The European Physical Journal Special Topics

, Volume 187, Issue 1, pp 127–134 | Cite as

Collective motion of active Brownian particles in one dimension

Article

Abstract

We analyze a model of active Brownian particles with non-linear friction and velocity coupling in one spatial dimension. The model exhibits two modes of motion observed in biological swarms: A disordered phase with vanishing mean velocity and an ordered phase with finite mean velocity. Starting from the microscopic Langevin equations, we derive mean-field equations of the collective dynamics. We identify the fixed points of the mean-field equations corresponding to the two modes and analyze their stability with respect to the model parameters. Finally, we compare our analytical findings with numerical simulations of the microscopic model.

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References

  1. 1.
    F. Schweitzer, L. Schimansky-Geier, Physica A 206, 359 (1994)CrossRefADSGoogle Scholar
  2. 2.
    W. Ebeling, F. Schweitzer, B. Tilch, Biosystems 49, 17 (1999)CrossRefGoogle Scholar
  3. 3.
    F. Schweitzer, Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences, 1 edition (Springer, 2003)Google Scholar
  4. 4.
    L. Schimansky-Geier, M. Mieth, H. Rose, H. Malchow, Phys. Lett. A 207, 140 (1995)MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    P. Romanczuk, U. Erdmann, H. Engel, L. Schimansky-Geier, Eur. Phys. J. Special Topics 157, 61 (2008)CrossRefADSGoogle Scholar
  6. 6.
    L. Schimansky-Geier, U. Erdmann, N. Komin, Physica A 351, 51 (2005)CrossRefADSGoogle Scholar
  7. 7.
    J. Strefler, U. Erdmann, L. Schimansky-Geier, Phys. Rev. E 78, 031927 (2008)CrossRefADSGoogle Scholar
  8. 8.
    N. Komin, U. Erdmann, L. Schimansky-Geier, Fluct. Noise Lett. 4, 151 (2004)CrossRefGoogle Scholar
  9. 9.
    B. Lindner, J. Stat. Mech.-Theory Exp. 2009, 01008 (2009)CrossRefGoogle Scholar
  10. 10.
    A.W. Liehr, H.U. Bödeker, M.C. Röttger, T.D. Frank, R. Friedrich, H.-G. Purwins, New J. Phys. 5, 89 (2003)CrossRefADSGoogle Scholar
  11. 11.
    Y. Sumino, N. Magome, T. Hamada, K. Yoshikawa, Phys. Rev. Lett. 94, 068301 (2005)CrossRefADSGoogle Scholar
  12. 12.
    G. Ruckner, R. Kapral, Phys. Rev. Lett. 98, 150603 (2007)CrossRefADSGoogle Scholar
  13. 13.
    J.K. Parrish, L. Edelstein-Keshet, Science 284, 99 (1999)CrossRefADSGoogle Scholar
  14. 14.
    I.D. Couzin, J. Krause, R. James, G.D. Ruxton, N.R. Franks, J. Theor. Biol. 218, 1 (2002)CrossRefMathSciNetGoogle Scholar
  15. 15.
    D.A. Paley, N.E. Leonard, R. Sepulchre, D. Grunbaum, IEEE Contr. Syst. Mag. (2007)Google Scholar
  16. 16.
    T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Phys. Rev. Lett. 75, 1226 (1995)CrossRefADSGoogle Scholar
  17. 17.
    J. Toner, Y. Tu, Phys. Rev. E 58, 4828 (1998)CrossRefMathSciNetADSGoogle Scholar
  18. 18.
    A. Czirók, A.-L. Barabási, T. Vicsek, Phys. Rev. Lett. 82, 209 (1999)CrossRefADSGoogle Scholar
  19. 19.
    R. Aditi Simha, S. Ramaswamy, Phys. Rev. Lett. 89, 058101 (2002)CrossRefADSGoogle Scholar
  20. 20.
    P. Romanczuk, I.D. Couzin, L. Schimansky-Geier, Phys. Rev. Lett. 102, 010602 (2009)CrossRefADSGoogle Scholar
  21. 21.
    F. Ginelli, F. Peruani, M. Bär, H. Chaté, Phys. Rev. Lett. 104, 184502 (2010)CrossRefADSGoogle Scholar
  22. 22.
    H.S. Niwa, J. Theor. Biol. 171, 123 (1994)CrossRefGoogle Scholar
  23. 23.
    M.R. D’Orsogna, Y.L. Chuang, A.L. Bertozzi, L.S. Chayes, Phys. Rev. Lett. 96, 104302 (2006)CrossRefADSGoogle Scholar
  24. 24.
    W. Ebeling, L. Schimansky-Geier, Eur. Phys. J. Special Topics 157, 17 (2008)CrossRefADSGoogle Scholar
  25. 25.
    J. Strefler, W. Ebeling, E. Gudowska-Nowak, L. Schimansky-Geier, Eur. Phys. J. B 72, 597 (2009)MATHCrossRefADSGoogle Scholar
  26. 26.
    U. Erdmann, W. Ebeling, L. Schimansky-Geier, F. Schweitzer, Eur. Phys. J. B 15, 105 (2000)CrossRefADSGoogle Scholar
  27. 27.
    U. Erdmann, W. Ebeling, V.S Anishchenko, Phys. Rev. E 65, 061106 (2002)CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    B. Lindner, E.M. Nicola, Eur. Phys. J. Special Topics 157, 43 (2008)CrossRefADSGoogle Scholar
  29. 29.
    R.F. Pawula, Phys. Rev. 162, 186 (1967)CrossRefADSGoogle Scholar
  30. 30.
    R.F. Pawula, Phys. Rev. A 36, 4996 (1987)CrossRefADSGoogle Scholar
  31. 31.
    M. Januszewski, M. Kostur, Comput. Phys. Commun. 181, 183 (2010)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences and Springer 2010

Authors and Affiliations

  1. 1.Department of PhysicsHumboldt Universität zu BerlinBerlinGermany
  2. 2.Helmholtz Association of German Research CentresBerlinGermany

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