Nils Berglund, Bastien Fernandez and Barbara Gentz
Metastability in Interacting Nonlinear
Stochastic Differential Equations II:
Large-N Behaviour
(1038K, pdf)
ABSTRACT. 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 he 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 ynchronised 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.