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  • Open Access

Dynamic modes of active Potts models with factorizable numbers of states

Hiroshi Noguchi*

  • Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

  • *Contact author: noguchi@issp.u-tokyo.ac.jp
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Phys. Rev. Research 7, 033243 – Published 11 September, 2025

DOI: https://doi.org/10.1103/w1fg-6qmv

Abstract

We studied the long-term nonequilibrium dynamics of q-state Potts models with q=4, 5, 6, and 8 using Monte Carlo simulations on a two-dimensional square lattice. When the contact energies between the nearest neighbors for the standard Potts models are used, cyclic changes in the q homogeneous phases and q-state coexisting wave mode appear at low and high flipping energies, respectively, for all values of q. However, for a factorizable q value, dynamic modes with skipping states emerge, depending on the contact energies. For q=6, a spiral wave mode with three domain types (one state dominant or two states mixed) and cyclic changes in three homogeneous phases are found. Although three states can coexist spatially under thermal equilibrium, the scaling exponents of the transitions to the wave modes are modified from the equilibrium values.

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References (67)

  1. G. Nicolis and I. Prigogine, Self-organization in Nonequilibrium Systems: From Dissipative Structures to Order Through Fluctuations (Wiley, New York, 1977).
  2. H. Haken, Synergetics: Introduction and Advanced Topics (Springer, Berlin, 2004).
  3. A. Mikhailov, Foundations of Synergetics I: Distributed Active Systems, 2nd ed. (Springer, Berlin, 1994).
  4. M. I. Rabinovich, A. B. Ezersky, and P. D. Weidman, The Dynamics of Patterns (World Scientific, Singapore, 2000).
  5. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984).
  6. J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).
  7. J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd ed. (Springer, New York, 2003).
  8. Z. You, A. Baskaran, and M. C. Marchetti, Nonreciprocity as a generic route to traveling states, Proc. Natl. Acad. Sci. USA 117, 19767 (2020).
  9. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  10. N. Rana and R. Golestanian, Defect solutions of the nonreciprocal Cahn-Hilliard model: Spirals and targets, Phys. Rev. Lett. 133, 078301 (2024).
  11. L. Guislain and E. Bertin, Collective oscillations in a three-dimensional spin model with non-reciprocal interactions, J. Stat. Mech. 093210 (2024).
  12. C. Beta and K. Kruse, Intracellular oscillations and waves, Annu. Rev. Condens. Matter Phys. 8, 239 (2017).
  13. S. Kondo, M. Watanabe, and S. Miyazawa, Studies of Turing pattern formation in zebrafish skin, Philos. Trans. R. Soc. A 379, 20200274 (2021).
  14. A. Bailles, E. W. Gehrels, and T. Lecuit, Mechanochemical principles of spatial and temporal patterns in cells and tissues, Annu. Rev. Cell Dev. Biol. 38, 321 (2022).
  15. H. Noguchi, Nonequilibrium membrane dynamics induced by active protein interactions and chemical reactions: A review, ChemSystemsChem 7, e202400042 (2025).
  16. F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
  17. R. B. Potts, Some generalized order-disorder transformations, Proc. Camb. Philos. Soc. 48, 106 (1952).
  18. R. J. Baxter, Potts model at the critical temperature, J. Phys. C: Solid State Phys. 6, L445 (1973).
  19. K. Binder, Static and dynamic critical phenomena of the two-dimensional q-state Potts model, J. Stat. Phys. 24, 69 (1981).
  20. K. Binder, Finite size scaling analysis of Ising model block distribution functions, Z. Phys. B 43, 119 (1981).
