Journal of Statistical Physics

, Volume 156, Issue 6, pp 1066–1092

# Langevin Dynamics, Large Deviations and Instantons for the Quasi-Geostrophic Model and Two-Dimensional Euler Equations

• Freddy Bouchet
• Jason Laurie
• Oleg Zaboronski
Article

## Abstract

We investigate a class of simple models for Langevin dynamics of turbulent flows, including the one-layer quasi-geostrophic equation and the two-dimensional Euler equations. Starting from a path integral representation of the transition probability, we compute the most probable fluctuation paths from one attractor to any state within its basin of attraction. We prove that such fluctuation paths are the time reversed trajectories of the relaxation paths for a corresponding dual dynamics, which are also within the framework of quasi-geostrophic Langevin dynamics. Cases with or without detailed balance are studied. We discuss a specific example for which the stationary measure displays either a second order (continuous) or a first order (discontinuous) phase transition and a tricritical point. In situations where a first order phase transition is observed, the dynamics are bistable. Then, the transition paths between two coexisting attractors are instantons (fluctuation paths from an attractor to a saddle), which are related to the relaxation paths of the corresponding dual dynamics. For this example, we show how one can analytically determine the instantons and compute the transition probabilities for rare transitions between two attractors.

## Keywords

Langevin dynamics Large deviations Fredilin–Wentzell theory Instanton Phase transitions Quasi-geostrophic dynamics

