Metastability in Interacting Nonlinear Stochastic Differential Equations. II: Large-N Behaviour

Nils Berglund (1, 2, 3) , Bastien Fernandez (1) , Barbara Gentz (4, 5)

1 CPT - Centre de Physique Théorique - UMR 6207
2 PHYMAT - PHYMAT [La Garde]
3 MAPMO - Mathématiques appliquées et physique mathématique d'Orléans
4 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]
5 Fakultät für Mathematik = Faculty of Mathematics [Bielefeld]

We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise, in the limit of large N. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. For strong coupling (of the order N^2), the system synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. In a previous paper, we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurcations of the system's stationary configurations, and analysed in particular the behaviour for coupling intensities slightly below the synchronisation threshold, for arbitrary N. Here we describe the behaviour for any positive coupling intensity \gamma of order N^2, provided the particle number N is sufficiently large (as a function of \gamma/N^2). In particular, we determine the transition time between synchronised states, as well as the shape of the "critical droplet", to leading order in 1/N. Our techniques involve the control of the exact number of periodic orbits of a near-integrable twist map, allowing us to give a detailed description of the system's potential landscape, in which the metastable behaviour is encoded.

Mots clés: Transitions times; Interacting diffusions; Stochastic differential equations; Open systems; Lattice dynamical systems; Spatially extended systems; Synchronisation; Diffusive coupling; Wentzell-Freidlin theory; Most probable transition paths; Large deviations; Metastability; Symmetry groups; Symplectic twist maps

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Nils Berglund, Bastien Fernandez, Barbara Gentz. Metastability in Interacting Nonlinear Stochastic Differential Equations. II: Large-N Behaviour. Nonlinearity, 2007, 20 (11), pp.2583-2614. ⟨10.1088/0951-7715/20/11/007⟩. ⟨hal-00115417v3⟩