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  • Letter
  • Open Access

Fluctuation dissipation relations for the nonreciprocal Cahn-Hilliard model

Martin Kjøllesdal Johnsrud1 and Ramin Golestanian1,2,*

  • *Contact author: ramin.golestanian@ds.mpg.de
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Phys. Rev. Research 7, L032054 – Published 9 September, 2025

DOI: https://doi.org/10.1103/flzv-lq7x

Abstract

Recent results demonstrate how deviations from equilibrium fluctuation–dissipation theorem can be quantified for active field theories by deriving exact fluctuations dissipation relations that involve the entropy production [M. K. Johnsrud and R. Golestanian, Phys. Rev. Res. 7, L032053 (2025)]. Here we develop and employ diagrammatic tools to perform perturbative calculations for a paradigmatic active field theory, the nonreciprocal Cahn-Hilliard (NRCH) model. We obtain analytical results, which serve as an illustration of how to implement the recently developed framework to active field theories, and help to illuminate the specific nonequilibrium characteristics of the NRCH field theory.

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Fluctuation dissipation relations for active field theories

Martin Kjøllesdal Johnsrud and Ramin Golestanian
Phys. Rev. Research 7, L032053 (2025)

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References (56)

  1. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena: Fifth Edition (Oxford University Press, Oxford, 2021).
  2. C. Aron, G. Biroli, and L. F. Cugliandolo, Symmetries of generating functionals of Langevin processes with colored multiplicative noise, J. Stat. Mech. (2010) P11018.
  3. L. M. Sieberer, A. Chiocchetta, A. Gambassi, U. C. Täuber, and S. Diehl, Thermodynamic equilibrium as a symmetry of the Schwinger-Keldysh action, Phys. Rev. B 92, 134307 (2015).
  4. C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78, 2690 (1997).
  5. G. E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60, 2721 (1999).
  6. U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Prog. Phys. 75, 126001 (2012).
  7. T. Harada and S.-i. Sasa, Equality connecting energy dissipation with a violation of the fluctuation-response relation, Phys. Rev. Lett. 95, 130602 (2005).
  8. T. Harada and S.-i. Sasa, Energy dissipation and violation of the fluctuation-response relation in nonequilibrium Langevin systems, Phys. Rev. E 73, 026131 (2006).
  9. R. Golestanian, Hydrodynamically consistent many-body Harada-Sasa relation, Phys. Rev. Lett. 134, 207101 (2025).
  10. G. Gompper, R. G. Winkler, T. Speck, A. Solon, C. Nardini, F. Peruani, H. Löwen, R. Golestanian, U. B. Kaupp, L. Alvarez et al., The 2020 motile active matter roadmap, J. Phys.: Condens. Matter 32, 193001 (2020).
  11. 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).
  12. R. Aditi Simha and S. Ramaswamy, Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles, Phys. Rev. Lett. 89, 058101 (2002).
  13. A. Gelimson and R. Golestanian, Collective dynamics of dividing chemotactic cells, Phys. Rev. Lett. 114, 028101 (2015).
  14. S. Mahdisoltani, R. B. A. Zinati, C. Duclut, A. Gambassi, and R. Golestanian, Nonequilibrium polarity-induced chemotaxis: Emergent Galilean symmetry and exact scaling exponents, Phys. Rev. Res. 3, 013100 (2021).
  15. R. Ben Alì Zinati, C. Duclut, S. Mahdisoltani, A. Gambassi, and R. Golestanian, Stochastic dynamics of chemotactic colonies with logistic growth, Europhys. Lett. 136, 50003 (2021).
  16. R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J. Allen, D. Marenduzzo, and M. E. Cates, Scalar φ4 field theory for active-particle phase separation, Nat. Commun. 5, 4351 (2014).
  17. E. Tjhung, C. Nardini, and M. E. Cates, Cluster phases and bubbly phase separation in active fluids: reversal of the Ostwald process, Phys. Rev. X 8, 031080 (2018).
  18. A. Tiribocchi, R. Wittkowski, D. Marenduzzo, and M. E. Cates, Active model h: scalar active matter in a momentum-conserving fluid, Phys. Rev. Lett. 115, 188302 (2015).
  19. M. Paoluzzi, Scaling of the entropy production rate in a φ 4 model of active matter, Phys. Rev. E 105, 044139 (2022).
  20. M. Paoluzzi, D. Levis, A. Crisanti, and I. Pagonabarraga, Noise-induced phase separation and time reversal symmetry breaking in active field theories driven by persistent noise, Phys. Rev. Lett. 133, 118301 (2024).
  21. S. Saha, J. Agudo-Canalejo, and R. Golestanian, Scalar active mixtures: the nonreciprocal Cahn-Hilliard model, Phys. Rev. X 10, 041009 (2020).
  22. Z. You, A. Baskaran, and M. C. Marchetti, Nonreciprocity as a generic route to traveling states, Proc. Natl. Acad. Sci. USA 117, 19767 (2020).
  23. G. Pisegna, S. Saha, and R. Golestanian, Emergent polar order in nonpolar mixtures with nonreciprocal interactions, Proc. Natl. Acad. Sci. USA 121, e2407705121 (2024).
  24. M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56, 889 (1986).
