  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.
