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Most probable path and invariant sets in noise-induced transition to turbulence

Yoshiki Hiruta1,*, Kento Yasuda2,†, and Kenta Ishimoto3,‡

  • 1Department of Physics and Astronomy, Faculty of Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan
  • 2Laboratory of Physics, College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan
  • 3Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan

  • *Contact author: hiruta@rs.tus.ac.jp
  • †Contact author: yasuda.kento@nihon-u.ac.jp
  • ‡Contact author: kenta.ishimoto@math.kyoto-u.ac.jp
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Phys. Rev. Research 8, 023303 – Published 16 June, 2026

DOI: https://doi.org/10.1103/qdws-1jc9

Abstract

Turbulence transition often arises from a subcritical transition between bistable states characterized by invariant sets of deterministic dynamical systems, and such transitions can be triggered by system noise as rare events. In this study, we employ the Onsager-Machlup (OM) formulation of stochastic dynamics to examine the Hamilton equations governing the most probable transition paths (MPPs). We introduce an effective potential function, termed the OM potential, which depends on the noise intensity. Focusing on the Dauchot-Manneville model as a minimal system with an edge state, we comprehensively analyze the MPP for different transition times between two attracting points corresponding to laminar and turbulent states. We find that the MPPs cross the separatrix at nearly the same point regardless of the transition time, and the obtained OM action values suggest that the transition to turbulence occurs more frequently than the transition to the laminar state. Moreover, we numerically demonstrate that the noise-induced transition paths follow the OM potential landscape and its bifurcation diagram, indicating that the qualitative behavior of the MPPs is determined by the OM potential. Our methodology formulated in general dynamical systems provides a theoretical basis for predicting noise-induced transitions among invariant sets of the dynamics.

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References (79)

