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

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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.

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Authors and Affiliations

  1. Laboratoire de Physique, École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 , Lyon, France

    Freddy Bouchet

  2. Department of Physics of Complex Systems, Weizmann Institute of Science, 234 Herzl Street, 76100 , Rehovot, Israel

    Jason Laurie

  3. Mathematics Institute, Warwick University, Gibbet Hill Road, Coventry , CV4 7AL, UK

    Oleg Zaboronski

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  1. Freddy Bouchet
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Bouchet, F., Laurie, J. & Zaboronski, O. Langevin Dynamics, Large Deviations and Instantons for the Quasi-Geostrophic Model and Two-Dimensional Euler Equations. J Stat Phys 156, 1066–1092 (2014). https://doi.org/10.1007/s10955-014-1052-5

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