52 research outputs found

### Fluctuations of large-scale jets in the stochastic 2D Euler equation

Two-dimensional turbulence in a rectangular domain self-organises into
large-scale unidirectional jets. While several results are present to
characterize the mean jets velocity profile, much less is known about the
fluctuations. We study jets dynamics in the stochastically forced
two-dimensional Euler equations. In the limit where the average jets velocity
profile evolves slowly with respect to turbulent fluctuations, we employ a
multi-scale (kinetic theory) approach, which relates jet dynamics to the
statistics of Reynolds stresses. We study analytically the Gaussian
fluctuations of Reynolds stresses and predict the spatial structure of the jets
velocity covariance. Our results agree qualitatively well with direct numerical
simulations, clearly showing that the jets velocity profile are enhanced away
from the stationary points of the average velocity profile. A numerical test of
our predictions at quantitative level seems out of reach at the present day

### 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

### 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 $1/\sqrt{N}$,
where $N$ 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 $\lambda$, starting
from a known quasi-potential for $\lambda=0$. 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

### Solvable model of a self-gravitating system

We introduce and discuss an effective model of a self-gravitating system
whose equilibrium thermodynamics can be solved in both the microcanonical and
the canonical ensemble, up to a maximization with respect to a single variable.
Such a model can be derived from a model of self-gravitating particles confined
on a ring, referred to as the self-gravitating ring (SGR) model, allowing a
quantitative comparison between the thermodynamics of the two models. Despite
the rather crude approximations involved in its derivation, the effective model
compares quite well with the SGR model. Moreover, we discuss the relation
between the effective model presented here and another model introduced by
Thirring forty years ago. The two models are very similar and can be considered
as examples of a class of minimal models of self-gravitating systems.Comment: 21 pages, 6 figures; submitted to JSTAT for the special issue on
long-range interaction

### Linear response theory for long-range interacting systems in quasistationary states

Long-range interacting systems, while relaxing to equilibrium, often get
trapped in long-lived quasistationary states which have lifetimes that diverge
with the system size. In this work, we address the question of how a long-range
system in a quasistationary state (QSS) responds to an external perturbation.
We consider a long-range system that evolves under deterministic Hamilton
dynamics. The perturbation is taken to couple to the canonical coordinates of
the individual constituents. Our study is based on analyzing the Vlasov
equation for the single-particle phase space distribution. The QSS represents
stable stationary solution of the Vlasov equation in the absence of the
external perturbation. In the presence of small perturbation, we linearize the
perturbed Vlasov equation about the QSS to obtain a formal expression for the
response observed in a single-particle dynamical quantity. For a QSS that is
homogeneous in the coordinate, we obtain an explicit formula for the response.
We apply our analysis to a paradigmatic model, the Hamiltonian mean-field
model, that involves particles moving on a circle under Hamilton dynamics. Our
prediction for the response of three representative QSSs in this model (the
water-bag QSS, the Fermi-Dirac QSS, and the Gaussian QSS) is found to be in
good agreement with $N$-particle simulations for large $N$. We also show the
long-time relaxation of the water-bag QSS to the Boltzmann-Gibbs equilibrium
state.Comment: 13 pages, 4 figures; v2: typos fixed; v3: small changes, close to the
published versio

### Fluctuating kinetic theory and fluctuating hydrodynamics of aligning active particles: the dilute limit

Kinetic and hydrodynamic theories are widely employed for describing the
collective behaviour of active matter systems. At the fluctuating level, these
have been obtained from explicit coarse-graining procedures in the limit where
each particle interacts weakly with many others, so that the total forces and
torques exerted on each of them is of order unity at all times. Such limit is
however not relevant for dilute systems that mostly interact via alignment;
there, collisions are rare and make the self-propulsion direction to change
abruptly. We derive a fluctuating kinetic theory, and the corresponding
fluctuating hydrodynamics, for aligning self-propelled particles in the limit
of dilute systems. We discover that fluctuations at kinetic level are not
Gaussian and depend on the interactions among particles, but that only their
Gaussian part survives in the hydrodynamic limit. At variance with fluctuating
hydrodynamics for weakly interacting particles, we find that the noise variance
at hydrodynamic level depends on the interaction rules among particles and is
proportional to the square of the density, reflecting the binary nature of the
aligning process. The results of this paper, which are derived for polar
self-propelled particles with polar alignment, could be straightforwardly
extended to polar particles with nematic alignment or to fully nematic systems.Comment: 23 pages, 4 figure

### Energy landscape and phase transitions in the self-gravitating ring model

We apply a recently proposed criterion for the existence of phase
transitions, which is based on the properties of the saddles of the energy
landscape, to a simplified model of a system with gravitational interactions,
referred to as the self-gravitating ring model. We show analytically that the
criterion correctly singles out the phase transition between a homogeneous and
a clustered phase and also suggests the presence of another phase transition,
not previously known. On the basis of the properties of the energy landscape we
conjecture on the nature of the latter transition

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