Stochasticity and Memort Effects in Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach


Valerio Lucarini

13:50:00 - 14:40:00

101 , Mathematics Research Center Building (ori. New Math. Bldg.)

We consider the problem of disentangling multi-level systems by connecting the seemingly unrelated approaches of the Mori-Zwanzig projection operator technique and of the Ruelle response theory, for which we propose a new derivation. We show that by using the Ruelle response theory on a weakly coupled system it is possible to construct a surrogate dynamics for the slow variables, 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, where both slow and fast variables are involved. The impact of the fast variables onto the slow variables is parametrized in the surrogate dynamics as the sum of a deterministic contribution, a stochastic forcing, and a memory term. Then, using a Dyson expansion, we show that such surrogate dynamics agrees up to second order to the effective dynamics one can derive by expanding perturbatively the Mori-Zwanzing projection operator, which creates, instead, an accurate representation of the trajectories of the slow variables. In the case of e.g. geophysical fluid dynamics, this implies that the parametrizations of unresolved processes suited for prediction (numerical weather forecast) and those suited for the representation of long term statistical properties (climate) are closely related, if one takes into account, in addition to the widely adopted stochastic forcing, the usually neglected memory effects. This bears relevance for the current trend of aiming at seamless prediction.

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