Active particle condensation by non-reciprocal and time-delayed interactions

  • Mihir Durve
  • Arnab SahaEmail author
  • Ahmed Sayeed
Regular Article


We consider the flocking of self-propelling agents in two dimensions, each of which communicates with its neighbors within a limited vision-cone. Also, the communication occurs with some time-delay. The communication among the agents are modeled by Vicsek rules. In this study we explore the combined effect of non-reciprocal interaction (induced by limited vision-cone) among the agents and the presence of delay in the interactions on the dynamical pattern formation within the flock. We find that under these two influences, without any position-based attractive interactions or confining boundaries, the agents can spontaneously condense into “drops”. Though the agents are in motion within the drop, the drop as a whole is pinned in space. We find that this novel state of the flock has a well-defined order and it is stabilized by the noise present in the system.

Graphical abstract


Flowing Matter: Liquids and Complex Fluids 

Supplementary material

10189_2018_11653_MOESM1_ESM.mp4 (36 mb)
Supplementary material


  1. 1.
    D. Bi, X. Yang, M.C. Marchetti, M.L. Manning, Phys. Rev. X 6, 021011 (2016)Google Scholar
  2. 2.
    H.P. Zhang, A. Be'er, R.S. Smith, E.-L. Florin, H.L. Swinney, EPL 87, 48011 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    N.C. Darnton, L. Tuner, S. Rojevsky, H.C. Berg, Biophys. J. 98, 2082 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    B.A. Camley, W.-J. Rappel, J. Phys. D: Appl. Phys. 50, 113002 (2017)ADSCrossRefGoogle Scholar
  5. 5.
    F. Peruani, J. Starruß, V. Jakovljevic, L. Søgaard-Anderson, A. Deutsch, M. Bär, Phys. Rev. Lett. 108, 098102 (2012)ADSCrossRefGoogle Scholar
  6. 6.
    H. Ke, S. Ye, R.L. Carroll, K. Showalter, J. Phys. Chem. A 114, 5462 (2010)CrossRefGoogle Scholar
  7. 7.
    I. Buttinoni, J. Bialké, F. Kümmel, H. Löwen, C. Bechinger, T. Speck, Phys. Rev. Lett. 110, 238301 (2013)ADSCrossRefGoogle Scholar
  8. 8.
    A. Kudrolli, G. Lumay, D. Volfson, L.S. Tsimring, Phys. Rev. Lett. 100, 058001 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    I.S. Aranson, D. Volfson, L.S. Tsimring, Phys. Rev. E 75, 051301 (2007)ADSCrossRefGoogle Scholar
  10. 10.
    P. Romanczuk, M. Bär, W. Ebeling, B. Lindner, L. Schimansky-Geier, Eur. Phys. J. ST 202, 1 (2012)CrossRefGoogle Scholar
  11. 11.
    C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, G. Volpe, Rev. Mod. Phys. 88, 045006 (2016)ADSCrossRefGoogle Scholar
  12. 12.
    I.D. Couzin, J. Krause, Adv. Study Behav. 32, 1 (2003)CrossRefGoogle Scholar
  13. 13.
    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, V. Zdravkovic, Anim. Behav. 76, 201 (2008)CrossRefGoogle Scholar
  14. 14.
    C. Becco, N. Vandewalle, J. Delcourt, P. Poncin, Physica A 367, 487 (2006)ADSCrossRefGoogle Scholar
  15. 15.
    A.E. Turgut, H. Çelikkanat, F. Gökçe, E. Sahin, Swarm Intell. 2, 97 (2008)CrossRefGoogle Scholar
  16. 16.
