- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Macroscopic traveling packet and soliton states of quasi-one-dimensional flocks

Nicholas Guttenberg and John Toner

Yuhai Tu

  • Institute for Theoretical Science and Department of Physics, University of Oregon, Eugene, Oregon 97403, USA

  • IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA

PDF

Phys. Rev. E 89, 052711 – Published 16 May, 2014

DOI: https://doi.org/10.1103/PhysRevE.89.052711

Abstract

Using a continuum model for inhomogeneous flocks, we show that a finite but arbitrarily large moving “packet” of active particles (e.g., moving creatures) can form in a background of a lower density disordered phase of these particles, like a liquid drop surrounded by vapor. The “vapor density” of the disordered background can be made arbitrarily low. We find three basic types of quasi-one-dimensional states: “longitudinal”, “transverse”, and “oblique” states, with their internal velocity fields, respectively, parallel, perpendicular, and oblique to the interface. The transitions between these states are also studied.

View figure in article

Article Text

References (33)

  1. L. Edelstein-Keshet, J. Watmough, and D. Grunbaum, J. Math. Biol. 36, 515 (1998).
  2. R. Lukeman, Y.-X. Li, and L. Edelstein-Keshet, Proc. Natl. Acad. Sci. USA 107, 12576 (2010).
  3. J. Vollmer, A. G. Vegh, C. Lange, and B. Eckhardt, Phys. Rev. E 73, 061924 (2006).
  4. M. Nagy, Z. Ákos, D. Biro, and T. Vicsek, Nature (London) 464, 890 (2010).
  5. Y. Wu, Y. Jiang, A. D. Kaiser, and M. Alber, Phys. Biol. 8, 055003 (2011).
  6. M. Dorigo et al., in Swarm Robotics (Springer, New York, 2005), pp. 31–44.
  7. A. F. Winfield, C. J. Harper, and J. Nembrini, in Swarm Robotics (Springer, Berlin, 2005), pp. 126–142.
  8. L. C. A. Pimenta, N. Michael, R. C. Mesquita, G. A. Pereira, and V. Kumar, in IEEE International Conference on Robotics and Automation (IEEE, Pasadena, CA, 2008), pp. 1948–1953.
  9. V. Narayan, S. Ramaswamy, and N. Menon, Science 317, 105 (2007).
  10. N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
  11. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995).
  12. H. Chaté, F. Ginelli, G. Grégoire, F. Peruani, and F. Raynaud, Eur. Phys. J. B 64, 451 (2008).
  13. H. Chaté, F. Ginelli, G. Grégoire, and F. Raynaud, Phys. Rev. E 77, 046113 (2008).
  14. J. Toner and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995).
  15. J. Toner, Y. Tu, and S. Ramaswamy, Ann. Phys. (NY) 318, 170 (2005).
  16. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013).
  17. E. Bertin, M. Droz, and G. Gregoire, Phys. Rev. E 74, 022101 (2006).
  18. E. Bertin, M. Droz, and G. Gregoire, J. Phys. A: Math. Theor. 42, 445001 (2009).
  19. S. Mishra, A. Baskaran, and M. C. Marchetti, Phys. Rev. E 81, 061916 (2010).
  20. F. D. C. Farrell, M. C. Marchetti, D. Marenduzzo, and J. Tailleur, Phys. Rev. Lett. 108, 248101 (2012).
  21. S. Yamanaka and T. Ohta, arXiv:1307.6709.
  22. J. Bialke, H. Lowen, and T. Speck, Europhys. Lett. 103, 30008 (2013).
  23. T. Ihle, Phys. Rev. E 88, 040303 (2013).
  24. J. B. Caussin, A. Solon, A. Peshkov, H. Chate, T. Dauxois, J. Tailleur, V. Vitelli, and D. Bartolo, arXiv:1401.1315v1. After our paper was submitted, this preprint, which also addresses this question (albeit in a somewhat different continuum model), appeared on the arXiv.
  25. Phase separation between two disordered, nonmoving (i.e., v⃗=0⃗) active phases has been investigated by J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Marenduzzo, and M. E. Cates, Phys. Rev. Lett. 111, 145702 (2013); J. Stenhammar, D. Marenduzzo, R. J. Allen, and M. E. Cates, Soft Matter 10, 1489 (2014).
  26. Some evidence for a “yes” answer to both of these questions has emerged from simulations; see, e.g., G. Gregoire, H. Chate, and Y. Tu, Physica D 181, 157 (2003); G. Gregoire and H. Chate, Phys. Rev. Lett. 92, 025702 (2004). However, it is unclear from simulations alone whether or not these flocks can be made arbitrarily large, or how internally homogeneous they are, since simulations must be quite large in order to reach the continuum limit. This is particularly true in the flocking problem, which exhibits giant number fluctuations.
  27. These equations can also be derived microscopically; see [17, 18, 19].
  28. J. Toner and Y. Tu, Phys. Rev. E 58, 4828 (1998).
  29. This sort of ρ dependence emerges from microscopic derivations of the hydrodnamic equations; see [17, 18, 19].
  30. J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Clarendon, Oxford, 1982).
  31. Both w and ρ∞ can be obtained by looking for steady-state traveling solutions ρ(x,t)=ρ(x−wt) and v⃗(x,t)=v⃗(x−wt) in our model. In this paper, we solve our model by direct simulations.
  32. The average velocity field of the packet is defined as 〈v⃗〉≡∫0Lv⃗(z)(ρ(z)−ρ∞)dz/∫0L(ρ(z)−ρ∞)dz, where ρ(z) and v⃗(z) (with z=x−wt) are the traveling steady-state density and velocity fields, respectively.
  33. In this paper, we use the words “transition” and “bifurcation” interchangeably.

Outline

Information

Phys. Rev. E 89, 052711– Published 16 May, 2014
  • Received 17 December 2013

©2014 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation