- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Numerical investigation of the Brownian q=2 Potts model

Letian Chen1,2,* and Gunnar Pruessner1,†

  • 1Department of Mathematics and Centre of Complexity Science, Imperial College London, London SW7 2AZ, United Kingdom
  • 2School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom

  • *Contact author: letian.chen@ed.ac.uk
  • †Contact author: g.pruessner@imperial.ac.uk
PDF

Phys. Rev. Research 7, 033295 – Published 29 September, 2025

DOI: https://doi.org/10.1103/zdfp-vgwk

Abstract

In active matter, such as the Vicsek model of flocking, particles possess an internal degree of freedom, such as their director, which is subject to interaction with other particles, provided they are within a certain range. In an effort to understand better the interplay between spatial and internal degrees of freedom, we study numerically a variation of the q=2 Potts model on and off the lattice, where particles are additionally subject to Brownian motion. The lack of a feedback of the internal degrees of freedom to the spatial degrees of freedom renders this model generically nonequilibrium. We confirm previous work that showed that the static exponents of the phase transition are unaffected by the diffusion. In contrast to previous work, we show that the formation of ordered clusters is not undermined by diffusion, but should rather be thought of as an effective form of interaction. We demonstrate how our numerical findings can be understood on the basis of the well-established Model A, B and C: Off lattice, the Brownian q=2 Potts model is Model C. On the lattice, it is Model A with an additional (irrelevant, conserved) Model B noise.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (63)