    M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini et al., Proc. Natl. Acad. Sci. U.S.A. 105, 1232 (2008)ADSCrossRefGoogle Scholar
  17. 17.
    J. Buhl, D.J. Sumpter, I.D. Couzin, J.J. Hale, E. Despland, E. Miller, S.J. Simpson, Science 312, 1402 (2006)ADSCrossRefGoogle Scholar
  18. 18.
    L. Dyson, C.A. Yates, J. Buhl, A.J. McKane, Phys. Rev. E 92, 052708 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Silvestri, M. Viale, A.M. Walczak, Proc. Natl. Acad. Sci. U.S.A. 109, 4786 (2012)ADSCrossRefGoogle Scholar
  20. 20.
    A. Cavagna, L. Del Castello, I. Giardina, T. Grigera, A. Jelic, S. Melillo, T. Mora, L. Parisi, E. Silvestri, M. Viale et al., J. Stat. Phys. 158, 601 (2015)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shoche, Phys. Rev. Lett. 75, 1226 (1995)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    H. Chaté, F. Ginelli, G. Grégoire, F. Raynaud, Phys. Rev. E 77, 046113 (2008)ADSCrossRefGoogle Scholar
  23. 23.
    G. Grégoire, H. Chaté, Y. Tu, Physica D 181, 157 (2003)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    J. Toner, Y. Tu, Phys. Rev. E 58, 4828 (1998)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    H. Levine, W.-J. Rappel, I. Cohen, Phys. Rev. E 63, 017101 (2000)ADSCrossRefGoogle Scholar
  26. 26.
    G. Grégoire, H. Chaté, Phys. Rev. Lett. 92, 025702 (2004)ADSCrossRefGoogle Scholar
  27. 27.
    H. Chaté, F. Ginelli, G. Gregoiré, F. Peruani, F. Raynaud, Eur. Phys. J. B 64, 451 (2008)ADSCrossRefGoogle Scholar
  28. 28.
    G. Salbreux, G. Charras, E. Paluch, Trends Cell Biol. 22, 536 (2012)CrossRefGoogle Scholar
  29. 29.
    M. Mayer, M. Depken, J.S. Bois, F. Jülicher, S.W. Grill, Nature 467, 617 (2010)ADSCrossRefGoogle Scholar
  30. 30.
    M. Behrndt, G. Salbreux, P. Campinho, R. Hauschild, F. Oswald, J. Roensch, S.W. Grill, C.-P. Heisenberg, Science 338, 257 (2012)ADSCrossRefGoogle Scholar
  31. 31.
    A. Saha, M. Nishikawa, M. Behrndt, C.-P. Heisenberg, F. Jülicher, S.W. Grill, Biophys. J. 110, 1421 (2016)ADSCrossRefGoogle Scholar
  32. 32.
    S.R. Naganathan, S. Fürthauer, M. Nishikawa, F. Jülicher, S.W. Grill, eLife 3, e04165 (2014)CrossRefGoogle Scholar
  33. 33.
    M. Nishikawa, S.R. Naganathan, F. Jülicher, S.W. Grill, eLife 6, e19595 (2017)Google Scholar
  34. 34.
    T. Vicsek, A. Zafeiris, Phys. Rep. 517, 71 (2012)ADSCrossRefGoogle Scholar
  35. 35.
    A. Menzel, Phys. Rep. 554, 1 (2015)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    R.D. Peacock, J.D. Averill, Pedestrian and Evacuation Dynamics (Springer Science & Business Media, 2011)Google Scholar
  37. 37.
    M. Mijalkov, A. McDaniel, J. Wehr, G. Volpe, Phys. Rev. X 6, 011008 (2016)Google Scholar
  38. 38.
    E. Forgoston, I.B. Schwartz, Phys. Rev. E 77, 035203 (2008)ADSCrossRefGoogle Scholar
  39. 39.
    J. Hindes, K. Szwaykowska, I.B. Schwartz, Phys. Rev. E 94, 032306 (2016)ADSCrossRefGoogle Scholar
  40. 40.