  21. R. V. Ditzian, J. R. Banavar, G. S. Grest, and L. P. Kadanoff, Phase diagram for the Ashkin-Teller model in three dimensions, Phys. Rev. B 22, 2542 (1980).
  22. G. S. Grest and M. Widom, n-color Ashkin-Teller model, Phys. Rev. B 24, 6508 (1981).
  23. J. A. Plascak and F. C. S. Barreto, Critical properties of the Ashkin-Teller model from the mean-field renormalisation group approach, J. Phys. A: Math. Gen. 19, 2195 (1986).
  24. N. Benayad, A. Benyoussef, N. Boccara, and A. E. Kenz, Renormalisation group recursion relations using the application of generalised Callen identities to the Ashkin-Teller model, J. Phys. C: Solid State Phys. 21, 5747 (1988).
  25. M. Kohmoto, M. den Nijs, and L. P. Kadanoff, Hamiltonian studies of the d=2 Ashkin-Teller model, Phys. Rev. B 24, 5229 (1981).
  26. E. Fradkin, n-color Ashkin-Teller model in two dimensions: Solution in the large-n limit, Phys. Rev. Lett. 53, 1967 (1984).
  27. T. Okubo, K. Oshikawa, H. Watanabe, and N. Kawashima, Scaling relation for dangerously irrelevant symmetry-breaking fields, Phys. Rev. B 91, 174417 (2015).
  28. M. Nishino and S. Miyashita, Anomalous finite-size effect due to quasidegenerate phases in triangular antiferromagnets with long-range interactions and mapping to the generalized six-state clock model, Phys. Rev. B 94, 184434 (2016).
  29. S. Iino, S. Morita, N. Kawashima, and A. W. Sandvik, Detecting signals of weakly first-order phase transitions in two-dimensional Potts models, J. Phys. Soc. Jpn. 88, 034006 (2019).
  30. G. Musiał, D. Jeziorek-Knioła, and Z. Wojtkowiak, Monte Carlo examination of first-order phase transitions in a system with many independent order parameters: Three-dimensional Ashkin-Teller model, Phys. Rev. E 103, 062124 (2021).
  31. U. Akinci, Nonequilibrium phase transitions in isotropic Ashkin-Teller model, Physica A 469, 740 (2017).
  32. J. Santos, R. Francisco, and D. Rosa, Dynamic magnetic properties and multicritical phase diagram of the spin-1/2 Ashkin-Teller model under a time dependent external field, J. Magn. Magn. Mater. 538, 168281 (2021).
  33. T. Martynec, S. H. L. Klapp, and S. A. M. Loos, Entropy production at criticality in a nonequilibrium Potts model, New J. Phys. 22, 093069 (2020).
  34. H. Noguchi, F. van Wijland, and J.-B. Fournier, Cycling and spiral-wave modes in an active cyclic Potts model, J. Chem. Phys. 161, 025101 (2024).
  35. H. Noguchi and J.-B. Fournier, Spatiotemporal patterns in the active cyclic Potts model, New J. Phys. 26, 093043 (2024).
  36. H. Noguchi, Spatiotemporal patterns in active four-state Potts models, Sci. Rep. 15, 674 (2025).
  37. A. Manacorda and E. Fodor, Diffusive oscillators capture the pulsating states of deformable particles, Phys. Rev. E 111, L053401 (2025).
  38. Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, Nonreciprocal Ising model, Phys. Rev. Lett. 134, 117103 (2025).
  39. Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, Dynamical phase transitions in the nonreciprocal Ising model, Phys. Rev. E 111, 034124 (2025).
  40. H. Noguchi, Spatiotemporal pattern formation of membranes induced by surface molecular binding/unbinding, Soft Matter 21, 1113 (2025).
  41. A. Szolnoki, M. Mobilia, L.-L. Jiang, B. Szczesny, A. M. Rucklidge, and M. Perc, Cyclic dominance in evolutionary games: A review, J. R. Soc. Interface 11, 20140735 (2014).
  42. G. Szabó and A. Szolnoki, Three-state cyclic voter model extended with Potts energy, Phys. Rev. E 65, 036115 (2002).
  43. T. Reichenbach, M. Mobilia, and E. Frey, Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games, Nature (London) 448, 1046 (2007).
  44. B. Szczesny, M. Mobilia, and A. M. Rucklidge, When does cyclic dominance lead to stable spiral waves? Europhys. Lett. 102, 28012 (2013).
