Thierry Dauxois

Dynamics and thermodynamics of systems with long-range interactions

Dynamics and Thermodynamics of Systems with Long-Range Interactions: An Introduction.- Dynamics and Thermodynamics of Systems with Long-Range Interactions: An Introduction.- Statistical Mechanics.- Thermo-statistics or Topology of the Microcanonical Entropy Surface.- Ensemble Inequivalence in Mean-Field Models of Magnetism.- Phase Transitions in Finite Systems.- Phase Transitions in Systems with 1/r ? Attractive Interactions.- Nonextensivity: From Low-Dimensional Maps to Hamiltonian Systems.- Astrophysics.- Statistical Mechanics of Gravitating Systems in Static and Cosmological Backgrounds.- Statistical Mechanics of Two-Dimensional Vortices and Stellar Systems.- Bose-Einstein Condensation.- Coherence and Superfluidity of Gaseous Bose-Einstein Condensates.- Ultracold Atoms and Bose-Einstein Condensates in Optical Lattices.- Canonical Statistics of Occupation Numbers for Ideal and Weakly Interacting Bose-Einstein Condensates.- New Regimes in Cold Gases via Laser-Induced Long-Range Interactions.- Nonlinear Dynamics.- Dynamics and Self-consistent Chaos in a Mean Field Hamiltonian Model.- Kinetic Theory for Plasmas and Wave-Particle Hamiltonian Dynamics.- Emergence of Fractal Clusters in Sequential Adsorption Processes.- The Hamiltonian Mean Field Model: From Dynamics to Statistical Mechanics and Back.

Statistical mechanics and dynamics of solvable models with long-range interactions

Abstract For systems with long-range interactions, the two-body potential decays at large distances as V ( r ) ∼ 1 / r α , with α ≤ d , where d is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties of the thermodynamics of short-range systems is at the origin of ensemble inequivalence. In turn, this inequivalence implies that specific heat can be negative in the microcanonical ensemble, and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameter space where ergodicity may be broken. Historically, negative specific heat had been found for gravitational systems and was thought to be a specific property of a system for which the existence of standard equilibrium statistical mechanics itself was doubted. Realizing that such properties may be present for a wider class of systems has renewed the interest in long-range interactions. Here, we present a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Remarkably, the entropy of all these models can be obtained using the method of large deviations. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and, what is more striking, the convergence towards quasi-stationary states. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. A statistical approach, founded on a variational principle introduced by Lynden-Bell, is shown to explain qualitatively and quantitatively some features of quasi-stationary states. Generalizations to models with both short and long-range interactions, and to models with weakly decaying interactions, show the robustness of the effects obtained for mean-field models.

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.

Modulational instability: first step towards energy localization in nonlinear lattices

We study the modulational instability in discrete lattices and we show how the discreteness drastically modifies the stability condition. Analytical and numerical results are in very good agreement. We predict also the evolution of a linear wave in the presence of noise and we show that modulational instability is the first step towards energy localization.

Near-critical reflection of internal waves

Using a matched asymptotic expansion we analyse the two-dimensional, near-critical reflection of a weakly nonlinear internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the amplitude of the incident wave, the dissipation, and the departure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience heals the singularity which is presented by the solution of Phillips (1966) in the precisely critical case. In the inviscid critical case, an explicit solution of the initial value problem shows that the buoyancy perturbation and the alongslope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as 1/ t . During the course of this scale reduction, the stratification is ‘overturned’ and the Miles–Howard condition for stratified shear flow stability is violated. However, for all slope angles, the ‘overturning’ occurs before the Miles–Howard stability condition is violated and so we argue that the first instability is convective. Solutions of the simplified dynamics resemble certain experimental visualizations of the reflection process. In particular, the buoyancy field computed from the analytic solution is in good agreement with visualizations reported by Thorpe & Haines (1987). One curious aspect of the weakly nonlinear theory is that the final reduced description is a linear equation (at the solvability order in the expansion all of the apparently resonant nonlinear contributions cancel amongst themselves). However, the reconstructed fields do contain nonlinearly driven second harmonics which are responsible for an important symmetry breaking in which alternate vortices differ in strength and size from their immediate neighbours.

Laboratory experiments on the generation of internal tidal beams over steep slopes

We designed a simple laboratory experiment to study internal tides generation. We consider a steep continental shelf, for which the internal tide is shown to be emitted from the critical point, which is clearly amphidromic. We also discuss the dependence of the width of the emitted beam on the local curvature of topography and on viscosity. Finally, we derive the form of the resulting internal tidal beam by drawing an analogy with an oscillating cylinder in a static fluid.

Order of the phase transition in models of DNA thermal denaturation.

We examine the behavior of a model which describes the melting of double-stranded DNA chains. The model, with displacement-dependent stiffness constants and a Morse on-site potential, is analyzed numerically; depending on the stiffness parameter, it is shown to have either (i) a second-order transition with nu( perpendicular) = -beta = 1,nu(||) = gamma/2 = 2 (characteristic of short-range attractive part of the Morse potential) or (ii) a first-order transition with finite melting entropy, discontinuous fraction of bound pairs, divergent correlation lengths, and critical exponents nu( perpendicular) = -beta = 1/2,nu(||) = gamma/2 = 1.

A novel internal waves generator

We present a new kind of generator of internal waves which has been designed for three purposes. First, the oscillating boundary conditions force the fluid particles to travel in the preferred direction of the wave ray, hence reducing the mixing due to forcing. Second, only one ray tube is produced so that all of the energy is in the beam of interest. Third, temporal and spatial frequency studies emphasize the high quality for temporal and spatial monochromaticity of the emitted beam. The greatest strength of this technique is therefore the ability to produce a large monochromatic and unidirectional beam.

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 0 ⩽ α < 1.

Physics of long-range interacting systems

PART I: STATIC AND EQUILIBRIUM PROPERTIES PART II: DYNAMICAL PROPERTIES PART III: APPLICATIONS