Journée de Dynamique Non Linéaire

10h30   Stefano Ruffo (Dipartimento di Energetica, Université de Florence, INFM et INFN)
Quasi-stationary states in a system of globally coupled rotators

11h35   Elena Floriani (Centre de Physique Théorique, Luminy)
A toy model for a system at a threshold of stability

11h55   Nicolas Bian (Dynamique des Systèmes Complexes, P.I.I.M.)
Propagation de bulles en convection

Résumé du séminaire de S. Ruffo

A system of N particles moving on a circle and interacting via a global repulsive cosine interaction is known to display spatially inhomogeneous structure of extraordinary stability starting from certain low energy initial conditions. We have recently shown how these structures arise and we have understood the origin of their long-time stability. We first explain the onset of this "collective mode" by approximating the short time dynamics by a forced Burgers equation.
We show how their emergence can be related to the presence of shock waves in the associated hydrodynamical equations. As for the long-time stability we have developed two independent analytical treatments which give compatible results: 1 - By a convenient canonical transformation we rewrite the Hamiltonian in such a way that fast and slow variables are singled out and the canonical variables of a collective mode are naturally introduced. If, initially, enough energy is put in this mode, its decay can be extremely slow. 2 - A very fast timescale can be separated from a slower motion; this enables us to use an adiabatic approximation to derive an effective Hamiltonian describing the long time dynamics. The collective mode then appears as the equilibrium state of this effective model, thus explaining its very long time stability, since the out-of-equilibrium states correspond to statistical equilibria of an effective mean-field dynamics.
References:
- T. Dauxois, P. Holdsworth and S. Ruffo:"Violation of ensemble equivalence in the antiferromagnetic mean-field XY model", European Physical Journal B, 16, 659 (2000).
- J. Barré, T. Dauxois and S. Ruffo:"Clustering in a model with repulsive long-range interactions", Physica A, 295, 254 (2001).
- F.Leyvraz, M.-C. Firpo and S. Ruffo:"Inhomogeneous quasi-stationary states in a mean-field model with repulsive cosine interactions", preprint, submitted to J. Phys. A (2001).
- J. Barré, F. Bouchet, T. Dauxois and S. Ruffo:"Birth dynamics and long-time stabilization of out-of-equilibrium coherent structures", preprint, submitted to Eur. Phys. J. B (2001).

Résumé du séminaire de E. Floriani

We propose a simple "toy" model for a system at a threshold of stability. We assume that the system loses stability when the parameter characterizing its state exceeds a threshold value. We treat both the state parameter and the threshold as random independent variables distributed on the unit interval with respect to different distribution laws. In our model, the threshold value is allowed to quench with a given probability, so that our natural control parameter is the relative frequency of threshold changes. We study the distribution of durations of laminar phases, whose asymptotic decay depends in an essential way from the value of the control parameter.

Résumé du séminaire de N. Bian

Je présenterai quelques résultats numériques sur le développement de l'instabilité d'une bulle en convection.