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Flocking and Turning: a New Model for Self-organized Collective Motion

Abstract

Birds in a flock move in a correlated way, resulting in large polarization of velocities. A good understanding of this collective behavior exists for linear motion of the flock. Yet observing actual birds, the center of mass of the group often turns giving rise to more complicated dynamics, still keeping strong polarization of the flock. Here we propose novel dynamical equations for the collective motion of polarized animal groups that account for correlated turning including solely social forces. We exploit rotational symmetries and conservation laws of the problem to formulate a theory in terms of generalized coordinates of motion for the velocity directions akin to a Hamiltonian formulation for rotations. We explicitly derive the correspondence between this formulation and the dynamics of the individual velocities, thus obtaining a new model of collective motion. In the appropriate overdamped limit we recover the well-known Vicsek model, which dissipates rotational information and does not allow for polarized turns. Although the new model has its most vivid success in describing turning groups, its dynamics is intrinsically different from previous ones in a wide dynamical regime, while reducing to the hydrodynamic description of Toner and Tu at very large length-scales. The derived framework is therefore general and it may describe the collective motion of any strongly polarized active matter system.

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References

  1. 1.

    Camazine, S., Deneubourg, J.-L., Franks, N.R., Sneyd, J., Theraulaz, G., Bonabeau, E.: Self-Organization in Biological Systems. Princeton University Press, Princeton (2001)

    Google Scholar 

  2. 2.

    Couzin, I.D.: Self-organization and collective behavior in vertebrates. J. Adv. Study Behav. 32, 1–75 (2003)

    Article  Google Scholar 

  3. 3.

    Giardina, I.: Collective behavior in animal groups: theoretical models and empirical studies. HFSP J. 2, 205–219 (2008)

    Article  Google Scholar 

  4. 4.

    Sumpter, D.J.T.: Collective Animal Behavior. Princeton University Press, Princeton (2010)

    Book  MATH  Google Scholar 

  5. 5.

    Cavagna, A., Giardina, I.: Bird Flocks as Condensed Matter. Ann. Rev. Cond. Matt. Phys. (2014). doi:10.1146/annurev-conmatphys-031113-133834

  6. 6.

    Aoki, I.: A simulation study on the schooling mechanism in fish. Bull. Jpn. Soc. Sci. Fish. 48, 1081–1088 (1982)

    Article  Google Scholar 

  7. 7.

    Reynolds, C.W.: Flocks, herds, and schools: a distributed behavioral model. Comput. Gr. 21, 25–33 (1987)

    Article  Google Scholar 

  8. 8.

    Huth, A., Wissel, C.: The simulation of the movement of fish schools. J. Theor. Biol. 156, 365–385 (1992)

    Article  Google Scholar 

  9. 9.

    Couzin, I.D., Krause, J., James, R., Ruxton, G.D., Franks, N.R.: Collective memory and spatial sorting in animal groups. J. Theor. Biol. 218, 1–11 (2002)

    Article  MathSciNet  Google Scholar 

  10. 10.

    Hildenbrandt, H., Carere, C., Hemelrijk, C.: Self-organized aerial displays of thousands of starlings: a model. Behav. Ecol. 21, 1349–1359 (2010)

    Article  Google Scholar 

  11. 11.

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

    ADS  Article  Google Scholar 

  12. 12.

    Toner, J., Tu, Y.: Long-range order in a two-dimensional dynamical XY model: how birds fly together Phys. Rev. Lett. 75, 4326–4329 (1995)

    ADS  Article  Google Scholar 

  13. 13.

    Grégoire, G., Chaté, H., Tu, Y.: Moving and staying together without a leader. Physica D 181, 157–170 (2003)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Grégoire, G., Chaté, H.: Onset of collective and cohesive motion. Phys. Rev. Lett. 92, 025702 (2004)

    ADS  Article  Google Scholar 

  15. 15.

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

    ADS  Article  Google Scholar 

  16. 16.

    Aldana, A., Dossetti, V., Huepe, C., Kenkre, V.M., Larralde, H.: Phase transitions in systems of self-propelled agents and related network models. Phys. Rev. Lett. 98, 095702 (2007)

    ADS  Article  Google Scholar 

  17. 17.

