Abstract:
It is reported that the numerical simulations of the Fermi-Pasta-Ulam problem were performed by a young lady, Mary Tsingou. After 50 years of omission, it is time for a proper recognition of her decisive contribution to the first ever numerical experiment, central in the solitons and chaos theories, but also one of the very first out-of-equilibrium statistical mechanics study. Let us quote from now on the Fermi-Pasta-Ulam-Tsingou problem.

Abstract:
Comment of the very interesting paper by Hilhorst & Schehr, J. Stat. Mech. P06003 (2007). The main point is that one should be extremely careful when interpreting non-Gaussian data in terms of q-Gaussians.

Abstract:
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.

Abstract:
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.

Abstract:
We study analytically the behavior of the largest Lyapunov exponent $\lambda_1$ for a one-dimensional chain of coupled nonlinear oscillators, by combining the transfer integral method and a Riemannian geometry approach. We apply the results to a simple model, proposed for the DNA denaturation, which emphasizes a first order-like or second order phase transition depending on the ratio of two length scales: this is an excellent model to characterize $\lambda_1$ as a dynamical indicator close to a phase transition.

Abstract:
We investigate a model of globally coupled conservative oscillators. Two different algebraic potentials are considered that display in the canonical ensemble either a second ($\phi^{4}$) or both a second and a first order phase transition separated by tricritical points ($\phi^{6}$). The stability of highly clustered states appearing in the low temperature/energy region is studied both analytically and numerically for the $\phi^{4}$-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 $\phi^{6}$-model show strong hysteretic effects and metastability near the first-order phase transition and a narrow region of negative specific heat.

Abstract:
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.

Abstract:
We study surface diffusion in the framework of a generalized Frenkel-Kontorova model with a nonconvex transverse degree of freedom. The model describes a lattice of atoms with a given concentration interacting by Morse-type forces, the lattice being subjected to a two-dimensional substrate potential which is periodic in one direction and nonconvex (Morse) in the transverse direction. The results are used to describe the complicated exchange-mediated diffusion mechanism recently observed in MD simulations [J.E. Black and Zeng-Ju Tian, Phys. Rev. Lett. {\bf 71}, 2445-2448(1993)].

Abstract:
We apply the Hilbert transform to the physics of internal waves in two-dimensional fluids. Using this demodulation technique, we can discriminate internal waves propagating in different directions: this is very helpful in answering several fundamental questions in the context of internal waves. We focus more precisely in this paper on phenomena associated with dissipation, diffraction and reflection of internal waves.