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Disordered yet directed: The emergence of polar flocks with disordered interactions

Eloise Lardet1, Raphaël Voituriez2,3, Silvia Grigolon2,*, and Thibault Bertrand1,†

  • *Contact author: silvia.grigolon@gmail.com
  • †Contact author: t.bertrand@imperial.ac.uk
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Phys. Rev. Research 8, 033279 – Published 4 September, 2026

DOI: https://doi.org/10.1103/tpxk-cltw

Abstract

Flocking is a prime example of how robust collective behavior can emerge from simple interaction rules. The flocking transition has been studied extensively since the inception of the original Vicsek model. Here, we introduce a self-propelled particle model with quenched disorder in the pairwise alignment interaction couplings akin to a spin-glass model. We find that increasing the variance of the coupling distribution can promote (rather than destroy) the emergence of global polar order. In particular, we show that our model can display a flocking phase even when the majority of the interaction couplings are antialigning. Activity is the key ingredient to reduce frustration in the system as it allows local particle clustering combined with self-organization of the particles to favor neighborhoods with strong cooperative interactions.

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References (93)

  1. S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010).
  2. M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft active matter, Rev. Mod. Phys. 85, 1143 (2013).
  3. C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Active particles in complex and crowded environments, Rev. Mod. Phys. 88, 045006 (2016).
  4. M. R. Shaebani, A. Wysocki, R. G. Winkler, G. Gompper, and H. Rieger, Computational models for active matter, Nat. Rev. Phys. 2, 181 (2020).
  5. T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75, 1226 (1995).
  6. J. Toner and Y. Tu, Long-range order in a two-dimensional dynamical XY model: How birds fly together, Phys. Rev. Lett. 75, 4326 (1995).
  7. J. Toner and Y. Tu, Flocks, herds, and schools: A quantitative theory of flocking, Phys. Rev. E 58, 4828 (1998).
  8. T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep. 517, 71 (2012).
  9. H. Chaté, Dry aligning dilute active matter, Annu. Rev. Condens. Matter Phys. 11, 189 (2020).
  10. C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, SIGGRAPH Comput. Graphics 21, 25 (1987).
  11. G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett. 92, 025702 (2004).
  12. Y. Tu, J. Toner, and M. Ulm, Sound waves and the absence of Galilean invariance in flocks, Phys. Rev. Lett. 80, 4819 (1998).
  13. R. Aditi Simha and S. Ramaswamy, Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles, Phys. Rev. Lett. 89, 058101 (2002).
  14. S. Ramaswamy, R. A. Simha, and J. Toner, Active nematics on a substrate: Giant number fluctuations and long-time tails, Europhys. Lett. 62, 196 (2003).
  15. J. Toner, Y. Tu, and S. Ramaswamy, Hydrodynamics and phases of flocks, Ann. Phys. 318, 170 (2005).
  16. J. Toner, Reanalysis of the hydrodynamic theory of fluid, polar-ordered flocks, Phys. Rev. E 86, 031918 (2012).
  17. J. Toner, Birth, death, and flight: A theory of Malthusian flocks, Phys. Rev. Lett. 108, 088102 (2012).
  18. A. Cavagna, L. Del Castello, S. Dey, I. Giardina, S. Melillo, L. Parisi, and M. Viale, Short-range interactions versus long-range correlations in bird flocks, Phys. Rev. E 92, 012705 (2015).
  19. B. Mahault, F. Ginelli, and H. Chaté, Quantitative assessment of the Toner and Tu theory of polar flocks, Phys. Rev. Lett. 123, 218001 (2019).
  20. N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17, 1133 (1966).
  21. P. C. Hohenberg, Existence of long-range order in one and two dimensions, Phys. Rev. 158, 383 (1967).