  45. E. D. Kelsic, J. Zhao, K. Vetsigian, and R. Kishony, Counteraction of antibiotic production and degradation stabilizes microbial communities, Nature (London) 521, 516 (2015).
  46. U. Dobramysl, M. Mobilia, M. Pleimling, and U. C. Täuber, Stochastic population dynamics in spatially extended predator–prey systems, J. Phys. A: Math. Theor. 51, 063001 (2018).
  47. G. Szabó and G. Arial Sznaider, Phase transition and selection in a four-species cyclic predator-prey model, Phys. Rev. E 69, 031911 (2004).
  48. G. Szabó, A. Szolnoki, and I. Borsos, Self-organizing patterns maintained by competing associations in a six-species predator-prey model, Phys. Rev. E 77, 041919 (2008).
  49. A. Roman, D. Konrad, and M. Pleimling, Cyclic competition of four species: Domains and interfaces, J. Stat. Mech. P07014 (2012).
  50. C. Rulquin and J. J. Arenzon, Globally synchronized oscillations in complex cyclic games, Phys. Rev. E 89, 032133 (2014).
  51. D. Bazeia, B. F. de Oliveira, and A. Szolnoki, Invasion-controlled pattern formation in a generalized multispecies predator-prey system, Phys. Rev. E 99, 052408 (2019).
  52. L. Zhong, L. Zhang, H. Li, Q. Dai, and J. Yang, Species coexistence in spatial cyclic game of five species, Chaos Solitons Fractals 156, 111806 (2022).
  53. R. K. Yang and J. Park, Evolutionary dynamics in the cyclic competition system of seven species: Common cascading dynamics in biodiversity, Chaos Solitons Fractals 175, 113949 (2023).
  54. A. Szolnoki and X. Chen, Emerging solutions from the battle of defensive alliances, Sci. Rep. 13, 8472 (2023).
  55. J. Takahashi and A. W. Sandvik, Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet, Phys. Rev. Res. 2, 033459 (2020).
  56. G. Ertl, Reactions at surfaces: From atoms to complexity (Nobel lecture), Angew. Chem. Int. Ed. 47, 3524 (2008).
  57. M. Brär, N. Gottschalk, M. Eiswirth, and G. Ertl, Spiral waves in a surface reaction: Model calculations, J. Chem. Phys. 100, 1202 (1994).
  58. V. Gorodetskii, J. Lauterbach, H.-H. Rotermund, J. H. Block, and G. Ertl, Coupling between adjacent crystal planes in heterogeneous catalysis by propagating reaction-diffusion waves, Nature (London) 370, 276 (1994).
  59. C. Barroo, Z.-J. Wang, R. Schlrögl, and M.-G. Willinger, Imaging the dynamics of catalysed surface reactions by in situ scanning electron microscopy, Nat. Catal. 3, 30 (2020).
  60. J. Zeininger et al., Pattern formation in catalytic H oxidation on Rh: Zooming in by correlative microscopy, ACS Catal. 12, 11974 (2022).
  61. Y. Tabe and H. Yokoyama, Coherent collective precession of molecular rotors with chiral propellers, Nat. Mater. 2, 806 (2003).
  62. Y. Miele, Z. Medveczky, G. Holló, B. Tegze, I. Derényi, Z. Hórvölgyi, E. Altamura, I. Lagzi, and F. Rossi, Self-division of giant vesicles driven by an internal enzymatic reaction, Chem. Sci. 11, 3228 (2020).
  63. G. Holló, Y. Miele, F. Rossi, and I. Lagzi, Shape changes and budding of giant vesicles induced by an internal chemical trigger: An interplay between osmosis and pH change, Phys. Chem. Chem. Phys. 23, 4262 (2021).
  64. H. Noguchi, Membrane domain formation induced by binding/unbinding of curvature-inducing molecules on both membrane surfaces, Soft Matter 19, 679 (2023).
  65. See Supplemental Material at http://link.aps.org/supplemental/10.1103/w1fg-6qmv for Movies S1–S6 of spatiotemporal patterns in active Potts models.
  66. H. Noguchi (unpublished).
  67. H. Noguchi, Dynamic modes of active Potts models with factorizable numbers of states, https://isspns-gitlab.issp.u-tokyo.ac.jp/hiroshi.noguchi/activePotts.

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Phys. Rev. Research 7, 033243– Published 11 September, 2025
  • Received 30 June 2025
  • Accepted 21 August 2025
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