  1. O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, Philos. Trans. R. Soc. London 174, 935 (1883).
  2. K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley, and B. Hof, The onset of turbulence in pipe flow, Science 333, 192 (2011).
  3. M. Avila, F. Mellibovsky, N. Roland, and B. Hof, Streamwise-localized solutions at the onset of turbulence in pipe flow, Phys. Rev. Lett. 110, 224502 (2013).
  4. R. R. Kerswell, Recent progress in understanding the transition to turbulence in a pipe, Nonlinearity 18, R17 (2005).
  5. M. Shimizu and S. Kida, A driving mechanism of a turbulent puff in pipe flow, Fluid Dyn. Res. 41, 045501 (2009).
  6. M. Avila, D. Barkley, and B. Hof, Transition to turbulence in pipe flow, Annu. Rev. Fluid Mech. 55, 575 (2023).
  7. M. Nagata, Three-dimensional finite-amplitude solutions in plane Couette flow: Bifurcation from infinity, J. Fluid Mech. 217, 519 (1990).
  8. J. Jiménez and P. Moin, The minimal flow unit in near-wall turbulence, J. Fluid Mech. 225, 213 (1991).
  9. J. M. Hamilton, J. Kim, and F. Waleffe, Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech. 287, 317 (1995).
  10. T. Itano and S. Toh, The dynamics of bursting process in wall turbulence, J. Phys. Soc. Jpn. 70, 703 (2001).
  11. G. Kawahara and S. Kida, Periodic motion embedded in plane Couette turbulence: Regeneration cycle and burst, J. Fluid Mech. 449, 291 (2001).
  12. J. F. Gibson, J. Halcrow, and P. Cvitanović, Visualizing the geometry of state space in plane Couette flow, J. Fluid Mech. 611, 107 (2008).
  13. E. Marensi, G. Yalnız, B. Hof, and N. B. Budanur, Symmetry-reduced dynamic mode decomposition of near-wall turbulence, J. Fluid Mech. 954, A10 (2023).
  14. M. Engel, O. Ashtari, T. M. Schneider, and M. Linkmann, Search for unstable relative periodic orbits in channel flow using symmetry-reduced dynamic mode decomposition, J. Fluid Mech. 1013, A45 (2025).
  15. Y. Hiruta and S. Toh, Solitary solutions including spatially localized chaos and their interactions in two-dimensional Kolmogorov flow, Phys. Rev. E 92, 063025 (2015).
  16. Y. Hiruta and S. Toh, Intermittent direction reversals of moving spatially localized turbulence observed in two-dimensional Kolmogorov flow, Phys. Rev. E 96, 063112 (2017).
  17. Y. Hiruta and S. Toh, Subcritical laminar–turbulent transition as nonequilibrium phase transition in two-dimensional kolmogorov flow, J. Phys. Soc. Jpn. 89, 044402 (2020).
  18. Y. Hiruta and S. Toh, A simple thermal convection system showing subcritical transition and localized turbulences in two-dimensional periodic domains, J. Phys. Soc. Jpn. 91, 013401 (2022).
  19. E. Pollak and P. Talkner, Reaction rate theory: What it was, where is it today, and where is it going? Chaos 15, 026116 (2005).
  20. L. S. Tsimring, Noise in biology, Rep. Prog. Phys. 77, 026601 (2014).
  21. X. Fang, K. Kruse, T. Lu, and J. Wang, Nonequilibrium physics in biology, Rev. Mod. Phys. 91, 045004 (2019).
  22. E. Forgoston and R. O. Moore, A primer on noise-induced transitions in applied dynamical systems, SIAM Rev. 60, 969 (2018).
  23. L. Xu, A. van Kan, C. Liu, and E. Knobloch, Fluctuation-induced transitions in anisotropic two-dimensional turbulence, Phys. Rev. Fluids 9, 064605 (2024).
  24. B. Podvin and A. Sergent, Precursor for wind reversal in a square Rayleigh-Bénard cell, Phys. Rev. E 95, 013112 (2017).
  25. J. Soons, T. Grafke, and H. A. Dijkstra, Most likely noise-induced tipping of the overturning circulation in a two-dimensional Boussinesq fluid model, J. Fluid Mech. 1009, A53 (2025).
  26. F. Bouchet, J. Rolland, and E. Simonnet, Rare event algorithm links transitions in turbulent flows with activated nucleations, Phys. Rev. Lett. 122, 074502 (2019).
  27. S. Gomé, L. S. Tuckerman, and D. Barkley, Extreme events in transitional turbulence, Philos. Trans. R. Soc. A 380, 20210036 (2022).
  28. J. D. Skufca, J. A. Yorke, and B. Eckhardt, Edge of chaos in a parallel shear flow, Phys. Rev. Lett. 96, 174101 (2006).
