Journal of Statistical Physics

, Volume 158, Issue 3, pp 601–627 | Cite as

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

  • Andrea Cavagna
  • Lorenzo Del Castello
  • Irene Giardina
  • Tomas Grigera
  • Asja Jelic
  • Stefania Melillo
  • Thierry Mora
  • Leonardo Parisi
  • Edmondo Silvestri
  • Massimiliano Viale
  • Aleksandra M. Walczak


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.


Collective behavior Flocking Self-organization Emergent behavior Animal groups 



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.

Supplementary material

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Andrea Cavagna
    • 1
    • 2
    • 3
  • Lorenzo Del Castello
    • 1
    • 2
  • Irene Giardina
    • 1
    • 2
    • 3
  • Tomas Grigera
    • 4
    • 5
  • Asja Jelic
    • 1
    • 2
  • Stefania Melillo
    • 1
    • 2
  • Thierry Mora
    • 6
  • Leonardo Parisi
    • 1
    • 2
    • 7
  • Edmondo Silvestri
    • 1
    • 2
    • 8
  • Massimiliano Viale
    • 1
    • 2
  • Aleksandra M. Walczak
    • 9
  1. 1.Istituto Sistemi Complessi (ISC-CNR)RomeItaly
  2. 2.Dipartimento di Fisica“Sapienza” Universitá di RomaRomeItaly
  3. 3.Initiative for the Theoretical Sciences, The Graduate CenterNew YorkUSA
  4. 4.Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA) and Departamento de Física, Facultad de Ciencias ExactasUniversidad Nacional de La PlataLa PlataArgentina
  5. 5.CONICET La Plata, Consejo Nacional de Investigaciones Científicas y TécnicasLa PlataArgentina
  6. 6.Laboratoire de Physique Statistique de l’École Normale SupérieureCNRS and Universites Paris VI and Paris VIIParis Cedex 05France
  7. 7.Dipartimento di Informatica“Sapienza” Universitá di RomaRomeItaly
  8. 8.Dipartimento di Matematica e FisicaUniversitá Roma TreRomeItaly
  9. 9.Laboratoire de Physique Théorique de l’École Normale SupérieureCNRS and University Paris VIParis Cedex 05France

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