## References

1. 1.
Arnold, V.I.: An a priori estimate in the theory of hydrodynamic stability. Am. Math. Soc. Transl. 19, 267–269 (1969)Google Scholar
2. 2.
Baiesi, M., Boksenbojm, E., Maes, C., Wynants, B.: Nonequilibrium linear response for markov dynamics, ii: inertial dynamics. J. Stat. Phys. 139(3), 492–505 (2010)
3. 3.
Baiesi, M., Maes, C.: Enstrophy dissipation in two-dimensional turbulence. Phys. Rev. E 72(5), 056314 (2005)
4. 4.
Berhanu, M., Monchaux, R., Fauve, S., Mordant, N., Petrelis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Marie, L., Ravelet, F., Bourgoin, M., Odier, P., Pinton, J., Volk, R.: Magnetic field reversals in an experimental turbulent dynamo. Eur. Phys. Lett. 77, 59001 (2007)
5. 5.
Bertini, L., de Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory for stationary non-equilibrium states. J. Stat. Phys. 107, 635–675 (2002). doi:
6. 6.
Bessaih, H., Ferrario, B.: Invariant measures of gaussian type for 2d turbulence. J. Stat. Phys. 149(2), 259–283 (2012)
7. 7.
Bessaih, H., Ferrario, B.: Inviscid limit of stochastic damped 2d navier-stokes equations. Nonlinearity 27(1), 1 (2014)
8. 8.
Boucher, C., Ellis, R.S., Turkington, B.: Derivation of maximum entropy principles in two-dimensional turbulence via large deviations. J. Stat. Phys. 98(5–6), 1235 (2000)
9. 9.
Bouchet, F.: Simpler variational problems for statistical equilibria of the 2d euler equation and other systems with long range interactions. Phys. D 237, 1976–1981 (2008). doi:
10. 10.
Bouchet, F., Barré, J.: Classification of phase transitions and ensemble inequivalence, in systems with long range interactions. J. Stat. Phys. 118, 1073–1105 (2005). doi:
11. 11.
Bouchet, F., Corvellec, M.: Invariant measures of the 2D Euler and Vlasov equations. J. Stat. Mech. 8, P08021 (2010)Google Scholar
12. 12.
Bouchet, F., Laurie, J.: Statistical mechanics approaches to self organization of 2d flows: fifty years after, where does onsager’s route lead to? RIMS Kôkyûroku. Res. Inst. Math. Sci. 1798, 42–58 (2012)Google Scholar
13. 13.
Bouchet, F., Laurie, J., Zaboronski, O.: Control and instanton trajectories for random transitions in turbulent flows. J. Phys. Conf. Ser. 318(2), 022041 (2011). doi:
14. 14.
Bouchet, F., Nardini, C., Tangarife, T.: Kinetic theory of jet dynamics in the stochastic barotropic and 2d navier-stokes equations. J. Stat. Phys. 153(4), 572–625 (2013)
15. 15.
Bouchet, F., Simonnet, E.: Random changes of flow topology in two-dimensional and geophysical turbulence. Phys. Rev. Lett. 102(9), 094504 (2009). doi:
16. 16.
Bouchet, F., Venaille, A.: Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227–295 (2012)
17. 17.
Brown, E., Ahlers, G.: Rotations and cessations of the large-scale circulation in turbulent Rayleigh-Bénard convection. J. Fluid Mech. 568, 351 (2006). doi:
18. 18.
Brzezniak, Z., Cerrai, S., Freidlin, M.: Quasipotential and exit time for 2d stochastic navier-stokes equations driven by space time white noise. arXiv preprint arXiv:1401.6299 (2014)
19. 19.
Chandra, M., Verma, M.: Dynamics and symmetries of flow reversals in turbulent convection. Phys. Rev. E 83(6), 7–10 (2011). doi:
20. 20.
Chavanis, P.: Fokker–Planck equations: applications to stellar dynamics and two-dimensional turbulence. Phys. Rev. E 68(3), 036108 (2003)
21. 21.
Chavanis, P.H.: Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations. Eur. Phys. J. B 70, 73–105 (2009). doi:
22. 22.
Chavanis, P.H., Sommeria, J.: Classification of self-organized vortices in two-dimensional turbulence: the case of a bounded domain. J. Fluid Mech. 314, 267–297 (1996)
23. 23.
Corvellec, M., Bouchet, F.: A complete theory of low-energy phase diagrams for two-dimensional turbulence steady states and equilibria. arXiv preprint arXiv:1207.1966 (2012)
24. 24.
Evans, D.J., Cohen, E.G.D., Morriss, G.P.: Probability of second law violations in shearing steady states. Phys. Rev. Lett. 71, 2401–2404 (1993). doi:
25. 25.
