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Disorder-to-order transition in one-dimensional nonreciprocal Cahn-Hilliard model

Navdeep Rana1,* and Ramin Golestanian1,2,†

  • *Contact author: navdeep.rana@ds.mpg.de
  • †Contact author: ramin.golestanian@ds.mpg.de
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Phys. Rev. Research 8, 023157 – Published 11 May, 2026

DOI: https://doi.org/10.1103/6n94-yyvn

Abstract

We present the phenomenology of the one-dimensional nonreciprocal Cahn-Hilliard model for varying nonreciprocity (α) and different boundary conditions. At small α, a perturbed uniform state evolves to a defect-laden configuration that lacks global polar order. Defects are the sources and sinks of traveling waves. For a given α, defects with a unique wave number that increases monotonically with α are selected. A critical threshold αc marks the onset of a transition to states with finite global polar order. For periodic boundary conditions, above αc, the system shows traveling waves that are completely ordered. In contrast, traveling waves are incompatible with the Neumann and Dirichlet boundary conditions. Instead, for α≳αc, we find fluctuating domains that show intermittent polar order, and at large α, the system partitions into two domains with opposite polar order.

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References (43)

  1. S. Saha, J. Agudo-Canalejo, and R. Golestanian, Scalar active mixtures: The nonreciprocal Cahn-Hilliard model, Phys. Rev. X 10, 041009 (2020).
  2. Z. You, A. Baskaran, and M. C. Marchetti, Nonreciprocity as a generic route to traveling states, Proc. Natl. Acad. Sci. USA 117, 19767 (2020).
  3. N. Rana and R. Golestanian, Defect solutions of the nonreciprocal Cahn-Hilliard model: Spirals and targets, Phys. Rev. Lett. 133, 078301 (2024).
  4. N. Rana and R. Golestanian, Defect interactions in the non-reciprocal Cahn–Hilliard model, New J. Phys. 26, 123008 (2024).
  5. L. Parkavousi, N. Rana, R. Golestanian, and S. Saha, Enhanced stability and chaotic condensates in multispecies nonreciprocal mixtures, Phys. Rev. Lett. 134, 148301 (2025).
  6. G. Pisegna, S. Saha, and R. Golestanian, Emergent polar order in nonpolar mixtures with nonreciprocal interactions, Proc. Natl. Acad. Sci. USA 121, e2407705121 (2024).
  7. G. Pisegna, N. Rana, R. Golestanian, and S. Saha, Nonreciprocal mixtures in suspension: The role of hydrodynamic interactions, Phys. Rev. Lett. 135, 108301 (2025).
  8. S. Saha and R. Golestanian, Effervescence in a binary mixture with nonlinear non-reciprocal interactions, Nat. Commun. 16, 7310 (2025).
  9. S. A. M. Loos and S. H. L. Klapp, Irreversibility, heat and information flows induced by non-reciprocal interactions, New J. Phys. 22, 123051 (2020).
  10. S. Osat and R. Golestanian, Non-reciprocal multifarious self-organization, Nat. Nanotechnol. 18, 79 (2023).
  11. S. Osat, J. Metson, M. Kardar, and R. Golestanian, Escaping kinetic traps using nonreciprocal interactions, Phys. Rev. Lett. 133, 028301 (2024).
  12. R. Soto and R. Golestanian, Self-assembly of catalytically active colloidal molecules: Tailoring activity through surface chemistry, Phys. Rev. Lett. 112, 068301 (2014).
  13. A. V. Ivlev, J. Bartnick, M. Heinen, C.-R. Du, V. Nosenko, and H. Löwen, Statistical mechanics where Newton's third law is broken, Phys. Rev. X 5, 011035 (2015).
  14. J. Agudo-Canalejo and R. Golestanian, Active phase separation in mixtures of chemically interacting particles, Phys. Rev. Lett. 123, 018101 (2019).
  15. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  16. R. Golestanian, Non-reciprocal active-matter: A tale of “loving hate, brawling love” across the scales, Europhys. News 55, 12 (2024).
  17. T. Frohoff-Hülsmann and U. Thiele, Localized states in coupled Cahn–Hilliard equations, IMA J. Appl. Math. 86, 924 (2021).
  18. T. Frohoff-Hülsmann, J. Wrembel, and U. Thiele, Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling, Phys. Rev. E 103, 042602 (2021).
  19. T. Frohoff-Hülsmann and U. Thiele, Nonreciprocal Cahn-Hilliard model emerges as a universal amplitude equation, Phys. Rev. Lett. 131, 107201 (2023).
