arXiv subjects

Freddy Bouchet

Publications and source records attributed to Freddy Bouchet.

At least 73 records · Page 4Linked to original sources

arXiv2012

Sampling microcanonical measures of the 2D Euler equations through Creutz's algorithm: a phase transition from disorder to order when energy is increased

The 2D Euler equations is the basic example of fluid models for which a microcanical measure can be constructed from first principles. This measure is defined through finite-dimensional approximations and a limiting procedure. Creutz's algorithm is a microcanonical generalization of the Metropolis-Hasting algorithm (to sample Gibbs measures, in the canonical ensemble). We prove that Creutz's algorithm can sample finite-dimensional approximations of the 2D Euler microcanonical measures (incorporating fixed energy and other invariants). This is essential as microcanonical and canonical measures are known to be inequivalent at some values of energy and vorticity distribution. Creutz's algorithm is used to check predictions from the mean-field statistical mechanics theory of the 2D Euler equations (the Robert-Sommeria-Miller theory). We found full agreement with theory. Three different ways to compute the temperature give consistent results. Using Creutz's algorithm, a first-order phase transition never observed previously, and a situation of statistical ensemble inequivalence are found and studied. Strikingly, and contrasting usual statistical mechanics interpretations, this phase transition appears from a disordered phase to an ordered phase (with less symmetries) when energy is increased. We explain this paradox.

cond-mat.stat-mech↗

arXiv2012

Kinetic theory of nonequilibrium stochastic long-range systems: Phase transition and bistability

We study long-range interacting systems driven by external stochastic forces that act collectively on all the particles constituting the system. Such a scenario is frequently encountered in the context of plasmas, self-gravitating systems, two-dimensional turbulence, and also in a broad class of other systems. Under the effect of stochastic driving, the system reaches a stationary state where external forces balance dissipation on average. These states have the invariant probability that does not respect detailed balance, and are characterized by non-vanishing currents of conserved quantities. In order to analyze spatially homogeneous stationary states, we develop a kinetic approach that generalizes the one known for deterministic long-range systems; we obtain a very good agreement between predictions from kinetic theory and extensive numerical simulations. Our approach may also be generalized to describe spatially inhomogeneous stationary states. We also report on numerical simulations exhibiting a first-order nonequilibrium phase transition from homogeneous to inhomogeneous states. Close to the phase transition, the system shows bistable behavior between the two states, with a mean residence time that diverges as an exponential in the inverse of the strength of the external stochastic forces, in the limit of low values of such forces.

cond-mat.stat-mech↗

arXiv2012

Vortex-density fluctuations, energy spectra and vortical regions in superfluid turbulence

Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations.

cond-mat.other↗

arXiv2012

The Equivalence of the Lagrangian-Averaged Navier-Stokes-α Model and the Rational LES model in Two Dimensions

In the Large Eddy Simulation (LES) framework for modeling a turbulent flow, when the large scale velocity field is defined by low-pass filtering the full velocity field, a Taylor series expansion of the full velocity field in terms of the large scale velocity field leads (at the leading order) to the nonlinear gradient model for the subfilter stresses. Motivated by the fact that while the nonlinear gradient model shows excellent a priori agreement in resolved simulations, the use of this model by itself is problematic, we consider two models that are related, but better behaved: The Rational LES model that uses a sub-diagonal Pade approximation instead of a Taylor series expansion and the Lagrangian Averaged Navier-Stokes-α model that uses a regulariza- tion approach to modeling turbulence. In this article, we show that these two latter models are identical in two dimensions.

physics.flu-dyn↗

arXiv2012

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).

cond-mat.stat-mech↗

arXiv2012

Non-classical large deviations for a noisy system with non-isolated attractors

We study the large deviations of a simple noise-perturbed dynamical system having continuous sets of steady states, which mimick those found in some partial differential equations related, for example, to turbulence problems. The system is a two-dimensional nonlinear Langevin equation involving a dissipative, non-potential force, which has the essential effect of creating a line of stable fixed points (attracting line) touching a line of unstable fixed points (repelling line). Using different analytical and numerical techniques, we show that the stationary distribution of this system satisfies in the low-noise limit a large deviation principle containing two competing terms: i) a classical but sub-dominant large deviation term, which can be derived from the Freidlin-Wentzell theory of large deviations by studying the fluctuation paths or instantons of the system near the attracting line, and ii) a dominant large deviation term, which does not follow from the Freidlin-Wentzell theory, as it is related to fluctuation paths of zero action, referred to as sub-instantons, emanating from the repelling line. We discuss the nature of these sub-instantons, and show how they arise from the connection between the attracting and repelling lines. We also discuss in a more general way how we expect these to arise in more general stochastic systems having connected sets of stable and unstable fixed points, and how they should determine the large deviation properties of these systems.