  29. T. Khapko, T. Kreilos, P. Schlatter, Y. Duguet, B. Eckhardt, and D. S. Henningson, Edge states as mediators of bypass transition in boundary-layer flows, J. Fluid Mech. 801, R2 (2016).
  30. G. Kawahara, M. Uhlmann, and L. van Veen, The significance of simple invariant solutions in turbulent flows, Annu. Rev. Fluid Mech. 44, 203 (2012).
  31. C. C. T. Pringle and R. R. Kerswell, Using nonlinear transient growth to construct the minimal seed for shear flow turbulence, Phys. Rev. Lett. 105, 154502 (2010).
  32. C. C. Pringle, A. P. Willis, and R. R. Kerswell, Minimal seeds for shear flow turbulence: Using nonlinear transient growth to touch the edge of chaos, J. Fluid Mech. 702, 415 (2012).
  33. R. Kerswell, Nonlinear nonmodal stability theory, Annu. Rev. Fluid Mech. 50, 319 (2018).
  34. T. M. Schneider, B. Eckhardt, and J. A. Yorke, Turbulence transition and the edge of chaos in pipe flow, Phys. Rev. Lett. 99, 034502 (2007).
  35. J. Wang, J. Gibson, and F. Waleffe, Lower branch coherent states in shear flows: Transition and control, Phys. Rev. Lett. 98, 204501 (2007).
  36. T. M. Schneider, J. F. Gibson, M. Lagha, F. De Lillo, and B. Eckhardt, Laminar-turbulent boundary in plane Couette flow, Phys. Rev. E 78, 037301 (2008).
  37. O. Dauchot and P. Manneville, Local versus global concepts in hydrodynamic stability theory, J. Phys. II 7, 371 (1997).
  38. F. Waleffe, Transition in shear flows. Nonlinear normality versus non‐normal linearity, Phys. Fluids 7, 3060 (1995).
  39. F. Waleffe, On a self-sustaining process in shear flows, Phys. Fluids 9, 883 (1997).
  40. M. Beneitez, Y. Duguet, P. Schlatter, and D. S. Henningson, Edge manifold as a Lagrangian coherent structure in a high-dimensional state space, Phys. Rev. Res. 2, 033258 (2020).
  41. M. Beneitez, Y. Duguet, and D. S. Henningson, Modeling the collapse of the edge when two transition routes compete, Phys. Rev. E 102, 053108 (2020).
  42. B. Kaszás, New methods for reduced-order modeling and uncertainty quantification for nonlinear analysis of fluid flows, Ph.D. thesis, ETH Zurich, 2023.
  43. M. Salmon, Closed-loop control of finite amplitude perturbations: Application to sub- and super-critical flow-bifurcations, Ph.D. thesis, École Nationale Supérieure des Arts et Métiers, 2024.
  44. X. Wan, H. Yu, and E. Weinan, Model the nonlinear instability of wall-bounded shear flows as a rare event: A study on two-dimensional Poiseuille flow, Nonlinearity 28, 1409 (2015).
  45. X. Wan, A minimum action method for small random perturbations of two-dimensional parallel shear flows, J. Comput. Phys. 235, 497 (2013).
  46. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer, New York, NY, 1984).
  47. J. Rolland, Rare, noise-induced, bypass transition in plane Couette flow can bypass instantons, Phys. Rev. E 110, 065106 (2024).
  48. J. Rolland, Collapse of transitional wall turbulence captured using a rare events algorithm, J. Fluid Mech. 931, A22 (2022).
  49. Y. Li, J. Duan, and X. Liu, Machine learning framework for computing the most probable paths of stochastic dynamical systems, Phys. Rev. E 103, 012124 (2021).
  50. J. Hu, X. Chen, and J. Duan, An Onsager–Machlup approach to the most probable transition pathway for a genetic regulatory network, Chaos 32, 041103 (2022).
  51. K. Yasuda and K. Ishimoto, Most probable path of an active Brownian particle, Phys. Rev. E 106, 064120 (2022).
  52. 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).
  53. K. Yasuda, K. Ishimoto, and S. Komura, Demon's variational principle for informational active matter, arXiv:2510.13145.
  54. J. Gladrow, U. F. Keyser, R. Adhikari, and J. Kappler, Experimental measurement of relative path probabilities and stochastic actions, Phys. Rev. X 11, 031022 (2021).
  55. L. Onsager and S. Machlup, Fluctuations and irreversible processes, Phys. Rev. 91, 1505 (1953).
  56. H. Risken, Fokker-Planck equation, in The Fokker-Planck Equation: Methods of Solution and Applications, edited by H. Haken (Springer, Berlin, 1989), pp. 63–95.
