TY - CHAP

T1 - Long term effects of small random perturbations on dynamical systems

T2 - Theoretical and computational tools

AU - Grafke, Tobias

AU - Schäfer, Tobias

AU - Vanden-Eijnden, Eric

N1 - Funding Information:
Acknowledgements We would like to thank M. Cates, A. Donev, and F. Bouchet for helpful discussions. This research was supported in part by the NSF grants DMR-1207432 (T. Grafke), DMS-1108780 and DMS-1522737 (T. Schäfer), and DMS-1522767 (E. Vanden-Eijnden).

PY - 2017

Y1 - 2017

N2 - Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. Here we review the theoretical and computational aspects behind these calculations, and propose an algorithm that simplifies the geometric minimum action method to minimize the action in the space of arc-length parametrized curves. We then illustrate this algorithm’s capabilities by applying it to various examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable.

AB - Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. Here we review the theoretical and computational aspects behind these calculations, and propose an algorithm that simplifies the geometric minimum action method to minimize the action in the space of arc-length parametrized curves. We then illustrate this algorithm’s capabilities by applying it to various examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable.

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U2 - 10.1007/978-1-4939-6969-2_2

DO - 10.1007/978-1-4939-6969-2_2

M3 - Chapter

AN - SCOPUS:85029211539

T3 - Fields Institute Communications

SP - 17

EP - 55

BT - Fields Institute Communications

PB - Springer New York LLC

ER -