  22. H. Chaté, F. Ginelli, G. Grégoire, and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E 77, 046113 (2008).
  23. H. Chaté, F. Ginelli, G. Grégoire, F. Peruani, and F. Raynaud, Modeling collective motion: Variations on the Vicsek model, Eur. Phys. J. B 64, 451 (2008).
  24. A. P. Solon, H. Chaté, and J. Tailleur, From phase to microphase separation in flocking models: The essential role of nonequilibrium fluctuations, Phys. Rev. Lett. 114, 068101 (2015).
  25. I. Giardina, Collective behavior in animal groups: Theoretical models and empirical studies, HFSP J. 2, 205 (2008).
  26. A. Cavagna, I. Giardina, and T. S. Grigera, The physics of flocking: Correlation as a compass from experiments to theory, Phys. Rep. 728, 1 (2018).
  27. F. Peruani, J. Starruß, V. Jakovljevic, L. Søgaard-Andersen, A. Deutsch, and M. Bär, Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria, Phys. Rev. Lett. 108, 098102 (2012).
  28. V. Schaller, C. Weber, C. Semmrich, E. Frey, and A. R. Bausch, Polar patterns of driven filaments, Nature (London) 467, 73 (2010).
  29. A. Bricard, J.-B. Caussin, N. Desreumaux, O. Dauchot, and D. Bartolo, Emergence of macroscopic directed motion in populations of motile colloids, Nature (London) 503, 95 (2013).
  30. A. Kaiser, A. Snezhko, and I. S. Aranson, Flocking ferromagnetic colloids, Sci. Adv. 3, e1601469 (2017).
  31. D. Geyer, A. Morin, and D. Bartolo, Sounds and hydrodynamics of polar active fluids, Nat. Mater. 17, 789 (2018).
  32. P. Romanczuk, I. D. Couzin, and L. Schimansky-Geier, Collective motion due to individual escape and pursuit response, Phys. Rev. Lett. 102, 010602 (2009).
  33. J. Deseigne, O. Dauchot, and H. Chaté, Collective motion of vibrated polar disks, Phys. Rev. Lett. 105, 098001 (2010).
  34. E. Ferrante, A. E. Turgut, M. Dorigo, and C. Huepe, Elasticity-based mechanism for the collective motion of self-propelled particles with springlike interactions: A model system for natural and artificial swarms, Phys. Rev. Lett. 111, 268302 (2013).
  35. M. Knežević, T. Welker, and H. Stark, Collective motion of active particles exhibiting non-reciprocal orientational interactions, Sci. Rep. 12, 19437 (2022).
  36. M. Casiulis and D. Levine, Emergent synchronization and flocking in purely repulsive self-navigating particles, Phys. Rev. E 106, 044611 (2022).
  37. L. Caprini and H. Löwen, Flocking without alignment interactions in attractive active Brownian particles, Phys. Rev. Lett. 130, 148202 (2023).
  38. S. Das, M. Ciarchi, Z. Zhou, J. Yan, J. Zhang, and R. Alert, Flocking by turning away, Phys. Rev. X 14, 031008 (2024).
  39. P. Baconnier, O. Dauchot, V. Démery, G. Düring, S. Henkes, C. Huepe, and A. Shee, Self-aligning polar active matter, Rev. Mod. Phys. 97, 015007 (2025).
  40. B. Ventejou, H. Chaté, R. Montagne, and X.-Q. Shi, Susceptibility of orientationally ordered active matter to chirality disorder, Phys. Rev. Lett. 127, 238001 (2021).
  41. A. M. Menzel, Collective motion of binary self-propelled particle mixtures, Phys. Rev. E 85, 021912 (2012).
  42. S. Chatterjee, M. Mangeat, C.-U. Woo, H. Rieger, and J. D. Noh, Flocking of two unfriendly species: The two-species Vicsek model, Phys. Rev. E 107, 024607 (2023).
  43. R. Kürsten, J. Mihatsch, and T. Ihle, Emergent flocking in mixtures of antialigning self-propelled particles, Phys. Rev. E 111, L023402 (2025).
  44. K. Copenhagen, D. A. Quint, and A. Gopinathan, Self-organized sorting limits behavioral variability in swarms, Sci. Rep. 6, 31808 (2016).
  45. G. Ariel, O. Rimer, and E. Ben-Jacob, Order–disorder phase transition in heterogeneous populations of self-propelled particles, J. Stat. Phys. 158, 579 (2015).
  46. F. Ginelli and H. Chaté, Relevance of metric-free interactions in flocking phenomena, Phys. Rev. Lett. 105, 168103 (2010).
  47. F. Ginelli, The physics of the Vicsek model, Eur. Phys. J. Spec. Top. 225, 2099 (2016).