cond-mat.stat-mech↗

arXiv2012

Kinetic theory for non-equilibrium stationary states in long-range interacting systems

We study long-range interacting systems perturbed by external stochastic forces. Unlike the case of short-range systems, where stochastic forces usually act locally on each particle, here we consider perturbations by external stochastic fields. The system reaches stationary states where external forces balance dissipation on average. These states do not respect detailed balance and support non-vanishing fluxes of conserved quantities. We generalize the kinetic theory of isolated long-range systems to describe the dynamics of this non-equilibrium problem. The kinetic equation that we obtain applies to plasmas, self-gravitating systems, and to a broad class of other systems. Our theoretical results hold for homogeneous states, but may also be generalized to apply to inhomogeneous states. We obtain an excellent agreement between our theoretical predictions and numerical simulations. We discuss possible applications to describe non-equilibrium phase transitions.

cond-mat.stat-mech↗

arXiv2011

Statistical mechanics of two-dimensional and geophysical flows

The theoretical study of the self-organization of two-dimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a self-contained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to applications to Jupiter's troposphere and ocean vortices and jets. Emphasize has been placed on examples with available analytical treatment in order to favor better understanding of the physics and dynamics. The equilibrium microcanonical measure is built from the Liouville theorem. On this theoretical basis, we predict the output of the long time evolution of complex turbulent flows as statistical equilibria. This is applied to make quantitative models of two-dimensional turbulence, the Great Red Spot and other Jovian vortices, ocean jets like the Gulf-Stream, and ocean vortices. We also present recent results for non-equilibrium situations, for the studies of either the relaxation towards equilibrium or non-equilibrium steady states.

physics.flu-dyn↗

arXiv2011

Oceanic rings and jets as statistical equilibrium states

Equilibrium statistical mechanics of two-dimensional flows provides an explanation and a prediction for the self-organization of large scale coherent structures. This theory is applied in this paper to the description of oceanic rings and jets, in the framework of a 1.5 layer quasi-geostrophic model. The theory predicts the spontaneous formation of regions where the potential vorticity is homogenized, with strong and localized jets at their interface. Mesoscale rings are shown to be close to a statistical equilibrium: the theory accounts for their shape, their drift, and their ubiquity in the ocean, independently of the underlying generation mechanism. At basin scale, inertial states presenting mid basin eastward jets (and then different from the classical Fofonoff solution) are described as marginally unstable states. These states are shown to be marginally unstable for the equilibrium statistical theory. In that case, considering a purely inertial limit is a first step toward more comprehensive out of equilibrium studies that would take into account other essential aspects, such as wind forcing.

physics.ao-ph↗

arXiv2011

Solvable phase diagrams and ensemble inequivalence for two-dimensional and geophysical turbulent flows

Using explicit analytical computations, generic occurrence of inequivalence between two or more statistical ensembles is obtained for a large class of equilibrium states of two-dimensional and geophysical turbulent flows. The occurrence of statistical ensemble inequivalence is shown to be related to previously observed phase transitions in the equilibrium flow topology. We find in these turbulent flow equilibria, two mechanisms for the appearance of ensemble equivalences, that were not observed in any physical systems before. These mechanisms are associated respectively with second-order azeotropy (simultaneous appearance of two second-order phase transitions), and with bicritical points (bifurcation from a first-order to two second-order phase transition lines). The important roles of domain geometry, of topography, and of a screening length scale (the Rossby radius of deformation) are discussed. It is found that decreasing the screening length scale (making interactions more local) surprisingly widens the range of parameters associated with ensemble inequivalence. These results are then generalized to a larger class of models, and applied to a complete description of an academic model for inertial oceanic circulation, the Fofonoff flow.

physics.flu-dyn↗

arXiv2010

Invariant measures of the 2D Euler and Vlasov equations

We discuss invariant measures of partial differential equations such as the 2D Euler or Vlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid for N-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measure where N goes to infinity. When only the energy and enstrophy invariants are taken into account, we give an explicit computation to prove the following result: the microcanonical measure is actually a Young measure corresponding to the maximization of a mean-field entropy. We explain why this result remains true for more general microcanonical measures, when all the dynamical invariants are taken into account. We give an explicit proof that these microcanonical measures are invariant measures for the dynamics of the 2D Euler equations. We describe a more general set of invariant measures, and discuss briefly their stability and their consequence for the ergodicity of the 2D Euler equations. The extension of these results to the Vlasov equations is also discussed, together with a proof of the uniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials. Even if we consider, in this paper, invariant measures only for Hamiltonian equations, with no fluxes of conserved quantities, we think this work is an important step towards the description of non-equilibrium invariant measures with fluxes.

cond-mat.stat-mech↗

arXiv2010

Thermodynamics and dynamics of systems with long-range interactions

Thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as $1/r^{d+σ}$ at large distances $r$ in $d$ dimensions, are reviewed. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed "strong" LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions affect the thermodynamic behavior of systems near phase transitions, where long-range correlations are naturally present. Long-range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems with local dynamics are considered within the framework outlined above.

cond-mat.stat-mech↗

arXiv2010

Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations

We study the asymptotic behavior and the asymptotic stability of the two-dimensional Euler equations and of the two-dimensional linearized Euler equations close to parallel flows. We focus on spectrally stable jet profiles $U(y)$ with stationary streamlines $y_{0}$ such that $U'(y_{0})=0$, a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence, is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of these flow velocities even in the absence of any dissipative mechanisms.

cond-mat.stat-mech↗

arXiv2008

Random changes of flow topology in two dimensional and geophysical turbulence

We study the two dimensional (2D) stochastic Navier Stokes (SNS) equations in the inertial limit of weak forcing and dissipation. The stationary measure is concentrated close to steady solutions of the 2D Euler equation. For such inertial flows, we prove that bifurcations in the flow topology occur either by changing the domain shape, the nonlinearity of the vorticity-stream function relation, or the energy. Associated to this, we observe in SNS bistable behavior with random changes from dipoles to unidirectional flows. The theoretical explanation being very general, we infer the existence of similar phenomena in experiments and in models of geophysical flows.

nlin.CD↗

arXiv2008

Equilibrium and out of equilibrium phase transitions in systems with long range interactions and in 2D flows

In self-gravitating stars, two dimensional or geophysical flows and in plasmas, long range interactions imply a lack of additivity for the energy; as a consequence, the usual thermodynamic limit is not appropriate. However, by contrast with many claims, the equilibrium statistical mechanics of such systems is a well understood subject. In this proceeding, we explain briefly the classical approach to equilibrium and non equilibrium statistical mechanics for these systems, starting from first principles. We emphasize recent and new results, mainly a classification of equilibrium phase transitions, new unobserved equilibrium phase transition, and out of equilibrium phase transitions. We briefly discuss what we consider as challenges in this field.

cond-mat.stat-mech↗

arXiv2007

Relaxation times of unstable states in systems with long range interactions

We consider several models with long-range interactions evolving via Hamiltonian dynamics. The microcanonical dynamics of the basic Hamiltonian Mean Field (HMF) model and perturbed HMF models with either global anisotropy or an on-site potential are studied both analytically and numerically. We find that in the magnetic phase, the initial zero magnetization state remains stable above a critical energy and is unstable below it. In the dynamically stable state, these models exhibit relaxation time scales that increase algebraically with the number $N$ of particles, indicating the robustness of the quasistationary state seen in previous studies. In the unstable state, the corresponding time scale increases logarithmically in $N$.

cond-mat.stat-mech↗

arXiv2007

Phase space gaps and ergodicity breaking in systems with long range interactions

We study a generalized isotropic XY-model which includes both two-spin and four-spin mean-field interactions. This model can be solved in the microcanonical ensemble. It is shown that in certain parameter regions the model exhibits gaps in the magnetization at fixed energy, resulting in ergodicity breaking. This phenomenon has previously been reported in anisotropic and discrete spin models. The entropy of the model is calculated and the microcanonical phase diagram is derived, showing the existence of first order phase transitions from the ferromagnetic to a paramagnetic disordered phase. It is found that ergodicity breaking takes place both in the ferromagnetic and the paramagnetic phases. As a consequence, the system can exhibit a stable ferromagnetic phase within the paramagnetic region, and conversely a disordered phase within the magnetically ordered region.

cond-mat.stat-mech↗

arXiv2007

Ensemble inequivalence, bicritical points and azeotropy for generalized Fofonoff flows

We present a theoretical description for the equilibrium states of a large class of models of two-dimensional and geophysical flows, in arbitrary domains. We account for the existence of ensemble inequivalence and negative specific heat in those models, for the first time using explicit computations. We give exact theoretical computation of a criteria to determine phase transition location and type. Strikingly, this criteria does not depend on the model, but only on the domain geometry. We report the first example of bicritical points and second order azeotropy in the context of systems with long range interactions.

cond-mat.stat-mech↗