- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

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
  • Editors' Suggestion
  • Open Access
  • Access by Xinjiang University

Nonreciprocal dynamics with weak noise: Aperiodic “Escher cycles” and their quasipotential landscape

Janik Schüttler1, Robert L. Jack1,2, and Michael E. Cates1

  • 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
  • 2Yusuf Hamied Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

PDF

Phys. Rev. E 114, 024104 – Published 3 August, 2026

DOI: https://doi.org/10.1103/slp9-rd27

Abstract

We present an explicit construction of the Freidlin-Wentzell quasipotential of a stochastic system with two degrees of freedom and nonreciprocal interactions. This model undergoes noise-induced transitions between four metastable attractors, forming recurrent but aperiodic “Escher cycles,” similar to the cyclic nucleation dynamics observed in the nonreciprocal Ising model. We calculate the quasipotential analytically to first order in nonreciprocality. We characterize it along a one-dimensional reaction coordinate that connects the attractors, and we also obtain the full two-dimensional landscape, at leading order in perturbation theory. The resulting landscapes feature flat regions and extended plateaus, together with nondifferentiable switching lines. These singular structures arise from two geometric mechanisms: the handover of dominance between competing transition paths, and the competition between basins of attraction. The system provides a rare case where the geometry of nonequilibrium rare events can be fully resolved, and a simple analytically tractable example of a quasipotential in more than one coordinate that captures a rich set of nonequilibrium features.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (54)