  25. C. Nardini, É. Fodor, E. Tjhung, F. van Wijland, J. Tailleur, and M. E. Cates, Entropy production in field theories without time-reversal symmetry: quantifying the non-equilibrium character of active matter, Phys. Rev. X 7, 021007 (2017).
  26. M. K. Johnsrud and R. Golestanian, preceding paper, Fluctuation dissipation relations for active field theories, Phys. Rev. Res. 7, L032053 (2025).
  27. U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge University Press, Cambridge, 2014).
  28. J. A. Hertz, Y. Roudi, and P. Sollich, Path integral methods for the dynamics of stochastic and disordered systems, J. Phys. A: Math. Theor. 50, 033001 (2017).
  29. A. Zee, Quantum Field Theory in a Nutshell (Princeton University Press, Princeton, NJ, 2003).
  30. R. Soto and R. Golestanian, Self-Assembly of catalytically active colloidal molecules: tailoring activity through surface Chemistry, Phys. Rev. Lett. 112, 068301 (2014).
  31. R. Golestanian, Non-reciprocal active-matter: A tale of “loving hate, brawling love” across the scales, Europhysics News 55, 12 (2024).
  32. N. Uchida and R. Golestanian, Synchronization and collective dynamics in a carpet of microfluidic rotors, Phys. Rev. Lett. 104, 178103 (2010).
  33. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  34. A. Dinelli, J. O'Byrne, A. Curatolo, Y. Zhao, P. Sollich, and J. Tailleur, Non-reciprocity across scales in active mixtures, Nat. Commun. 14, 7035 (2023).
  35. Y. Duan, J. Agudo-Canalejo, R. Golestanian, and B. Mahault, Dynamical pattern formation without self-attraction in quorum-sensing active matter: the interplay between nonreciprocity and motility, Phys. Rev. Lett. 131, 148301 (2023).
  36. S. Osat, J. Metson, M. Kardar, and R. Golestanian, Escaping kinetic traps using nonreciprocal interactions, Phys. Rev. Lett. 133, 028301 (2024).
  37. J. Agudo-Canalejo and R. Golestanian, Active phase separation in mixtures of chemically interacting particles, Phys. Rev. Lett. 123, 018101 (2019).
  38. A. Melzer, V. A. Schweigert, I. V. Schweigert, A. Homann, S. Peters, and A. Piel, Structure and stability of the plasma crystal, Phys. Rev. E 54, R46 (1996).
  39. R. Golestanian, T. B. Liverpool, and A. Ajdari, Propulsion of a molecular machine by asymmetric distribution of reaction products, Phys. Rev. Lett. 94, 220801 (2005).
  40. R. Golestanian, Phoretic active matter, in Active Matter and Nonequilibrium Statistical Physics: Lecture Notes of the Les Houches Summer School: Volume 112, September 2018 (Oxford University Press, Oxford, 2022).
  41. S. Saha, R. Golestanian, and S. Ramaswamy, Clusters, asters, and collective oscillations in chemotactic colloids, Phys. Rev. E 89, 062316 (2014).
  42. G. Tucci, R. Golestanian, and S. Saha, Nonreciprocal collective dynamics in a mixture of phoretic janus colloids, New J. Phys. 26, 073006 (2024).
  43. J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys. 28, 258 (1958).
  44. J. W. Cahn, Free energy of a nonuniform system. II. thermodynamic basis, J. Chem. Phys. 30, 1121 (1959).
  45. J. W. Cahn, Phase separation by spinodal decomposition in isotropic systems, J. Chem. Phys. 42, 93 (1965).
  46. S. Saha and R. Golestanian, Effervescent waves in a binary mixture with non-reciprocal couplings, arXiv:2208.14985 [cond-mat] [Nat. Commun. (to be published)].
  47. N. Rana and R. Golestanian, Defect solutions of the nonreciprocal Cahn-Hilliard model: spirals and targets, Phys. Rev. Lett. 133, 078301 (2024).
  48. N. Rana and R. Golestanian, Defect interactions in the non-reciprocal Cahn-Hilliard model, New J. Phys. 26, 123008 (2024).
  49. L. Parkavousi, N. Rana, R. Golestanian, and S. Saha, Enhanced stability and chaotic condensates in multispecies nonreciprocal mixtures, Phys. Rev. Lett. 134, 148301 (2025).
  50. T. Suchanek, K. Kroy, and S. A. M. Loos, Entropy production in the nonreciprocal Cahn-Hilliard model, Phys. Rev. E 108, 064610 (2023).
  51. H. Alston, L. Cocconi, and T. Bertrand, Irreversibility across a nonreciprocal PT-symmetry-breaking phase transition, Phys. Rev. Lett. 131, 258301 (2023).
  52. I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys. 74, 99 (2002).
  53. We denote δq+q′≡(2π)dδd(q+q′) and δω+ω′≡2πδ(ω+ω′) for brevity.
  54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/flzv-lq7x for details on bubble diagrams, calculation of the FDRs using explicit forms of the propagators, renormalization, and calculation of Σ using explicit matrix inversion.
  55. M. K. Johnsrud and R. Golestanian, Phase diagram of the non-reciprocal Cahn-Hilliard model and the effects of symmetry, arXiv:2503.07579 [cond-mat].
  56. M. K. Johnsrud, G. Pisegna, and R. Golestanian, Critical behavior of the non-recirprocal Cahn-Hilliard model (unpublished).

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Phys. Rev. Research 7, L032054– Published 9 September, 2025
  • Received 5 February 2025
  • Accepted 13 June 2025
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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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