 170-244 | MR 2148645 | Zbl 1126.82347

[55] Y. Tu, J. Toner and M. Ulm, Sound waves and the absence of Galilean invariance in flocks, Phys. Rev. Lett., 80 (1998) 4819-4822.

[56] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995) 1226-1229.

[57] T. Vicsek, A. Zafeiris, Collective motion, Phys. Rep., 517 (2012) 71-140.