Intermittency, metastability and coarse graining for coupled deterministic-stochastic lattice systems
Author
Summary, in English
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 con-
sisting 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 ODEs. 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 cor-
responding 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 com-
putationally 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 auto- and 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.
emanating from the micro-/sub-grid scale, in the context of a simple prototype hybrid system con-
sisting 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 ODEs. 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 cor-
responding 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 com-
putationally 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 auto- and 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.
Publishing year
2006
Language
English
Pages
1021-1047
Publication/Series
Nonlinearity
Volume
19
Issue
5
Document type
Journal article
Publisher
London Mathematical Society / IOP Science
Topic
- Mathematics
Status
Published
ISBN/ISSN/Other
- ISSN: 0951-7715