2006/11/30 by Nils Berglund, Bastien Fernandez, Barbara Gentz · 2 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Quantum chaos and dynamical systems #math-ph #math.DS #math.MP #math.PR #msc:37G40 #msc:37H20 #msc:37L60 #msc:60K35 #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/0951-7715/20/11/007
published as Nonlinearity 20, 11 (2007) 2583-2614
arxiv created 2007/07/06 · openalex publication_date 2007/10/04 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
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 synchronizes, in the sense that all particles assume almost the same position in their respective local potential most of the time. In a previous work ( Berglund et al 2007 Nonlinearity 20 2551 ), we showed that the transition from strong to weak coupling involves a sequence of symmetry-breaking bifurcations of the system's stationary configurations. We analysed, for arbitrary N , the behaviour for coupling intensities slightly below the synchronization threshold. 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 synchronized 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.