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Thierry Dauxois

Publications and source records attributed to Thierry Dauxois.

At least 73 records · Page 4Linked to original sources
arXiv2006

A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model

A generic feature of systems with long-range interactions is the presence of {\it quasi-stationary} states with non-Gaussian single particle velocity distributions. For the case of the Hamiltonian Mean Field (HMF) model, we demonstrate that a maximum entropy principle applied to the associated Vlasov equation explains known features of such states for a wide range of initial conditions. We are able to reproduce velocity distribution functions with an analytical expression which is derived from the theory with no adjustable parameters. A normal diffusion of angles is detected and a new dynamical effect, two oscillating clusters surrounded by a halo, is also found and theoretically justified.

cond-mat.stat-mech↗
arXiv2006

Quantitative laboratory observations of internal wave reflection on ascending slopes

Internal waves propagate obliquely through a stratified fluid with an angle that is fixed with respect to gravity. Upon reflection on a sloping bed, striking phenomena are expected to occur close to the slope. We present here laboratory observations at moderately large Reynolds number. A particle image velocimetry (PIV) technique is used to provide time resolved velocity fields in large volumes. The generation of the second and third harmonic frequencies are clearly demonstrated in the impact zone. The mechanism for nonlinear wavelength selection is also discussed. Evanescent waves with frequency larger than the Brunt-Väisälä frequency are detected and experimental results agree very well with theoretical predictions. The amplitude of the different harmonics after reflection are also obtained.

nlin.PS↗
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

The Fermi-Pasta-Ulam "numerical experiment": history and pedagogical perspectives

The Fermi-Pasta-Ulam (FPU) pioneering numerical experiment played a major role in the history of computer simulation because it introduced this concept for the first time. Moreover, it raised a puzzling question which was answered more than 10 years later. After an introduction to this problem, we briefly review its history and then suggest some simple numerical experiments, with a provided Matlab code, to study various aspects of the ``FPU'' problem.

nlin.PS↗
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

The Anti-FPU Problem

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi-Pasta-Ulam (FPU) lattices. Following this instability, a process of relaxation to equipartition takes place, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, at variance with the original FPU problem (low frequency excitations of the lattice). This process leads to the formation of chaotic breathers in both one and two dimensions. Finally, the system relaxes to energy equipartition on time scales which increase as the energy density is decreased. We show that breathers formed when cooling the lattice at the edges, starting from a random initial state, bear strong qualitative similarities with chaotic breathers.

nlin.PS↗
arXiv2004

Links between nonlinear dynamics and statistical mechanics in a simple one-dimensional model

We consider the links between nonlinear dynamics and thermodynamics in the framework of a simple nonlinear model for DNA. Two analyses of the phase transition, either with the transfer integral approach or by considering the instability of a nonlinear particular solution, are discussed. Conversely, the computation of the largest Lyapunov exponent is obtained within a thermodynamic treatment. Differences with the Peyrard-Bishop model are also discussed.

cond-mat.stat-mech↗
arXiv2004

Statistical theory of high-gain free-electron laser saturation

We propose a new approach, based on statistical mechanics, to predict the saturated state of a single-pass, high-gain free-electron laser (FEL). In analogy with the violent relaxation process in self-gravitating systems and in the Euler equation of 2D turbulence, the initial relaxation of the laser can be described by the statistical mechanics of an associated Vlasov equation. The Laser field intensity and the electron bunching parameter reach quasi-stationary values that are well fitted by a Vlasov stationary state if the number of electrons $N$ is sufficiently large. Finite $N$ effects (granularity) finally drive the system to Boltzmann-Gibbs statistical equilibrium, but this occurs on times that are unphysical (i.e. excessively long undulators). All theoretical predictions are successfully tested in finite $N$ numerical experiments. Our results may provide important hints to the optimization of the length of the FEL undulator.

cond-mat.stat-mech↗
arXiv2003

Observation of near-critical reflection of internal waves in a stably stratified fluid

An experimental study is reported of the near-critical reflection of internal gravity waves over sloping topography in a stratified fluid. An overturning instability close to the slope and triggering the boundary-mixing process is observed and characterized. These observations are found in good agreement with a recent nonlinear theory.

physics.flu-dyn↗
arXiv2003

Clustering and ensembles inequivalence in the phi-4 and phi-6 mean-field Hamiltonian models

We investigate a model of globally coupled conservative oscillators. Two different algebraic potentials are considered that display in the canonical ensemble either a second ($ϕ^{4}$) or both a second and a first order phase transition separated by tricritical points ($ϕ^{6}$). The stability of highly clustered states appearing in the low temperature/energy region is studied both analytically and numerically for the $ϕ^{4}$-model. Moreover, long-lived out-of-equilibrium states appear close to the second order phase transition when starting with "water-bag" initial conditions, in analogy with what has been found for the Hamiltonian Mean Field (HMF) model. The microcanonical simulations of the $ϕ^{6}$-model show strong hysteretic effects and metastability near the first-order phase transition and a narrow region of negative specific heat.

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

The Hamiltonian Mean Field Model: from Dynamics to Statistical Mechanics and back

The thermodynamics and the dynamics of particle systems with infinite-range coupling display several unusual and new features with respect to systems with short-range interactions. The Hamiltonian Mean Field (HMF) model represents a paradigmatic example of this class of systems. The present study addresses both attractive and repulsive interactions, with a particular emphasis on the description of clustering phenomena from a thermodynamical as well as from a dynamical point of view. The observed clustering transition can be first or second order, in the usual thermodynamical sense. In the former case, ensemble inequivalence naturally arises close to the transition, i.e. canonical and microcanonical ensembles give different results. In particular, in the microcanonical ensemble negative specific heat regimes and temperature jumps are observed. Moreover, having access to dynamics one can study non-equilibrium processes. Among them, the most striking is the emergence of coherent structures in the repulsive model, whose formation and dynamics can be studied either by using the tools of statistical mechanics or as a manifestation of the solutions of an associated Vlasov equation. The chaotic character of the HMF model has been also analyzed in terms of its Lyapunov spectrum.

cond-mat.stat-mech↗
arXiv2002

Dynamics and Thermodynamics of Systems with Long Range Interactions: an Introduction

We review theoretical results obtained recently in the framework of statistical mechanics to study systems with long range forces. This fundamental and methodological study leads us to consider the different domains of applications in a trans-disciplinary perspective (astrophysics, nuclear physics, plasmas physics, metallic clusters, hydrodynamics,...) with a special emphasis on Bose-Einstein condensates.

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↗
arXiv2001

Thermodynamic instabilities in one dimension: correlations, scaling and solitons

Many thermodynamic instabilities in one dimension (e.g. DNA thermal denaturation, wetting of interfaces) can be described in terms of simple models involving harmonic coupling between nearest neighbors and an asymmetric on-site potential with a repulsive core, a stable minimum and a flat top. The paper deals with the case of the Morse on-site potential, which can be treated exactly in the continuum limit. Analytical expressions for correlation functions are derived; they are shown to obey scaling; numerical transfer-integral values obtained for a discrete version of the model exhibit the same critical behavior. Furthermore, it is shown in detail that the onset of the transition can be characterized by an entropic stabilization of an -otherwise unstable-, nonlinear field configuration, a soliton-like domain wall (DW) with macroscopic energy content. The statistical mechanics of the DW provides an exact estimate of the critical temperature for a wide range of the discretization parameter; this suggests that the transition can be accurately viewed as being "driven" by a nonlinear entity.

cond-mat.stat-mech↗