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Prototype hybrid couplings of macroscopic deterministic models and microscopic stochastic lattice dynamics

Katsoulakis, M.A.; Majda, A.J. and Sopasakis, Alexandros LU (2007) In Contemporary Mathematics 429. p.143-187
Abstract
We study a class of model prototype hybrid systems comprised of a microscopic Arrhenius surface process describing adsorption/desorption and/or surface diffusion of particles coupled to an ordinary differential equation displaying bifurcations triggered by the microscopic process. The models proposed here are caricatures of realistic systems arising in diverse applications ranging from surface processes and catalysis to atmospheric and oceanic sciences. Furthermore, we derive and study closures of this hybrid system by employing three different methods: deterministic closures through an averaging principle, mean field approximation and stochastic closures by employing a hierarchy of coarse-grained models. We focus on analyzing the impact... (More)
We study a class of model prototype hybrid systems comprised of a microscopic Arrhenius surface process describing adsorption/desorption and/or surface diffusion of particles coupled to an ordinary differential equation displaying bifurcations triggered by the microscopic process. The models proposed here are caricatures of realistic systems arising in diverse applications ranging from surface processes and catalysis to atmospheric and oceanic sciences. Furthermore, we derive and study closures of this hybrid system by employing three different methods: deterministic closures through an averaging principle, mean field approximation and stochastic closures by employing a hierarchy of coarse-grained models. We focus on analyzing the impact of microscopic fluctuations and interactions on the overall system's transient and long-time dynamics. For example, fluctuation-driven rare events may effect regimes exhibiting metastability. This type of phenomena can occur in several parameter regimes and typically they cannot be accounted for by the deterministic closures. In contrast, the stochastic coarse-grained closure gives rise to computationally inexpensive reduced hybrid models that capture correctly the behavior of the full microscopic system. This is further demonstrated in several Monte Carlo simulations testing a variety of parameter regimes and displaying numerically the extent, limitations and validity of the theory. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Contemporary Mathematics
volume
429
pages
143 - 187
publisher
American Mathematical Society (AMS)
ISSN
1098-3627
language
English
LU publication?
no
id
291af424-8460-4ec5-9809-d2150311577b (old id 2201608)
date added to LUP
2011-12-30 18:34:28
date last changed
2016-06-29 08:57:18
@article{291af424-8460-4ec5-9809-d2150311577b,
  abstract     = {We study a class of model prototype hybrid systems comprised of a microscopic Arrhenius surface process describing adsorption/desorption and/or surface diffusion of particles coupled to an ordinary differential equation displaying bifurcations triggered by the microscopic process. The models proposed here are caricatures of realistic systems arising in diverse applications ranging from surface processes and catalysis to atmospheric and oceanic sciences. Furthermore, we derive and study closures of this hybrid system by employing three different methods: deterministic closures through an averaging principle, mean field approximation and stochastic closures by employing a hierarchy of coarse-grained models. We focus on analyzing the impact of microscopic fluctuations and interactions on the overall system's transient and long-time dynamics. For example, fluctuation-driven rare events may effect regimes exhibiting metastability. This type of phenomena can occur in several parameter regimes and typically they cannot be accounted for by the deterministic closures. In contrast, the stochastic coarse-grained closure gives rise to computationally inexpensive reduced hybrid models that capture correctly the behavior of the full microscopic system. This is further demonstrated in several Monte Carlo simulations testing a variety of parameter regimes and displaying numerically the extent, limitations and validity of the theory.},
  author       = {Katsoulakis, M.A. and Majda, A.J. and Sopasakis, Alexandros},
  issn         = {1098-3627},
  language     = {eng},
  pages        = {143--187},
  publisher    = {American Mathematical Society (AMS)},
  series       = {Contemporary Mathematics},
  title        = {Prototype hybrid couplings of macroscopic deterministic models and microscopic stochastic lattice dynamics},
  volume       = {429},
  year         = {2007},
}