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Disentangling multi-level systems: averaging, correlations and memory

Disentangling multi-level systems: averaging, correlations and memory We consider two weakly coupled systems and adopt a perturbative approach based on theRuelle response theory to study their interaction. We propose a systematic way ofparameterizing the effect of the coupling as a function of only the variables of a system ofinterest. Our focus is on describing the impacts of the coupling on the long term statisticsrather than on the finite-time behavior. By direct calculation, we find that, at firstorder, the coupling can be surrogated by adding a deterministic perturbation tothe autonomous dynamics of the system of interest. At second order, there areadditionally two separate and very different contributions. One is a term takinginto account the second-order contributions of the fluctuations in the coupling,which can be parameterized as a stochastic forcing with given spectral properties.The other one is a memory term, coupling the system of interest to its previoushistory, through the correlations of the second system. If these correlations areknown, this effect can be implemented as a perturbation with memory on the singlesystem. In order to treat this case, we present an extension to Ruelles responsetheory able to deal with integral operators. We discuss our results in the context ofother methods previously proposed for disentangling the dynamics of two coupledsystems. We emphasize that our results do not rely on assuming a time scaleseparation, and, if such a separation exists, can be used equally well to studythe statistics of the slow variables and that of the fast variables. By recursivelyapplying the technique proposed here, we can treat the general case of multi-levelsystems. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Statistical Mechanics: Theory and Experiment IOP Publishing

Disentangling multi-level systems: averaging, correlations and memory

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Copyright
Copyright IOP Publishing Ltd
eISSN
1742-5468
DOI
10.1088/1742-5468/2012/03/P03003
Publisher site
See Article on Publisher Site

Abstract

We consider two weakly coupled systems and adopt a perturbative approach based on theRuelle response theory to study their interaction. We propose a systematic way ofparameterizing the effect of the coupling as a function of only the variables of a system ofinterest. Our focus is on describing the impacts of the coupling on the long term statisticsrather than on the finite-time behavior. By direct calculation, we find that, at firstorder, the coupling can be surrogated by adding a deterministic perturbation tothe autonomous dynamics of the system of interest. At second order, there areadditionally two separate and very different contributions. One is a term takinginto account the second-order contributions of the fluctuations in the coupling,which can be parameterized as a stochastic forcing with given spectral properties.The other one is a memory term, coupling the system of interest to its previoushistory, through the correlations of the second system. If these correlations areknown, this effect can be implemented as a perturbation with memory on the singlesystem. In order to treat this case, we present an extension to Ruelles responsetheory able to deal with integral operators. We discuss our results in the context ofother methods previously proposed for disentangling the dynamics of two coupledsystems. We emphasize that our results do not rely on assuming a time scaleseparation, and, if such a separation exists, can be used equally well to studythe statistics of the slow variables and that of the fast variables. By recursivelyapplying the technique proposed here, we can treat the general case of multi-levelsystems.

Journal

Journal of Statistical Mechanics: Theory and ExperimentIOP Publishing

Published: Mar 1, 2012

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