    Ginelli, F., Chaté, H.: Relevance of metric-free interactions in flocking phenomena. Phys. Rev. Lett. 105, 168103 (2010)

    ADS  Article  Google Scholar 

  18. 18.

    Justh, E.W., Krishnaprasad, P.S.: Equilibria and steering laws for planar formations. Syst. Controls Lett. 52, 25–38 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Tanner, H.G., Jadbabaie, A., Pappas, G.J.: Flocking in fixed and switching networks. IEEE Trans. Autom. Control 52, 863–868 (2007)

    Article  MathSciNet  Google Scholar 

  20. 20.

    Toner, J., Tu, Y.: Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E 58, 4828–4858 (1998)

    ADS  Article  MathSciNet  Google Scholar 

  21. 21.

    Vicsek, T., Zafeiris, A.: Collective motion. Phys. Rep. 517, 71–140 (2012)

    ADS  Article  Google Scholar 

  22. 22.

    Ramaswamy, S.: The mechanics and statistics of active matter. Ann. Rev. Cond. Matt. Phys. 1, 301 (2010)

    Article  Google Scholar 

  23. 23.

    Marchetti, M.C., Joanny, J.F., Ramaswamy, S., Liverpool, T.P., Prost, J., et al.: Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143 (2013)

    ADS  Article  Google Scholar 

  24. 24.

    Attanasi, A., Cavagna, A., Del Castello, L., Giardina, I., Jelic, A., Melillo, S., Parisi, L., Shen, E., Viale, M.: Information transfer and behavioural inertia in starling flocks. Nat. Phys. 10, 691–696 (2014)

    Article  Google Scholar 

  25. 25.

    Bialek, W., Cavagna, A., Giardina, I., Mora, T., Silvestri, E., Viale, M., Walczak, A.M.: Statistical mechanics for natural flocks of birds. Proc. Natl. Acad. Sci. USA 109, 4786–4791 (2012)

    ADS  Article  Google Scholar 

  26. 26.

    Cavagna, A., Cimarelli, A., Giardina, I., Parisi, G., Santagati, R., Stefanini, F., Viale, M.: Scale-free correlations in starling flocks. Proc. Natl. Acad. Sci. USA 107, 11865–11870 (2010)

    ADS  Article  Google Scholar 

  27. 27.

    Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procaccini, A., et al.: Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl. Acad. Sci. USA 105, 1232–1237 (2008)

    ADS  Article  Google Scholar 

  28. 28.

    Tu, Y., Toner, J., Ulm, M.: Sound waves and the absence of galilean invariance in flocks. Phys. Rev. Lett. 80, 4819 (1998)

    ADS  Article  Google Scholar 

  29. 29.

    Goldstein, H.: Classical Mechanics. Addison-Wesley Publishing Company, Reading (1980)

    MATH  Google Scholar 

  30. 30.

    Fetter, A.L., Walecka, J.D.: Theoretical Mechanics of Particles and Continua. Courier Dover Publications, New York (2012)

    Google Scholar 

  31. 31.

    Pomeroy, H., Heppner, F.: Structure of turning in airborne rock dove (Columba livia) flocks. Auk 109, 256–267 (1992)

    Article  Google Scholar 

  32. 32.

    Cavagna, A., Duarte Queirós, S.M., Giardina, I., Stefanini, F.,Viale, M.: Diffusion of individual birds in starling flocks. Proc. R. Soc. B 280, 20122484 (2013)

  33. 33.

    Matsubara, T., Matsuda, H.: A lattice model of Liquid Helium, I. Prog. Theor. Phys. 16, 569–582 (1956)

    ADS  Article  MATH  Google Scholar 

  34. 34.

    Halperin, B.I., Hohenberg, P.C.: Hydrodynamic theory of spin waves. Phys. Rev. 188, 898–918 (1969)

    ADS  Article  Google Scholar 

  35. 35.

