Journal of Statistical Physics

, Volume 156, Issue 6, pp 1066–1092 | Cite as

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

  • Freddy Bouchet
  • Jason LaurieEmail author
  • Oleg Zaboronski


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.


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


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Authors and Affiliations

  1. 1.Laboratoire de Physique, École Normale Supérieure de LyonLyonFrance
  2. 2.Department of Physics of Complex SystemsWeizmann Institute of ScienceRehovotIsrael
  3. 3.Mathematics InstituteWarwick UniversityCoventry UK

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