Faris, W., Jona-Lasinio, G.: Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 15, 3025–3055 (1982)
26. 26.
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, Berlin (1984)
27. 27.
Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74, 2694–2697 (1995). doi:
28. 28.
Gallet, B., Herault, J., Laroche, C., Pétrélis, F., Fauve, S.: Reversals of a large-scale field generated over a turbulent background. Geophys. Astrophys. Fluid Dyn. 106(4–5), 468–492 (2012)
29. 29.
Gourcy, M.: A large deviation principle for 2d stochastic navier-stokes equation. Stoch. Process. Appl. 117(7), 904–927 (2007)
30. 30.
Graham, R.: Macroscopic potentials, bifurcations and noise in dissipative systems. In: Garrido, L. (ed.) Fluctuations and Stochastic Phenomena in Condensed Matter, pp. 1–34. Springer, New York (1987)Google Scholar
31. 31.
Hairer, M., Maas, J., Weber, H.: Approximating rough stochastic pdes. Commun. Pure Appl. Math. 67, 1129–1214 (2013)
32. 32.
Hairer, M., Weber, H.: Large deviations for white-noise driven, nonlinear stochastic pdes in two and three dimensions. arXiv preprint arXiv:1404.5863 (2014)
33. 33.
Haussmann, U.G., Pardoux, E.: Time reversal of diffusions. Ann. Probab. 14, 1188–1205 (1986)
34. 34.
Herbert, C.: Additional invariants and statistical equilibria for the 2d euler equations on a spherical domain. J. Stat. Phys. 152(6), 1084–1114 (2013)
35. 35.
Herbert, C., Dubrulle, B., Chavanis, P.H., Paillard, D.: Statistical mechanics of quasi-geostrophic flows on a rotating sphere. J. Stat. Mech. 2012(05), P05023 (2012)
36. 36.
Herbert, C., Marino, R., Pouquet, A.: Statistical equilibrium and inverse cascades of vortical modes for rotating and stratified flows. Bull. Am. Phys. Soc. 58, 209–230 (2013)Google Scholar
37. 37.
Heymann, M., Vanden-Eijnden, E.: The geometric minimum action method: A least action principle on the space of curves. Commun. Pure Appl. Math. 61(8), 1052–1117 (2008). doi:
38. 38.
Holm, D.D., Marsden, J.E., Ratiu, T.S.: The Euler–Poincare equations and semidirect products with applications to continuum theories. p. 1015 (1998). arXiv:chao-dyn/9801015
39. 39.
Jaksic, V., Nersesyan, V., Pillet, C.A., Shirikyan, A.: Large deviations from a stationary measure for a class of dissipative pde’s with random kicks. arXiv preprint arXiv:1212.0527 (2012)
40. 40.
Jaksic, V., Nersesyan, V., Pillet, C.A., Shirikyan, A.: Large deviations and gallavotti-cohen principle for dissipative pde’s with rough noise. arXiv preprint arXiv:1312.2964 (2013)
41. 41.
Janssen, H.: Field-theoretic method applied to critical dynamics. In: Enz, C.P. (ed.) Dynamical critical phenomena and related topics, pp. 25–47. Springer, New York (1979)Google Scholar
42. 42.
Jona-Lasinio, G., Mitter, P.: On the stochastic quantization of field theory. Commun. Math. Phys. 101(3), 409–436 (1985)
43. 43.
Kraichnan, R.H., Montgomery, D.: Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619 (1980)
44. 44.
Kuksin, S.B.: The eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys. 115, 469–492 (2004)
45. 45.
Kuksin, S.B., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence. Cambridge University Press, Cambridge (2012)
46. 46.
Luchinsky, D., McClintock, P.V., Dykman, M.: Analogue studies of nonlinear systems. Rep. Prog. Phys. 61(8), 889 (1998)
47. 47.
Maassen, S.R., Clercx, H.J.H., Van Heijst, G.J.F.: Self-organization of decaying quasi-two-dimensional turbulence in stratified fluid in rectangular containers. J. Fluid Mech. 495, 19–33 (2003). doi:
48. 48.
Maes, C., Netočný, K.: Canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states. Europhys. Lett. 82, 30003 (2008). doi:
49. 49.
Michel, J., Robert, R.: Large deviations for young measures and statistical mechanics of infinite dimensional dynamical systems with conservation law. Commun. Math. Phys. 159, 195–215 (1994)
50. 50.
Miller, J.: Statistical mechanics of euler equations in two dimensions. Phys. Rev. Lett. 65(17), 2137–2140 (1990). doi:
51. 51.
