Metastability in interacting nonlinear stochastic differential equations II: Large-N behaviour

Nils Berglund, Bastien Fernandez and Barbara Gentz
Nonlinearity 20:2583-2614 (2007)

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 γ of order N ² , provided the particle number N is sufficiently large (as a function of γ/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.

Mathematical Subject Classification: 37H20, 37L60 (primary), 37G40, 60K35 (secondary)

Keywords and phrases: Spatially extended systems, lattice dynamical systems, open systems, stochastic differential equations, interacting diffusions, transitions times, most probable transition paths, large deviations, Wentzell-Freidlin theory, diffusive coupling, synchronisation, metastability, symmetry groups, symplectic twist maps.

 

Journal Homepage Published article: 10.1088/0951-7715/20/11/007 MR2361247 Zbl1140.60350

PDF file (772 Kb) hal-00115417 arXiv/math.PR/0611648