  57. The noise intensity D may differ from the standard definition of the diffusion constant by a factor of 2 [56].
  58. T. Grafke, R. Grauer, and T. Schäfer, The instanton method and its numerical implementation in fluid mechanics, J. Phys. A: Math. Theor. 48, 333001 (2015).
  59. J. Wang, K. Zhang, and E. Wang, Kinetic paths, time scale, and underlying landscapes: A path integral framework to study global natures of nonequilibrium systems and networks, J. Chem. Phys. 133, 125103 (2010).
  60. M. Sano and K. Tamai, A universal transition to turbulence in channel flow, Nat. Phys. 12, 249 (2016).
  61. G. Lemoult, L. Shi, K. Avila, S. V. Jalikop, M. Avila, and B. Hof, Directed percolation phase transition to sustained turbulence in Couette flow, Nat. Phys. 12, 254 (2016).
  62. L. S. Tuckerman, M. Chantry, and D. Barkley, Patterns in wall-bounded shear flows, Annu. Rev. Fluid Mech. 52, 343 (2020).
  63. S. G. Huisman, R. C. A. van der Veen, C. Sun, and D. Lohse, Multiple states in highly turbulent Taylor–Couette flow, Nat. Commun. 5, 3820 (2014).
  64. K. Sugiyama, R. Ni, R. J. A. M. Stevens, T. S. Chan, S.-Q. Zhou, H.-D. Xi, C. Sun, S. Grossmann, K.-Q. Xia, and D. Lohse, Flow reversals in thermally driven turbulence, Phys. Rev. Lett. 105, 034503 (2010).
  65. P. K. Mishra, J. Herault, S. Fauve, and M. K. Verma, Dynamics of reversals and condensates in two-dimensional Kolmogorov flows, Phys. Rev. E 91, 053005 (2015).
  66. B. Suri, Predictive framework for flow reversals and excursions in turbulence, Phys. Rev. Lett. 133, 154002 (2024).
  67. E. Shoji, H. Nishimori, A. Awazu, S. Izumi, and M. Iima, Localized bioconvection patterns and their initial state dependency in Euglena gracilis suspensions in an annular container, J. Phys. Soc. Jpn. 83, 043001 (2014).
  68. H. Yamashita, T. Yamaguchi, N. J. Suematsu, S. Izumi, and M. Iima, Formation of a single bioconvection spot in Euglena suspension induced by negative phototaxis, Phys. Fluids 37, 071909 (2025).
  69. Y. Hiruta and K. Ishimoto, Localization and bistability of bioconvection in a doubly periodic domain, J. Phys. Soc. Jpn. 94, 104401 (2025).
  70. J. Dunkel, S. Heidenreich, K. Drescher, H. H. Wensink, M. Bär, and R. E. Goldstein, Fluid dynamics of bacterial turbulence, Phys. Rev. Lett. 110, 228102 (2013).
  71. Y. Hiruta and K. Ishimoto, Reciprocal microswimming in fluctuating and confined environments, Phys. Rev. Res. 6, 013117 (2024).
  72. D. Nishiguchi, S. Shiratani, K. A. Takeuchi, and I. S. Aranson, Vortex reversal is a precursor of confined bacterial turbulence, Proc. Natl. Acad. Sci. USA 122, e2414446122 (2025).
  73. A. Mazzolo and C. Monthus, Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator, Phys. Rev. E 107, 014101 (2023).
  74. K. Yasuda, K. Ishimoto, A. Kobayashi, L.-S. Lin, I. Sou, Y. Hosaka, and S. Komura, Time-correlation functions for odd Langevin systems, J. Chem. Phys. 157, 095101 (2022).
  75. K. Ishimoto, C. Moreau, and K. Yasuda, Odd elastohydrodynamics: Non-reciprocal living material in a viscous fluid, PRX Life 1, 023002 (2023).
  76. M. Raissi, P. Perdikaris, and G. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686 (2019).
  77. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Learning representations by back-propagating errors, Nature (London) 323, 533 (1986).
  78. Y. Hiruta and K. Ishimoto, Autoencoder for limit cycle in Kolmogorov flow, J. Phys. Soc. Jpn. 94, 064401 (2025).
  79. A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al., PyTorch: An imperative style, high-performance deep learning library, in Advances in Neural Information Processing Systems, edited by H. Wallach, H. Larochelle, A. Beygelzimer, F. d’Alché-Buc, E. Fox, and R. Garnett (Curran Associates, Inc., New York, 2019), Vol. 32.

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Phys. Rev. Research 8, 023303– Published 16 June, 2026
  • Received 7 January 2026
  • Accepted 20 May 2026
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