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

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

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

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