  1. 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).
  2. A. P. Solon and J. Tailleur, Revisiting the flocking transition using active spins, Phys. Rev. Lett. 111, 078101 (2013).
  3. M. R. Shaebani, A. Wysocki, R. G. Winkler, G. Gompper, and H. Rieger, Computational models for active matter, Nat. Rev. Phys. 2, 181 (2020).
  4. J. Deseigne, O. Dauchot, and H. Chaté, Collective motion of vibrated polar disks, Phys. Rev. Lett. 105, 098001 (2010).
  5. P. Romanczuk, I. D. Couzin, and L. Schimansky-Geier, Collective motion due to individual escape and pursuit response, Phys. Rev. Lett. 102, 010602 (2009).
  6. M. Knežević, T. Welker, and H. Stark, Collective motion of active particles exhibiting non-reciprocal orientational interactions, Sci. Rep. 12, 19437 (2022).
  7. S. Das, M. Ciarchi, Z. Zhou, J. Yan, J. Zhang, and R. Alert, Flocking by turning away, Phys. Rev. X 14, 031008 (2024).
  8. L. Caprini and H. Löwen, Flocking without alignment interactions in attractive active brownian particles, Phys. Rev. Lett. 130, 148202 (2023).
  9. E. Lardet, L. Chen, and T. Bertrand, Flocking beyond one species: Novel phase coexistence in a generalized two-species Vicsek model, arXiv:2503.17617.
  10. Y. Chen and J. Zhang, Anomalous flocking in nonpolar granular Brownian vibrators, Nat. Commun. 15, 6032 (2024).
  11. H. Yuan, L. X. Cui, L. T. Chen, and C. P. Sun, Quantum Vicsek model for active matter, arXiv:2407.09860.
  12. 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).
  13. H. Tasaki, Hohenberg-Mermin-Wagner-type theorems for equilibrium models of flocking, Phys. Rev. Lett. 125, 220601 (2020).
  14. F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
  15. C.-U. Woo, H. Rieger, and J. D. Noh, Suppression of discontinuous phase transitions by particle diffusion, Phys. Rev. E 105, 054144 (2022).
  16. 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).
  17. S. A. M. Loos, S. H. L. Klapp, and T. Martynec, Long-range order and directional defect propagation in the nonreciprocal XY model with vision cone interactions, Phys. Rev. Lett. 130, 198301 (2023).
  18. S. Bandyopadhyay, S. Chatterjee, A. K. Dutta, M. Karmakar, H. Rieger, and R. Paul, Ordering kinetics in the active Ising model, Phys. Rev. E 109, 064143 (2024).
  19. A. P. Solon and J. Tailleur, Flocking with discrete symmetry: The two-dimensional active Ising model, Phys. Rev. E 92, 042119 (2015).
  20. C.-U. Woo and J. D. Noh, Nonequilibrium phase transitions in a Brownian p-state clock model, Phys. Rev. E 109, 014105 (2024).
  21. P. Gaspard, Time-Reversed dynamical entropy and irreversibility in Markovian random processes, J. Stat. Phys. 117, 599 (2004).
  22. L. Cocconi, R. Garcia-Millan, Z. Zhen, B. Buturca, and G. Pruessner, Entropy production in exactly solvable systems, Entropy 22, 1252 (2020).
  23. M. P. Nightingale and H. W. J. Blöte, Dynamic exponent of the two-dimensional Ising model and monte carlo computation of the subdominant eigenvalue of the stochastic matrix, Phys. Rev. Lett. 76, 4548 (1996).
  24. R. J. Baxter, Potts model at the critical temperature, J. Phys. C: Solid State Phys. 6, L445 (1973).
  25. M. S. S. Challa, D. P. Landau, and K. Binder, Finite-size effects at temperature-driven first-order transitions, Phys. Rev. B 34, 1841 (1986).
  26. G. Grégoire and H. Chaté, Onset of collective and cohesive motion, Phys. Rev. Lett. 92, 025702 (2004).
  27. P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435 (1977).
  28. U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and non-Equilibrium Scaling Behavior (Cambridge University Press, Cambridge, 2008).
  29. M. Kardar, Statistical Physics of Particles (Cambridge University Press, Cambridge, 2007).
  30. W. Krauth, Statistical Mechanics: Algorithms and Computations (Oxford University Press, Oxford, 2006).
  31. D. P. Landau and K. K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 2nd ed. (Cambridge University Press, Cambridge, 2005).
  32. T. W. Anderson, The Statistical Analysis of Time Series (John Wiley & Sons, New York, NY, 1971).
  33. M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Clarendon Press, Oxford, 1999).
  34. H. Hinrichsen and M. Antoni, Identification of domain walls in coarsening systems at finite temperature, Phys. Rev. E 57, 2650 (1998).
  35. M. Le Bellac, Quantum and Statistical Field Theory (Oxford University Press, Oxford, 1991).
  36. V. K. Akkineni and U. C. Täuber, Nonequilibrium critical dynamics of the relaxational models C and D, Phys. Rev. E 69, 036113 (2004).
  37. S. Mertens and C. Moore, Continuum percolation thresholds in two dimensions, Phys. Rev. E 86, 061109 (2012).
  38. J. Villain, R. Bidaux, J. P. Carton, and R. L. Conte, Order as an effect of disorder, J. Phys. France 41, 1263 (1980).
  39. Y. Han, X. Huang, Z. Komargodski, A. Lucas, and F. K. Popov, Entropic order, arXiv:2503.22789.
  40. Z. B. Li, L. Schülke, and B. Zheng, Dynamic Monte Carlo measurement of critical exponents, Phys. Rev. Lett. 74, 3396 (1995).
  41. B. Zheng, Monte Carlo simulations of short-time critical dynamics, Int. J. Mod. Phys. B 12, 1419 (1998).
  42. G.-P. Zheng and M. Li, Effect of disorder on critical short-time dynamics, Phys. Rev. E 65, 036130 (2002).
  43. M. Scandolo, J. Pausch, and M. E. Cates, Active Ising models of flocking: A field-theoretic approach, Eur. Phys. J. B 46, 103 (2023).
  44. J. M. Yeomans, Statistical Mechanics of Phase Transitions (Oxford Science Publications, Clarendon Press, Oxford, England, 1992).
  45. A. B. Harris, Effect of random defects on the critical behaviour of Ising models, J. Phys. C Solid State Phys. 7, 1671 (1974).
  46. H. A. Brooks, The “Harris criterion” lives on, J. Phys. Condens. Matter 28, 421006 (2016).
  47. A. Weinrib and B. I. Halperin, Critical phenomena in systems with long-range-correlated quenched disorder, Phys. Rev. B 27, 413 (1983).
  48. A. Weinrib, Long-range correlated percolation, Phys. Rev. B 29, 387 (1984).
  49. T. Vojta and R. Dickman, Spatiotemporal generalization of the Harris criterion and its application to diffusive disorder, Phys. Rev. E 93, 032143 (2016).
  50. J. J. Alonso and M. A. Muñoz, Temporally disordered Ising models, Europhys. Lett. 56, 485 (2001).
  51. H. Dashti, A. A. Saberi, S. Rahbari, and J. Kurths, Emergence of rigidity percolation in flowing granular systems, Sci. Adv. 9, eadh5586 (2023).
  52. H. E. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, International Series of Monographs on Physics (Oxford University Press, New York, 1971).
  53. D. Mesterházy, J. H. Stockemer, L. F. Palhares, and J. Berges, Dynamic universality class of model C from the functional renormalization group, Phys. Rev. B 88, 174301 (2013).
  54. S. Tang and D. P. Landau, Monte Carlo study of dynamic universality in two-dimensional Potts models, Phys. Rev. B 36, 567 (1987).
  55. E. Vatansever, G. T. Barkema, and N. G. Fytas, Dynamical critical behavior of the two-dimensional three-state Potts model, Phys. Rev. E 110, 014135 (2024).
  56. R. da Silva, N. A. Alves, and J. R. Drugowich de Felício, Mixed initial conditions to estimate the dynamic critical exponent in short-time Monte Carlo simulation, Phys. Lett. A 298, 325 (2002).
  57. T. Olson and A. P. Young, Monte carlo study of the critical behavior of random bond Potts models, Phys. Rev. B 60, 3428 (1999).
  58. R. L. Greenblatt, M. Aizenman, and J. L. Lebowitz, On spin systems with quenched randomness: Classical and quantum, Physica A 389, 2902 (2010).
  59. M. Aizenman and J. Wehr, Rounding of first-order phase transitions in systems with quenched disorder, Phys. Rev. Lett. 62, 2503 (1989).
  60. J. Cardy and J. L. Jacobsen, Critical behavior of random-bond Potts models, Phys. Rev. Lett. 79, 4063 (1997).
  61. Y. Imry and M. Wortis, Influence of quenched impurities on first-order phase transitions, Phys. Rev. B 19, 3580 (1979).
  62. K. Hui and A. N. Berker, Random-field mechanism in random-bond multicritical systems, Phys. Rev. Lett. 62, 2507 (1989).
  63. P. Villa Martín, J. A. Bonachela, and M. A. Muñoz, Quenched disorder forbids discontinuous transitions in nonequilibrium low-dimensional systems, Phys. Rev. E 89, 012145 (2014).

Outline

Information

Phys. Rev. Research 7, 033295– Published 29 September, 2025
  • Received 7 March 2025
  • Accepted 5 September 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation