Freddy Bouchet

Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model

We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field model, a prototype for long-range interactions in N-particle dynamics. In particular, we point out the role played by the infinity of stationary states of the associated N→∞ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, finite N, dynamics. We then propose, and verify numerically, a scenario for the relaxation process, relying on the Vlasov equation. When starting from a nonstationary or a Vlasov unstable stationary state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via nonstationary states: we characterize numerically this dynamical instability in the finite N system by introducing appropriate indicators. This first step of the evolution towards Boltzmann–Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through different stable stationary states of the Vlasov equation. If the finite N system is initialized in a Vlasov stable homogeneous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N1.7. Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be fitted by the q-exponential distributions derived from Tsallis statistics.

Thermodynamics and dynamics of systems with long-range interactions

We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as 1/rd+σ at large distances r in d dimensions. 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.

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.

Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model

Abstract.We discuss the dynamics and thermodynamics of then Hamiltonian Mean Field model (HMF) which is a prototypical systemn with long-range interactions. The HMF model can be seen as the onen Fourier component of a one-dimensional self-gravitatingn system. Interestingly, it exhibits many features of realn self-gravitating systems (violent relaxation, persistence ofn metaequilibrium states, slow collisional dynamics, phasen transitions,...) while avoiding complicated problems posed by then singularity of the gravitational potential at short distances andn by the absence of a large-scale confinement. We stress the deepn analogy between the HMF model and self-gravitating systems byn developing a complete parallel between these two systems. Thisn allows us to apply many technics introduced in plasma physics andn astrophysics to a new problem and to see how the results depend onn the dimension of space and on the form of the potential ofn interaction. This comparative study brings new light in then statistical mechanics of self-gravitating systems. We also mentionn simple astrophysical applications of the HMF model in relation withn the formation of bars in spiral galaxies.n

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 equations. 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 with this, we observe bistable behavior in SNS 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 some regimes of geophysical flows.

Large deviation techniques applied to systems with long-range interactions

AbstractWe discuss a method to solve models with long-range interactions in the microcanonical and canonical ensemble. The method closely follows the one introduced by R.S. Ellis, Physica D133:106 (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-equilibrium effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the α-Ising model in one-dimension with 0n⩽ α < 1.

Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations

Abstract We study the asymptotic behavior and the asymptotic stability of the 2D Euler equations and of the 2D linearized Euler equations close to parallel flows. We focus on flows with spectrally stable profiles U ( y ) and with stationary streamlines y = 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 this ensemble of flow profiles even in the absence of any dissipative mechanisms.

Statistical Ensemble Inequivalence and Bicritical Points for Two-Dimensional Flows and Geophysical Flows

X iv :0 71 0. 56 06 v1 [ co nd -m at .s ta tm ec h] 3 0 O ct 2 00 7 Ensemble inequivalen e, bi riti al points and azeotropy for generalized Fofono ows Antoine Venaille and Freddy Bou het∗ Laboratoire des É oulements Géophysiques et Industriels, UJF, INPG, CNRS ; BP 53, 38041 Grenoble, Fran e and Institut Non Linéaire de Ni e , CNRS, UNSA, 1361 route des lu ioles, 06 560 Valbonne Sophia Antipolis, Fran e (Dated: February 2, 2008) We present a theoreti al des ription for the equilibrium states of a large lass of models of twodimensional and geophysi al ows, in arbitrary domains. We a ount for the existen e of ensemble inequivalen e and negative spe i heat in those models, for the rst time using expli it omputations. We give exa t theoreti al omputation of a riteria to determine phase transition lo ation and type. Strikingly, this riteria does not depend on the model, but only on the domain geometry. We report the rst example of bi riti al points and se ond order azeotropy in the ontext of systems with long range intera tions. PACS numbers: 05.20.-y, 05.70.Fh, 47.32.-y. In many elds of physi s, the parti le or elds dynami s is not governed by lo al intera tions. For instan e for self gravitating stars in astrophysi s [1, 2℄, for vorti es in two dimensional and geophysi al ows [3, 4, 5℄, for uns reened plasma or models des ribing intera tions between waves and parti les [6℄, the intera tion potential is not integrable [7℄. Re ently, a new light was shed on the equilibrium statisti al me hani s of su h systems with long range intera tions : there has been a mathemati al hara terization of ensemble inequivalen e [8℄, a study of several simple models [9, 10℄, and a full lassi ation of phase transitions and of ensemble inequivalen e [11℄ . One of the promising eld of appli ation for the statisti al me hani s of systems with long range intera tions, is the statisti al predi tion of large s ale geophysi al ows. For instan e, the stru ture of Jupiters troposphere has been su essfully explained using the Robert-SommeriaMiller (RSM) equilibrium theory [12℄ [13℄. One of the major s ope of this eld is to go towards earth o ean appli ations. All textbook in o eanography present the Fofono ows whi h have played an important histori al role in that eld [14℄. In this letter, we propose a theoreti al des ription of su h ows in the ontext of the statistial theories whi h, for the rst time, relates its properties to phase transitions (see Fig. 1), negative spe i heat and ensemble inequivalen e. One of the striking features of the equilibrium theory of systems with long range intera tions is the generi existen e of negative spe i heat. This strange phenomena is possible as a onsequen e of the la k of additivity of the energy and is related to the inequivalen e between the mi ro anoni al and anoni al ensemble of statistial physi s. This was rst predi ted in the ontext of astrophysi s [15℄. For two dimensional ows, existen e of su h inequivalen e has been matemati ally proven for point vorti es [16℄ (without expli it omputation), and numeri ally observed in a parti ular situation of a QuasiGeostrophi (QG) model [17℄, and in a Monte Carlo study of points vorti es in a disk [18℄. One of the novelty 0 0

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.

Simpler variational problems for statistical equilibria of the 2D Euler equation and other systems with long range interactions

Abstract 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.