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Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach

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Abstract

We consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system, which are weakly coupled to the remaining variables. In a previous paper we have used the Ruelle response theory on such a weakly coupled system to construct a surrogate dynamics, such that the expectation value of any observable agrees, up to second order in the coupling strength, to its expectation evaluated on the full dynamics. We show here that such surrogate dynamics agree up to second order to an expansion of the Mori-Zwanzig projected dynamics. This implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related, if one takes into account, in addition to the widely adopted stochastic forcing, the often neglected memory effects.

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Authors and Affiliations

  1. Meteorologisches Institut, University of Hamburg, Grindelberg 7, 20144, Hamburg, Germany

    Jeroen Wouters & Valerio Lucarini

  2. Department of Mathematics and Statistics, University of Reading, Reading, RG6 6AX, UK

    Valerio Lucarini

Authors
  1. Jeroen Wouters
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  2. Valerio Lucarini
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Correspondence to Jeroen Wouters.

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Wouters, J., Lucarini, V. Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach. J Stat Phys 151, 850–860 (2013). https://doi.org/10.1007/s10955-013-0726-8

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  • DOI: https://doi.org/10.1007/s10955-013-0726-8

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