  20. R. Golestanian, Phoretic active matter, in Active Matter and Nonequilibrium Statistical Physics: Lecture Notes of the Les Houches Summer School: September 2018, edited by J. Tailleur, G. Gompper, M. C. Marchetti, J. M. Yeomans, and C. Salomon (Oxford University Press, Oxford, UK, 2022), Vol. 112, pp. 230–293.
  21. G. Tucci, R. Golestanian, and S. Saha, Nonreciprocal collective dynamics in a mixture of phoretic Janus colloids, New J. Phys. 26, 073006 (2024).
  22. N. Uchida and R. Golestanian, Synchronization and collective dynamics in a carpet of microfluidic rotors, Phys. Rev. Lett. 104, 178103 (2010).
  23. L. F. A., W. Hugo, B. Tobias, and B. Clemens, Group formation and cohesion of active particles with visual perception–dependent motility, Science 364, 70 (2019).
  24. J. A. Cohen and R. Golestanian, Emergent cometlike swarming of optically driven thermally active colloids, Phys. Rev. Lett. 112, 068302 (2014).
  25. U. Khadka, V. Holubec, H. Yang, and F. Cichos, Active particles bound by information flows, Nat. Commun. 9, 3864 (2018).
  26. R. Soto and R. Golestanian, Self-assembly of active colloidal molecules with dynamic function, Phys. Rev. E 91, 052304 (2015).
  27. S. Saha, S. Ramaswamy, and R. Golestanian, Pairing, waltzing and scattering of chemotactic active colloids, New J. Phys. 21, 063006 (2019).
  28. L. P. Dadhichi, J. Kethapelli, R. Chajwa, S. Ramaswamy, and A. Maitra, Nonmutual torques and the unimportance of motility for long-range order in two-dimensional flocks, Phys. Rev. E 101, 052601 (2020).
  29. V. Ouazan-Reboul, R. Golestanian, and J. Agudo-Canalejo, Network effects lead to self-organization in metabolic cycles of self-repelling catalysts, Phys. Rev. Lett. 131, 128301 (2023).
  30. R. Wittkowski, J. Stenhammar, and M. E. Cates, Nonequilibrium dynamics of mixtures of active and passive colloidal particles, New J. Phys. 19, 105003 (2017).
  31. A. M. Tayar, F. Caballero, T. Anderberg, O. A. Saleh, M. Cristina Marchetti, and Z. Dogic, Controlling liquid–liquid phase behaviour with an active fluid, Nat. Mater. 22, 1401 (2023).
  32. F. Brauns, G. Pawlik, J. Halatek, J. Kerssemakers, E. Frey, and C. Dekker, Bulk-surface coupling identifies the mechanistic connection between min-protein patterns in vivo and in vitro, Nat. Commun. 12, 3312 (2021).
  33. L. Würthner, F. Brauns, G. Pawlik, J. Halatek, J. Kerssemakers, C. Dekker, and E. Frey, Bridging scales in a multiscale pattern-forming system, Proc. Natl. Acad. Sci. USA 119, e2206888119 (2022).
  34. I. S. Aranson, L. Kramer, and A. Weber, Theory of interaction and bound states of spiral waves in oscillatory media, Phys. Rev. E 47, 3231 (1993).
  35. I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys. 74, 99 (2002).
  36. P. S. Hagan, Target patterns in reaction-diffusion systems, Adv. Appl. Math. 2, 400 (1981).
  37. P. S. Hagan, Spiral waves in reaction-diffusion equations, SIAM J. Appl. Math. 42, 762 (1982).
  38. Mathematical Models for Phase Change Problems: Proceedings of the European Workshop Held at Óbidos, Portugal, October 1–3, 1988, International Series of Numerical Mathematics Vol. 88, edited by J.-F. Rodrigues (Birkhäuser Verlag, Basel, 1989).
  39. S. Cox and P. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176, 430 (2002).
  40. K. J. Burns, G. M. Vasil, J. S. Oishi, D. Lecoanet, and B. P. Brown, Dedalus: A flexible framework for numerical simulations with spectral methods, Phys. Rev. Res. 2, 023068 (2020).
  41. https://github.com/navdeeprana/DedalusScripts.
  42. W. Zimmermann, Stability of traveling waves for a conserved field, Physica A 237, 405 (1997).
  43. L. Angheluta, P. Jeraldo, and N. Goldenfeld, Anisotropic velocity statistics of topological defects under shear flow, Phys. Rev. E 85, 011153 (2012).

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Phys. Rev. Research 8, 023157– Published 11 May, 2026
  • Received 13 October 2025
  • Accepted 20 April 2026
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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. Open access publication funded by Max Planck Society.

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