    Lane, C.T., Fairbank, H.A., Fairbank, W.M.: Second sound in Liquid Helium II. Phys. Rev. 71, 600–605 (1947)

    ADS  Article  Google Scholar 

  36. 36.

    Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977)

    ADS  Article  Google Scholar 

  37. 37.

    Sonin, E.B.: Spin currents and spin superfluidity. Adv. Phys. 59, 181–255 (2010)

    ADS  Article  Google Scholar 

  38. 38.

    Justh, E.W., Krishnaprasad, P.S.: Equilibria and steering laws for planar formations. Syst. Controls Lett. 52, 25–38 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  39. 39.

    Szabo, P., Nagy, M., Vicsek, T.: Transitions in a self-propelled-particles model with coupling of accelerations. Phys. Rev. E 79, 021908 (2009)

    ADS  Article  Google Scholar 

  40. 40.

    Hemelrijk, C.K., Hildenbrandt, H.: Some causes of the variable shape of flocks of birds. PLoS ONE 6, e22479 (2011)

    ADS  Article  Google Scholar 

  41. 41.

    Gautrais, J., Ginelli, F., Fournier, R., Blanco, S., Soria, M., Chateé, H., Theraulaz, G.: Deciphering interactions in moving animal groups. Plos Comp. Biol. 8, e1002678 (2012)

    ADS  Article  MathSciNet  Google Scholar 

  42. 42.

    Sumino, Y., Nagai, K.H., Shitaka, Y., Tanaka, D., Yoshikawa, K., Chaté, H., Oiwa, K.: Large-scale vortex lattice emerging from collectively moving microtubules. Nature 483, 448–452 (2012)

    ADS  Article  Google Scholar 

  43. 43.

    Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, Oxford (2001)

    MATH  Google Scholar 

  44. 44.

    Gardiner, C.W.: Handbook of Stochastic Methods, vol. 3. Springer, Berlin (1985)

    Google Scholar 

  45. 45.

    Goldstone, J.: Field theories with Superconductor solutions. Il Nuovo Cimento 19, 154–164 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  46. 46.

    Ramaswamy, S., Simha, R.A.: Hydrodynamics fluctuations and instabilities in ordered suspensions of self- propelled particles. Phys. Rev. Lett. 89, 058101 (2002)

    ADS  Article  Google Scholar 

  47. 47.

    Bertin, E., Droz, M., Gregoire, G.: Boltzmann and hydrodynamic description for self-propelled particles. Phys. Rev. E 74, 022101 (2006)

    ADS  Article  Google Scholar 

  48. 48.

    Ihle, T.: Kinetic theory of flocking: derivation of hydrodynamic equations. Phys. Rev. E 83, 030901 (2011)

    ADS  Article  Google Scholar 

  49. 49.

    Bialek, W., Cavagna, A., Giardina, I., Mora, T., Pohl, O., Silvestri, E., Viale, M., Walczak, A.M.: Social interactions dominate speed control in poising natural flocks near criticality. Proc. Natl. Acad. Sci. USA 111, 7212–7217 (2014)

    ADS  Article  Google Scholar 

  50. 50.

    Cavagna, A., Giardina, I., Ginelli, I., Mora, T., Piovani, D., Tavarone, R., Walczak, A.: M. Dynamical maximum entropy approach to flocking. Phys. Rev. E 89, 042707 (2014)

    ADS  Article  Google Scholar 

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Acknowledgments

We thank William Bialek, Serena Bradde, Paul Chaikin and Dov Levine for discussions. Work in Rome was supported by Grants IIT–Seed Artswarm, ERC–StG n.257126 and US-AFOSR - FA95501010250 (through the University of Maryland). Work in Paris was supported by Grant ERC–StG n. 306312.

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Correspondence to Andrea Cavagna.

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Cavagna, A., Del Castello, L., Giardina, I. et al. Flocking and Turning: a New Model for Self-organized Collective Motion. J Stat Phys 158, 601–627 (2015). https://doi.org/10.1007/s10955-014-1119-3

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Keywords

  • Collective behavior
  • Flocking
  • Self-organization
  • Emergent behavior
  • Animal groups