  48. D. Martin, H. Chaté, C. Nardini, A. Solon, J. Tailleur, and F. Van Wijland, Fluctuation-induced phase separation in metric and topological models of collective motion, Phys. Rev. Lett. 126, 148001 (2021).
  49. D. Martin, G. Spera, H. Chaté, C. Duclut, C. Nardini, J. Tailleur, and F. van Wijland, Fluctuation-induced first order transition to collective motion, J. Stat. Mech. (2024) 084003.
  50. O. Chepizhko, E. G. Altmann, and F. Peruani, Optimal noise maximizes collective motion in heterogeneous media, Phys. Rev. Lett. 110, 238101 (2013).
  51. A. Morin, N. Desreumaux, J.-B. Caussin, and D. Bartolo, Distortion and destruction of colloidal flocks in disordered environments, Nat. Phys. 13, 63 (2017).
  52. J. Codina, B. Mahault, H. Chaté, J. Dobnikar, I. Pagonabarraga, and X.-Q. Shi, Small obstacle in a large polar flock, Phys. Rev. Lett. 128, 218001 (2022).
  53. A. Solon, H. Chaté, J. Toner, and J. Tailleur, Susceptibility of polar flocks to spatial anisotropy, Phys. Rev. Lett. 128, 208004 (2022).
  54. G. Baglietto, E. V. Albano, and J. Candia, Gregarious versus individualistic behavior in Vicsek swarms and the onset of first-order phase transitions, Physica A 392, 3240 (2013).
  55. D. Yllanes, M. Leoni, and M. C. Marchetti, How many dissenters does it take to disorder a flock? New J. Phys. 19, 103026 (2017).
  56. B. Benvegnen, O. Granek, S. Ro, R. Yaacoby, H. Chaté, Y. Kafri, D. Mukamel, A. Solon, and J. Tailleur, Metastability of discrete-symmetry flocks, Phys. Rev. Lett. 131, 218301 (2023).
  57. F. D. C. Farrell, M. C. Marchetti, D. Marenduzzo, and J. Tailleur, Pattern formation in self-propelled particles with density-dependent motility, Phys. Rev. Lett. 108, 248101 (2012).
  58. B. Liebchen and D. Levis, Collective behavior of chiral active matter: Pattern formation and enhanced flocking, Phys. Rev. Lett. 119, 058002 (2017).
  59. A. Martín-Gómez, D. Levis, A. Díaz-Guilera, and I. Pagonabarraga, Collective motion of active Brownian particles with polar alignment, Soft Matter 14, 2610 (2018).
  60. Y. Zhao, T. Ihle, Z. Han, C. Huepe, and P. Romanczuk, Phases and homogeneous ordered states in alignment-based self-propelled particle models, Phys. Rev. E 104, 044605 (2021).
  61. F. Peruani and I. S. Aranson, Cold active motion: How time-independent disorder affects the motion of self-propelled agents, Phys. Rev. Lett. 120, 238101 (2018).
  62. J. Toner, N. Guttenberg, and Y. Tu, Swarming in the dirt: Ordered flocks with quenched disorder, Phys. Rev. Lett. 121, 248002 (2018).
  63. J. Toner, N. Guttenberg, and Y. Tu, Hydrodynamic theory of flocking in the presence of quenched disorder, Phys. Rev. E 98, 062604 (2018).
  64. Y. Duan, B. Mahault, Y.-Q. Ma, X.-Q. Shi, and H. Chaté, Breakdown of ergodicity and self-averaging in polar flocks with quenched disorder, Phys. Rev. Lett. 126, 178001 (2021).
  65. L. Chen, C. F. Lee, A. Maitra, and J. Toner, Packed swarms on dirt: Two-dimensional incompressible flocks with quenched and annealed disorder, Phys. Rev. Lett. 129, 188004 (2022).
  66. L. Chen, C. F. Lee, A. Maitra, and J. Toner, Hydrodynamic theory of two-dimensional incompressible polar active fluids with quenched and annealed disorder, Phys. Rev. E 106, 044608 (2022).
  67. L. Chen, C. F. Lee, A. Maitra, and J. Toner, Incompressible polar active fluids with quenched random field disorder in dimensions d>2, Phys. Rev. Lett. 129, 198001 (2022).
  68. S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F: Met. Phys. 5, 965 (1975).
  69. D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett. 35, 1792 (1975).
  70. K. Binder and A. P. Young, Spin glasses: Experimental facts, theoretical concepts, and open questions, Rev. Mod. Phys. 58, 801 (1986).
  71. K. H. Fischer and J. A. Hertz, Spin glasses, Cambridge Studies in Magnetism (Cambridge University Press, Cambridge, 1991).