  1. O. J. O'Loan and M. R. Evans, Alternating steady state in one-dimensional flocking, J. Phys. A: Math. Gen. 32, L99 (1999).
  2. M. Besse, H. Chaté, and A. Solon, Metastability of constant-density flocks, Phys. Rev. Lett. 129, 268003 (2022).
  3. B. Benvegnen, O. Granek, S. Ro, R. Yaacoby, H. Chaté, Y. Kafri, D. Mukamel, A. Solon, and J. Tailleur, Metastability of discrete-symmetry flocks, Phys. Rev. Lett. 131, 218301 (2023).
  4. T. Grafke, M. E. Cates, and E. Vanden-Eijnden, Spatiotemporal self-organization of fluctuating bacterial colonies, Phys. Rev. Lett. 119, 188003 (2017).
  5. H. Noguchi and J.-B. Fournier, Spatiotemporal patterns in the active cyclic Potts model, New J. Phys. 26, 093043 (2024).
  6. H. Noguchi, Spatiotemporal patterns in active four-state Potts models, Sci. Rep. 15, 674 (2025).
  7. Y. Avni, M. Fruchart, D. Martin, D. Seara, and V. Vitelli, Nonreciprocal Ising model, Phys. Rev. Lett. 134, 117103 (2025).
  8. 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).
  9. Z. You, A. Baskaran, and M. C. Marchetti, Nonreciprocity as a generic route to traveling states, Proc. Natl. Acad. Sci. USA 117, 19767 (2020).
  10. M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, Non-reciprocal phase transitions, Nature (London) 592, 363 (2021).
  11. S. Shankar, A. Souslov, M. J. Bowick, M. C. Marchetti, and V. Vitelli, Topological active matter, Nat. Rev. Phys. 4, 380 (2022).
  12. B. Peters, Reaction Rate Theory and Rare Events Simulations (Elsevier, Amsterdam, 2017).
  13. M. E. Cates and C. Nardini, Classical nucleation theory for active fluid phase separation, Phys. Rev. Lett. 130, 098203 (2023).
  14. E. Woillez, Y. Zhao, Y. Kafri, V. Lecomte, and J. Tailleur, Activated escape of a self-propelled particle from a metastable state, Phys. Rev. Lett. 122, 258001 (2019).
  15. D. Santolin, N. Freitas, M. Esposito, and G. Falasco, Bridging Freidlin-Wentzell large deviations theory and stochastic thermodynamics, Phys. Rev. E 111, 024106 (2025).
  16. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren der mathematischen Wissenschaften (Springer, Berlin, 2012), Vol. 260.
  17. M. I. Dykman, E. Mori, J. Ross, and P. M. Hunt, Large fluctuations and optimal paths in chemical kinetics, J. Chem. Phys. 100, 5735 (1994).
  18. A. Lazarescu, T. Cossetto, G. Falasco, and M. Esposito, Large deviations and dynamical phase transitions in stochastic chemical networks, J. Chem. Phys. 151, 064117 (2019).
  19. J. Schüttler, R. L. Jack, and M. E. Cates, Effects of phase separation on extinction times in population models, J. Stat. Mech. (2024) 083209.
  20. E. R. Heller and D. T. Limmer, Evaluation of transition rates from nonequilibrium instantons, Phys. Rev. Res. 6, 043110 (2024).
  21. A. Crisanti and M. Paoluzzi, Most probable path of active Ornstein–Uhlenbeck particles, Phys. Rev. E 107, 034110 (2023).
  22. K. Yasuda and K. Ishimoto, Most probable path of an active Brownian particle, Phys. Rev. E 106, 064120 (2022).
  23. K. Goswami and R. Metzler, Effects of active noise on transition-path dynamics, J. Phys. Complex. 4, 025005 (2023).
  24. H. Jauslin, Nondifferentiable potentials for nonequilibrium steady states, Physica A 144, 179 (1987).
  25. M. V. Day, Recent progress on the small parameter exit problem, Stochastics 20, 121 (1987).
  26. R. Graham and T. Tél, Weak-noise limit of Fokker-Planck models and nondifferentiable potentials for dissipative dynamical systems, Phys. Rev. A 31, 1109 (1985).
  27. R. Graham and T. Tél, Nonequilibrium potential for coexisting attractors, Phys. Rev. A 33, 1322 (1986).
  28. R. S. Maier and D. L. Stein, Transition-rate theory for nongradient drift fields, Phys. Rev. Lett. 69, 3691 (1992).
  29. R. S. Maier and D. L. Stein, Escape problem for irreversible systems, Phys. Rev. E 48, 931 (1993).
  30. R. S. Maier and D. L. Stein, Effect of focusing and caustics on exit phenomena in systems lacking detailed balance, Phys. Rev. Lett. 71, 1783 (1993).
  31. H. Ge and H. Qian, Landscapes of non-gradient dynamics without detailed balance: Stable limit cycles and multiple attractors, Chaos 22, 023140 (2012).
  32. P. Zhou and T. Li, Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond, J. Chem. Phys. 144, 094109 (2016).
  33. L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, and C. Landim, Lagrangian phase transitions in nonequilibrium thermodynamic systems, J. Stat. Mech. (2010) L11001.
  34. Y. Baek and Y. Kafri, Singularities in large deviation functions, J. Stat. Mech. (2015) P08026.
  35. T. Nemoto, R. L. Jack, and V. Lecomte, Finite-size scaling of a first-order dynamical phase transition: Adaptive population dynamics and an effective model, Phys. Rev. Lett. 118, 115702 (2017).
  36. Y. Baek, Y. Kafri, and V. Lecomte, Dynamical phase transitions in the current distribution of driven diffusive channels, J. Phys. A: Math. Theor. 51, 105001 (2018).
  37. A. Aminov, G. Bunin, and Y. Kafri, Singularities in large deviation functionals of bulk-driven transport models, J. Stat. Mech. (2014) P08017.
  38. F. Bouchet, K. Gawędzki, and C. Nardini, Perturbative calculation of quasi-potential in non-equilibrium diffusions: A mean-field example, J. Stat. Phys. 163, 1157 (2016).
  39. T. Grafke and E. Vanden-Eijnden, Numerical computation of rare events via large deviation theory, Chaos 29, 063118 (2019).
  40. L. Kikuchi, R. Singh, M. E. Cates, and R. Adhikari, Ritz method for transition paths and quasipotentials of rare diffusive events, Phys. Rev. Res. 2, 033208 (2020).
  41. F. Bouchet and J. Reygner, Generalisation of the Eyring–Kramers transition rate formula to irreversible diffusion processes, Ann. Henri Poincaré 17, 3499 (2016).
  42. R. S. Maier and D. L. Stein, A scaling theory of bifurcations in the symmetric weak-noise escape problem, J. Stat. Phys. 83, 291 (1996).
  43. J. Feng and T. Kurtz, Large Deviations for Stochastic Processes, Mathematical Surveys and Monographs (American Mathematical Society, Providence, RI, 2006).
  44. T. Tél, R. Graham, and G. Hu, Nonequilibrium potentials and their power-series expansions, Phys. Rev. A 40, 4065 (1989).
  45. N. Freitas, G. Falasco, and M. Esposito, Linear response in large deviations theory: A method to compute non-equilibrium distributions, New J. Phys. 23, 093003 (2021).
  46. M. Colangeli, C. Maes, and B. Wynants, A meaningful expansion around detailed balance, J. Phys. A: Math. Theor. 44, 095001 (2011).
  47. L. Rey-Bellet and K. Spiliopoulos, Improving the convergence of reversible samplers, J. Stat. Phys. 164, 472 (2016).
  48. M. Kaiser, R. L. Jack, and J. Zimmer, Acceleration of convergence to equilibrium in Markov chains by breaking detailed balance, J. Stat. Phys. 168, 259 (2017).
  49. R. Börner, R. Deeley, R. Römer, T. Grafke, V. Lucarini, and U. Feudel, Saddle avoidance of noise-induced transitions in multiscale systems, Phys. Rev. Res. 6, L042053 (2024).
  50. W. E, W. Ren, and E. Vanden-Eijnden, Minimum action method for the study of rare events, Commun. Pure Appl. Math. 57, 637 (2004).
  51. M. Heymann and E. Vanden-Eijnden, The geometric minimum action method: A least action principle on the space of curves, Commun. Pure Appl. Math. 61, 1052 (2008).
  52. T. Grafke, T. Schäfer, and E. Vanden-Eijnden, Long term effects of small random perturbations on dynamical systems: Theoretical and computational tools, in Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, edited by R. Melnik (Springer, New York, 2017), pp. 17–55.
  53. M. Heymann and E. Vanden-Eijnden, Pathways of maximum likelihood for rare events in nonequilibrium systems: Application to nucleation in the presence of shear, Phys. Rev. Lett. 100, 140601 (2008).
  54. R. Zakine and E. Vanden-Eijnden, Minimum-action method for nonequilibrium phase transitions, Phys. Rev. X 13, 041044 (2023).

Outline

Information

Phys. Rev. E 114, 024104– Published 3 August, 2026
  • Received 20 December 2025
  • Accepted 25 June 2026
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