    K. Szwaykowska, I.B. Schwartz, L.M.-y.-T. Romero, C.R. Heckman, D. Mox, M.A. Hsieh, Phys. Rev. E 93, 032307 (2016)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    J.D. Biggs, D.J. Bennet, S.K. Dadzie, Phys. Rev. E 85, 016105 (2012)ADSCrossRefGoogle Scholar
  42. 42.
    U. Erdmann, W. Ebeling, A.S. Mikhailov, Phys. Rev. E 71, 051904 (2005)ADSCrossRefGoogle Scholar
  43. 43.
    K. Dholakia, P. Zemánek, Rev. Mod. Phys. 82, 1767 (2010)ADSCrossRefGoogle Scholar
  44. 44.
    E.R. Shanblatt, D.G. Grier, Opt. Express 19, 5833 (2011)ADSCrossRefGoogle Scholar
  45. 45.
    B. Sabass, U. Seifert, Phys. Rev. Lett. 105, 218103 (2010)ADSCrossRefGoogle Scholar
  46. 46.
    R. Soto, R. Golestanian, Phys. Rev. Lett. 112, 068301 (2014)ADSCrossRefGoogle Scholar
  47. 47.
    J. Dzubiella, H. Löwen, C. Likos, Phys. Rev. Lett. 91, 248301 (2003)ADSCrossRefGoogle Scholar
  48. 48.
    A.S. Khair, J.F. Brady, Proc. R. Soc. London A: Math. Phys. Eng. Sci. 463, 223 (2007)ADSCrossRefGoogle Scholar
  49. 49.
    C. Mejía-Monasteriom G. Oshanin, Soft Matter 7, 993 (2011)ADSCrossRefGoogle Scholar
  50. 50.
    V.N. Tsytovich, Phys. Usp. 40, 53 (1997)ADSCrossRefGoogle Scholar
  51. 51.
    G.E. Morfill, A.V. Ivlev, Rev. Mod. Phys. 81, 1353 (2009)ADSCrossRefGoogle Scholar
  52. 52.
    M. Chaudhuri, A.V. Ivlev, S.A. Khrapak, H.M. Thomas, G.E. Morfill, Soft Matter 7, 1287 (2011)ADSCrossRefGoogle Scholar
  53. 53.
    D. Helbing, I. Farkas, T. Vicsek, Nature 407, 487 (2000)ADSCrossRefGoogle Scholar
  54. 54.
    D. Helbing, P. Molnar, Phys. Rev. E 51, 4282 (1995)ADSCrossRefGoogle Scholar
  55. 55.
    J. Gao, S. Havlin, X. Xu, H.E. Stanley, Phys. Rev. E 84, 046115 (2011)ADSCrossRefGoogle Scholar
  56. 56.
    M. Durve, A. Sayeed, Phys. Rev. E 93, 052115 (2016)ADSCrossRefGoogle Scholar
  57. 57.
    B.-M. Tian, H.-X. Yang, W. Li, W.-X. Wang, B.-H. Wang, T. Zhou, Phys. Rev. E 79, 052102 (2009)ADSCrossRefGoogle Scholar
  58. 58.
    L. Barberis, F. Peruani, Phys. Rev. Lett. 117, 248001 (2016)ADSCrossRefGoogle Scholar
  59. 59.
    V. Schaller, C.A. Weber, B. Hammerich, E. Frey, A.R. Bausch, Proc. Natl. Acad. Sci. U.S.A. 108, 19183 (2011)ADSCrossRefGoogle Scholar
  60. 60.
    Y. Sumino, K.H. Nagai, Y. Shitaka, D. Tanaka, K. Yoshikawa, H. Chaté, K. Oiwa, Nature 483, 448 (2012)ADSCrossRefGoogle Scholar
  61. 61.
    K.H. Nagai, Y. Sumino, R. Montagne, I.S. Aranson, H. Chaté, Phys. Rev. Lett. 114, 168001 (2015)ADSCrossRefGoogle Scholar
  62. 62.
    M. Durve, A. Saha, A. Sayeed, unpublishedGoogle Scholar
  63. 63.
    O. Chepizhko, E.G. Altmann, F. Peruani, Phys. Rev. Lett. 110, 238101 (2013)ADSCrossRefGoogle Scholar
  64. 64.
    M.E. Cates, J. Tailleur, Annu. Rev. Condens. Matter Phys. 6, 219 (2015)ADSCrossRefGoogle Scholar
  65. 65.
    A. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, H. Löwen, Phys. Rev. X 5, 011035 (2015)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversità degli studi di TriesteTriesteItaly
  2. 2.The Abdus Salam International Centre for Theoretical PhysicsTriesteItaly
  3. 3.Department of PhysicsSavitribai Phule Pune UniversityPuneIndia

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