Nardini, C., Gupta, S., Ruffo, S., Dauxois, T., Bouchet, F.: Kinetic theory for non-equilibrium stationary states in long-range interacting systems. J. Stat. Mech. 1, L01002 (2012). doi: Google Scholar
52. 52.
Nardini, C., Gupta, S., Ruffo, S., Dauxois, T., Bouchet, F.: Kinetic theory of nonequilibrium stochastic long-range systems: phase transition and bistability. J. Stat. Mech. 2012(12), P12010 (2012)
53. 53.
Naso, A., Chavanis, P.H., Dubrulle, B.: Statistical mechanics of Fofonoff flows in an oceanic basin (2009). arXiv:1007.0164
54. 54.
Naso, A., Chavanis, P.H., Dubrulle, B.: Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states. Eur. Phys. J. B 77, 187–212 (2010)
55. 55.
Niemela, J.J., Skrbek, L., Sreenivasan, K.R., Donnelly, R.J.: The wind in confined thermal convection. J. Fluid Mech. 449, 169 (2001). doi:
56. 56.
Onsager, L., Machlup, S.: Fluctuations and Irreversible Processes. Phys. Rev. 91, 1505–1512 (1953). doi:
57. 57.
Pétrélis, F., Fauve, S.: Mechanisms for magnetic field reversals. Philos. Trans. R. Soc. A 368(1916), 1595–1605 (2010)
58. 58.
Potters, M., Vaillant, T., Bouche, F.: Sampling microcanonical measures of the 2d euler equations through creutz’s algorithm: a phase transition from disorder to order when energy is increased. J. Stat. Mech. 2013(02), P02017 (2013)
59. 59.
Qiu, B., Miao, W.: Kuroshio path variations South of Japan: bimodality as a self-sustained internal oscillation. J. Phys. Oceanogr. 30, 2124–2137 (2000)
60. 60.
Rahmstorf, S., et al.: Ocean circulation and climate during the past 120,000 years. Nature 419(6903), 207–214 (2002)
61. 61.
Ravelet, F., Marié, L., Chiffaudel, A., Daviaud, F.: Multistability and memory effect in a highly turbulent flow: experimental evidence for a global bifurcation. Phys. Rev. Lett. 93(16), 164501 (2004). doi:
62. 62.
Ren, W., Vanden-Eijnden, E., et al.: Minimum action method for the study of rare events. Commun. Pure Appl. Math. 57(5), 637–656 (2004)
63. 63.
Robert, R.: Etats d’équilibre statistique pour l’écoulement bidimensionnel d’un fluide parfait. C. R. Acad. Sci. 1(311), 575–578 (1990)Google Scholar
64. 64.
Robert, R.: A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Stat. Phys. 65, 531–553 (1991)
65. 65.
Robert, R., Sommeria, J.: Statistical equilibrium states for two-dimensional flows. J. Fluid Mech. 229, 291–310 (1991). doi:
66. 66.
Schmeits, M.J., Dijkstra, H.A.: Bimodal Behavior of the Kuroshio and the Gulf Stream. J. Phys. Oceanogr. 31, 3435–3456 (2001). doi:
67. 67.
Sommeria, J.: Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–68 (1986)Google Scholar
68. 68.
Sugiyama, K., Ni, R., Stevens, R., Chan, T., Zho, S.Q., Xi, H.D., Sun, C., Grossmann, S., Xia, K.Q., Lohse, D.: Flow reversals in thermally driven turbulence. Phys. Rev. Lett. 105(3), 1–4 (2010). doi:
69. 69.
Tailleur, J., Kurchan, J., Lecomte, V.: Mapping out-of-equilibrium into equilibrium in one-dimensional transport models. J. Phys. A 41, 5001 (2008). doi:
70. 70.
Thalabard, S., Dubrulle, B., Bouchet, F.: Statistical mechanics of the 3d axisymmetric euler equations in a taylor-couette geometry. arXiv preprint arXiv:1306.1081 (2013)
71. 71.
Venaille, A.: Bottom-trapped currents as statistical equilibrium states above topographic anomalies. arXiv preprint arXiv:1202.6155 (2012)
72. 72.
Venaille, A., Bouchet, F.: Statistical ensemble inequivalence and bicritical points for two-dimensional flows and geophysical flows. Phys. Rev. Lett. 102(2), 104501 (2009). doi:
73. 73.
Venaille, A., Vallis, G.K., Griffies, S.M.: The catalytic role of the beta effect in barotropization processes. J. Fluid Mech. 709, 490–515 (2012). doi:
74. 74.
Weeks, E.R., Tian, Y., Urbach, J.S., Ide, K., Swinney, H.L., Ghil, M.: Transitions between blocked and zonal flows in a rotating annulus. Science 278, 1598 (1997)
75. 75.
Weichman, P.B.: Long-range correlations and coherent structures in magnetohydrodynamic equilibria. Phys. Rev. Lett. 109(23), 235002 (2012)
76. 76.
Zinn-Justin, J., et al.: Quantum Field Theory and Critical Phenomena, vol. 142. Clarendon Press, Oxford (2002)