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Freddy Bouchet

Publications and source records attributed to Freddy Bouchet.

86 records · Page 5Linked to original sources
arXiv2007

Simpler Variational Problem for Statistical Equilibria of the 2D Euler Equation and Other Systems with Long Range Interactions

The Robert-Sommeria-Miller equilibrium statistical mechanics predicts the final organization of two dimensional flows. This powerful theory is difficult to handle practically, due to the complexity associated with an infinite number of constraints. Several alternative simpler variational problems, based on Casimir's or stream function functionals, have been considered recently. We establish the relations between all these variational problems, justifying the use of simpler formulations.

cond-mat.stat-mech↗
arXiv2007

Algebraic Correlation Function and Anomalous Diffusion in the HMF model

In the quasi-stationary states of the Hamiltonian Mean-Field model, we numerically compute correlation functions of momenta and diffusion of angles with homogeneous initial conditions. This is an example, in a N-body Hamiltonian system, of anomalous transport properties characterized by non exponential relaxations and long-range temporal correlations. Kinetic theory predicts a striking transition between weak anomalous diffusion and strong anomalous diffusion. The numerical results are in excellent agreement with the quantitative predictions of the anomalous transport exponents. Noteworthy, also at statistical equilibrium, the system exhibits long-range temporal correlations: the correlation function is inversely proportional to time with a logarithmic correction instead of the usually expected exponential decay, leading to weak anomalous transport properties.

cond-mat.stat-mech↗
arXiv2005

The Vlasov equation and the Hamiltonian Mean-Field model

We show that the quasi-stationary states observed in the $N$-particle dynamics of the Hamiltonian Mean-Field (HMF) model are nothing but Vlasov stable homogeneous (zero magnetization) states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis $q$-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite $N$ effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann-Gibbs equilibrium.

cond-mat.stat-mech↗
arXiv2005

Prediction of anomalous diffusion and algebraic relaxations for long-range interacting systems, using classical statistical mechanics

We explain the ubiquity and extremely slow evolution of non gaussian out-of-equilibrium distributions for the Hamiltonian Mean-Field model, by means of traditional kinetic theory. Deriving the Fokker-Planck equation for a test particle, one also unambiguously explains and predicts striking slow algebraic relaxation of the momenta autocorrelation, previously found in numerical simulations. Finally, angular anomalous diffusion are predicted for a large class of initial distributions. Non Extensive Statistical Mechanics is shown to be unnecessary for the interpretation of these phenomena.

cond-mat.stat-mech↗
arXiv2005

Large deviation techniques applied to systems with long-range interactions

We discuss a method to solve models with long-range interactions in the microcanonical and canonical ensemble. The method closely follows the one introduced by Ellis, Physica D 133, 106 (1999), which uses large deviation techniques. We show how it can be adapted to obtain the solution of a large class of simple models, which can show ensemble inequivalence. The model Hamiltonian can have both discrete (Ising, Potts) and continuous (HMF, Free Electron Laser) state variables. This latter extension gives access to the comparison with dynamics and to the study of non-equilibri um effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the alpha-Ising model in one-dimension with $0\leqα<1$.

cond-mat.stat-mech↗
arXiv2005

Thermodynamics of the self-gravitating ring model

We present the phase diagram, in both the microcanonical and the canonical ensemble, of the Self-Gravitating-Ring (SGR) model, which describes the motion of equal point masses constrained on a ring and subject to 3D gravitational attraction. If the interaction is regularized at short distances by the introduction of a softening parameter, a global entropy maximum always exists, and thermodynamics is well defined in the mean-field limit. However, ensembles are not equivalent and a phase of negative specific heat in the microcanonical ensemble appears in a wide intermediate energy region, if the softening parameter is small enough. The phase transition changes from second to first order at a tricritical point, whose location is not the same in the two ensembles. All these features make of the SGR model the best prototype of a self-gravitating system in one dimension. In order to obtain the stable stationary mass distribution, we apply a new iterative method, inspired by a previous one used in 2D turbulence, which ensures entropy increase and, hence, convergence towards an equilibrium state.

cond-mat.stat-mech↗
arXiv2004

Classification of phase transitions and ensemble inequivalence, in systems with long range interactions

Systems with long range interactions in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors. We propose a classification of microcanonical phase transitions, of their link to canonical ones, and of the possible situations of ensemble inequivalence. We discuss previously observed phase transitions and inequivalence in self-gravitating, two-dimensional fluid dynamics and non-neutral plasmas. We note a number of generic situations that have not yet been observed in such systems.

