Intermittency, metastability and coarse graining for coupled deterministic–stochastic lattice systems

We study the role of strong particle/particle interactions and stochastic fluctuations emanating from the micro-/sub-grid scale, in the context of a simple prototype hybrid system consisting of a scalar linear ordinary differential equation (ODE), coupled to a microscopic spin flip Ising lattice system. Due to the presence of strong interactions in the lattice model, the mean-field approximation of this system is a Fitzhugh–Nagumo-type system of ODE. However, microscopic noise and local interactions will significantly alter the deterministic and spatially homogeneous mean-field Fitzhugh–Nagumo behaviours (excitable, bistable and oscillatory) and will yield corresponding regimes with phenomena driven by the interaction of nonlinearity and noise across scales, such as strong intermittency, metastability and random oscillations. Motivated by these observations we consider a class of stochastic numerical approximations based on systematic coarse-grainings of stochastic lattice dynamics. The resulting stochastic closures give rise to computationally inexpensive reduced hybrid models that capture correctly the transient and long-time behaviour of the full system; this is demonstrated by detailed time series analysis that includes comparisons of power spectra and autoand cross-correlations in time and space, especially in examples dominated by strong interactions between scales and fluctuations, such as nucleation, intermittent and random oscillation regimes. Mathematics Subject Classification: 82B26, 82B27, 82B80, 82B20, 82B31, 82B22, 34C55, 34C15, 60G99 (Some figures in this article are in colour only in the electronic version) 0951-7715/06/051021+27$30.00 © 2006 IOP Publishing Ltd and London Mathematical Society Printed in the UK 1021 1022 M A Katsoulakis et al