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

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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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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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Authors and Affiliations

  1. Istituto Sistemi Complessi (ISC-CNR), Via dei Taurini 19, 00185, Rome, Italy

    Andrea Cavagna, Lorenzo Del Castello, Irene Giardina, Asja Jelic, Stefania Melillo, Leonardo Parisi, Edmondo Silvestri & Massimiliano Viale

  2. Dipartimento di Fisica, “Sapienza” Universitá di Roma, P.le Aldo Moro 2, 00185, Rome, Italy

    Andrea Cavagna, Lorenzo Del Castello, Irene Giardina, Asja Jelic, Stefania Melillo, Leonardo Parisi, Edmondo Silvestri & Massimiliano Viale

  3. Initiative for the Theoretical Sciences, The Graduate Center, 365 Fifth Avenue, New York, NY, 10016, USA

    Andrea Cavagna & Irene Giardina

  4. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA) and Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, c.c. 16, suc. 4, 1900, La Plata, Argentina

    Tomas Grigera

  5. CONICET La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, La Plata, Argentina

    Tomas Grigera

  6. Laboratoire de Physique Statistique de l’École Normale Supérieure, CNRS and Universites Paris VI and Paris VII, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Thierry Mora

  7. Dipartimento di Informatica, “Sapienza” Universitá di Roma, 00198, Rome, Italy

    Leonardo Parisi

  8. Dipartimento di Matematica e Fisica, Universitá Roma Tre, Via della Vasca Navale, 84, 00146, Rome, Italy

    Edmondo Silvestri

  9. Laboratoire de Physique Théorique de l’École Normale Supérieure, CNRS and University Paris VI, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Aleksandra M. Walczak

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  1. Andrea Cavagna
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  2. Lorenzo Del Castello
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  11. Aleksandra M. Walczak
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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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  • DOI: https://doi.org/10.1007/s10955-014-1119-3

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