cond-mat.stat-mech↗
arXiv2004

Minimal Stochastic Model for Fermi's Acceleration

We introduce a simple stochastic system able to generate anomalous diffusion both for position and velocity. The model represents a viable description of the Fermi's acceleration mechanism and it is amenable to analytical treatment through a linear Boltzmann equation. The asymptotic probability distribution functions (PDF) for velocity and position are explicitly derived. The diffusion process is highly non-Gaussian and the time growth of moments is characterized by only two exponents $ν_x$ and $ν_v$. The diffusion process is anomalous (non Gaussian) but with a defined scaling properties i.e. $P(|{\bf x}|,t) = 1/t^{ν_x}F_x(|{\bf x}|/t^{ν_x})$ and similarly for velocity.

cond-mat.stat-mech↗
arXiv2003

Parameterization of two-dimensional turbulence using an anisotropic maximum entropy production principle

We consider the modeling of the effect of unresolved scales, for two-dimensional and geophysical flows. We first show that the effect of small scales on a coarse-grained field, can be approximated at leading order, by the effect of the strain tensor on the gradient of the vorticity, which exactly conserves the energy. We show that this approximation would lead to unstable numerical code. In order to propose a stable parameterization, while taking into account of these dynamical properties, we apply a maximum entropy production principle. The parameterization acts as a selective diffusion proportional to the mean strain, in the contraction direction, while conserving the energy. We show on numerical computation that the obtained \foreignlanguage{french}{anisotropic relaxation equations} give an important predictability improvement, with respect to Navier-Stokes, Smagorinsky or hyperviscous parameterizations.

cond-mat.stat-mech↗
arXiv2003

Emergence of Jupiter's vortices and jets from random initial conditions

We explain the emergence and stability of the most important jets and vortices, in the highly turbulent Jupiter's atmosphere, by a statistical mechanics of the potential vorticity mixing. Using the Quasi-Geostrophic 1-1/2 layer, with topography, when the Rossby deformation radius is small, we predict strong jets. These jets can be either zonal, or closed into a ring structure like the Great Red Spot one. We reproduce the GRS observed velocity field to a very good quantitative accuracy. For smaller vortices, or for stronger topography curvature, we reproduce the characteristics properties of the White Ovals or of the cyclonic Brown Barges. The link between their shape, topography and surrounding shear is explicitly described. We obtain very strong qualitative results for the Jupiter's vortices. For instance, any of these vortices must be on topography extrema (in the reference frame moving with the structure), the shear in the active layer is larger than the shear in the deep layer. On a same latitudinal band, the velocity of the vortex is related to their latitude. These theoretical predictions are in accordance with the observed properties of Jovian vortices.

cond-mat.stat-mech↗
arXiv2003

The stochastic process of equilibrium fluctuations, of a system with long range interactions

The relaxation towards equilibrium of systems with long range interactions is not yet fully understood. As a step towards such a comprehension, we propose the study of the dynamical equilibrium fluctuations in a model system with long range interaction. We compute analytically, from the microscopic dynamics, the autocorrelation function of the order parameter. From this result, we derive analytically a Fokker Planck equation which describes the stochastic process of the impulsion of a single particle in an equilibrium bath. The diffusion coefficient is explicitly computed.

cond-mat.stat-mech↗
arXiv2002

Out-of-equilibrium states as statistical equilibria of an effective dynamics

We study the formation of coherent structures in a system with long-range interactions where particles moving on a circle interact through a repulsive cosine potential. Non equilibrium structures are shown to correspond to statistical equilibria of an effective dynamics, which is derived using averaging techniques. This simple behavior might be a prototype of others observed in more complicated systems with long-range interactions, like two-dimensional incompressible fluids or self-gravitating systems.

cond-mat.stat-mech↗
arXiv2002

Birth and long-time stabilization of out-of-equilibrium coherent structures

We study an analytically tractable model with long-range interactions for which an out-of-equilibrium very long-lived coherent structure spontaneously appears. The dynamics of this model is indeed very peculiar: a bicluster forms at low energy and is stable for very long time, contrary to statistical mechanics predictions. We first explain the onset of the structure, by approximating the short time dynamics with a forced Burgers equation. The emergence of the bicluster is the signature of the shock waves present in the associated hydrodynamical equations. The striking quantitative agreement with the dynamics of the particles fully confirms this procedure. We then show that a very fast timescale can be singled out from a slower motion. This enables us to use an adiabatic approximation to derive an effective Hamiltonian that describes very well the long time dynamics. We then get an explanation of the very long time stability of the bicluster: this out-of-equilibrium state corresponds to a statistical equilibrium of an effective mean-field dynamics.

cond-mat.stat-mech↗