  72. M. Mezard, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1986).
  73. M. Paoluzzi, D. Levis, and I. Pagonabarraga, From flocking to glassiness in dense disordered polar active matter, Commun. Phys. 7, 57 (2024).
  74. R. Hanai, Nonreciprocal frustration: Time crystalline order-by-disorder phenomenon and a spin-glass-like state, Phys. Rev. X 14, 011029 (2024).
  75. Indeed, No long-range order would be observed in a Vicsek model with a nonrandom coupling whose value is set to the same average coupling strength.
  76. O. Chepizhko, D. Saintillan, and F. Peruani, Revisiting the emergence of order in active matter, Soft Matter 17, 3113 (2021).
  77. S. Kirkpatrick and D. Sherrington, Infinite-ranged models of spin-glasses, Phys. Rev. B 17, 4384 (1978).
  78. G. Grégoire, H. Chaté, and Y. Tu, Moving and staying together without a leader, Physica D 181, 157 (2003).
  79. E. Sese-Sansa, I. Pagonabarraga, and D. Levis, Velocity alignment promotes motility-induced phase separation, Europhys. Lett. 124, 30004 (2018).
  80. See Supplemental Material at http://link.aps.org/supplemental/10.1103/tpxk-cltw for further analytical and computational details, including supplementary movies, which contains Ref. [91].
  81. J. Choi, J. D. Noh, and H. Rieger, Flocking with random nonreciprocal interactions, Phys. Rev. Res. 7, L042059 (2025).
  82. S. Chakraborti and P. Pradhan, Additivity and density fluctuations in Vicsek-like models of self-propelled particles, Phys. Rev. E 99, 052604 (2019).
  83. R. Großmann, P. Romanczuk, M. Bär, and L. Schimansky-Geier, Pattern formation in active particle systems due to competing alignment interactions, Eur. Phys. J. Spec. Top. 224, 1325 (2015).
  84. Toner and Tu predicted the connected, equal-time, velocity autocorrelation to scale C(r)∝r2χ/ζ for sufficiently large r, with χ=−1/5, ζ=3/5 in d=2 [7]. These scaling exponents were later shown not to be exact [16]; however, recent work still shows the same scaling of ω=2χ/ζ=−2/3 [92, 93]. This exponent was found to be approximately −0.65 by others numerically [19].
  85. G. Ariel, A. Ayali, A. Be’er, and D. Knebel, Variability and heterogeneity in natural swarms: Experiments and modeling, in Active Particles: Advances in Theory, Models, and Applications, edited by N. Bellomo, J. A. Carrillo, and E. Tadmor (Springer International Publishing, Cham, 2022), Vol. 3, pp. 1–33.
  86. C. M. Topaz, M. R. D’Orsogna, L. Edelstein-Keshet, and A. J. Bernoff, Locust dynamics: Behavioral phase change and swarming, PLoS Comput. Biol. 8, e1002642 (2012).
  87. W. Zuo and Y. Wu, Dynamic motility selection drives population segregation in a bacterial swarm, Proc. Natl. Acad. Sci. USA 117, 4693 (2020).
  88. S. Peled, S. D. Ryan, S. Heidenreich, M. Bär, G. Ariel, and A. Be’er, Heterogeneous bacterial swarms with mixed lengths, Phys. Rev. E 103, 032413 (2021).
  89. T. Bertrand, J. d’Alessandro, A. Maitra, S. Jain, B. Mercier, R.-M. Mège, B. Ladoux, and R. Voituriez, Clustering and ordering in cell assemblies with generic asymmetric aligning interactions, Phys. Rev. Res. 6, 023022 (2024).
  90. E. Lardet, R. Voituriez, S. Grigolon, and T. Bertrand, Dataset for “Disordered yet directed: The emergence of polar flocks with disordered interactions,” [Dataset], Zenodo, 2026, https://doi.org/10.5281/zenodo.20799646.
  91. S. Torquato, Hyperuniform states of matter, Phys. Rep. 745, 1 (2018).
  92. P. Jentsch and C. F. Lee, New universality class describes Vicsek’s flocking phase in physical dimensions, Phys. Rev. Lett. 133, 128301 (2024).
  93. H. Chaté and A. Solon, Dynamic scaling of two-dimensional polar flocks, Phys. Rev. Lett. 132, 268302 (2024).

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Phys. Rev. Research 8, 033279– Published 4 September, 2026
  • Received 11 September 2024
  • Revised 13 November 2025
  • Accepted 6 July 2026
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