158 research outputs found
A complete theory of low-energy phase diagrams for two-dimensional turbulence steady states and equilibria
For the 2D Euler equations and related models of geophysical flows, minima of
energy--Casimir variational problems are stable steady states of the equations
(Arnol'd theorems). The same variational problems also describe sets of
statistical equilibria of the equations. In this paper, we make use of
Lyapunov--Schmidt reduction in order to study the bifurcation diagrams for
these variational problems, in the limit of small energy or, equivalently, of
small departure from quadratic Casimir functionals. We show a generic
occurrence of phase transitions, either continuous or discontinuous. We derive
the type of phase transitions for any domain geometry and any model analogous
to the 2D Euler equations. The bifurcations depend crucially on a_4, the
quartic coefficient in the Taylor expansion of the Casimir functional around
its minima. Note that a_4 can be related to the fourth moment of the vorticity
in the statistical mechanics framework. A tricritical point (bifurcation from a
continuous to a discontinuous phase transition) often occurs when a_4 changes
sign. The bifurcations depend also on possible constraints on the variational
problems (circulation, energy). These results show that the analytical results
obtained with quadratic Casimir functionals by several authors are non-generic
(not robust to a small change in the parameters)
Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations
We discuss the dynamics of zonal (or unidirectional) jets for barotropic
flows forced by Gaussian stochastic fields with white in time correlation
functions. This problem contains the stochastic dynamics of 2D Navier-Stokes
equation as a special case. We consider the limit of weak forces and
dissipation, when there is a time scale separation between the inertial time
scale (fast) and the spin-up or spin-down time (large) needed to reach an
average energy balance. In this limit, we show that an adiabatic reduction (or
stochastic averaging) of the dynamics can be performed. We then obtain a
kinetic equation that describes the slow evolution of zonal jets over a very
long time scale, where the effect of non-zonal turbulence has been integrated
out. The main theoretical difficulty, achieved in this work, is to analyze the
stationary distribution of a Lyapunov equation that describes quasi-Gaussian
fluctuations around each zonal jet, in the inertial limit. This is necessary to
prove that there is no ultraviolet divergence at leading order in such a way
that the asymptotic expansion is self-consistent. We obtain at leading order a
Fokker--Planck equation, associated to a stochastic kinetic equation, that
describes the slow jet dynamics. Its deterministic part is related to well
known phenomenological theories (for instance Stochastic Structural Stability
Theory) and to quasi-linear approximations, whereas the stochastic part allows
to go beyond the computation of the most probable zonal jet. We argue that the
effect of the stochastic part may be of huge importance when, as for instance
in the proximity of phase transitions, more than one attractor of the dynamics
is present
Controversy about the applicability of Tsallis statistics to the HMF model
Comment to "Nonextensive Thermodynamics and Glassy Behaviour in Hamiltonian
Systems" by A. Rapisarda and A. Pluchino, Europhysics News 36, 202 (2005)
Equilibrium statistical mechanics and energy partition for the shallow water model
The aim of this paper is to use large deviation theory in order to compute
the entropy of macrostates for the microcanonical measure of the shallow water
system. The main prediction of this full statistical mechanics computation is
the energy partition between a large scale vortical flow and small scale
fluctuations related to inertia-gravity waves. We introduce for that purpose a
discretized model of the continuous shallow water system, and compute the
corresponding statistical equilibria. We argue that microcanonical equilibrium
states of the discretized model in the continuous limit are equilibrium states
of the actual shallow water system. We show that the presence of small scale
fluctuations selects a subclass of equilibria among the states that were
previously computed by phenomenological approaches that were neglecting such
fluctuations. In the limit of weak height fluctuations, the equilibrium state
can be interpreted as two subsystems in thermal contact: one subsystem
corresponds to the large scale vortical flow, the other subsystem corresponds
to small scale height and velocity fluctuations. It is shown that either a
non-zero circulation or rotation and bottom topography are required to sustain
a non-zero large scale flow at equilibrium. Explicit computation of the
equilibria and their energy partition is presented in the quasi-geostrophic
limit for the energy-enstrophy ensemble. The possible role of small scale
dissipation and shocks is discussed. A geophysical application to the Zapiola
anticyclone is presented.Comment: Journal of Statistical Physics, Springer Verlag, 201
Perturbative calculation of quasi-potential in non-equilibrium diffusions: a mean-field example
In stochastic systems with weak noise, the logarithm of the stationary
distribution becomes proportional to a large deviation rate function called the
quasi-potential. The quasi-potential, and its characterization through a
variational problem, lies at the core of the Freidlin-Wentzell large deviations
theory%.~\cite{freidlin1984}.In many interacting particle systems, the particle
density is described by fluctuating hydrodynamics governed by Macroscopic
Fluctuation Theory%, ~\cite{bertini2014},which formally fits within
Freidlin-Wentzell's framework with a weak noise proportional to ,
where is the number of particles. The quasi-potential then appears as a
natural generalization of the equilibrium free energy to non-equilibrium
particle systems. A key physical and practical issue is to actually compute
quasi-potentials from their variational characterization for non-equilibrium
systems for which detailed balance does not hold. We discuss how to perform
such a computation perturbatively in an external parameter , starting
from a known quasi-potential for . In a general setup, explicit
iterative formulae for all terms of the power-series expansion of the
quasi-potential are given for the first time. The key point is a proof of
solvability conditions that assure the existence of the perturbation expansion
to all orders. We apply the perturbative approach to diffusive particles
interacting through a mean-field potential. For such systems, the variational
characterization of the quasi-potential was proven by Dawson and Gartner%.
~\cite{dawson1987,dawson1987b}. Our perturbative analysis provides new explicit
results about the quasi-potential and about fluctuations of one-particle
observables in a simple example of mean field diffusions: the
Shinomoto-Kuramoto model of coupled rotators%. ~\cite{shinomoto1986}. This is
one of few systems for which non-equilibrium free energies can be computed and
analyzed in an effective way, at least perturbatively
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