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Many applications of contemporary science involve multiscale dynamics, whichare typically characterized by the time and space scale separation of patternsof motion, with fewer slowly evolving variables and much larger set of fasterevolving variables. This time-space scale separation causes direct numericalsimulation of the evolution of the dynamics to be computationally expensive,due both to the large number of variables and the necessity to choose a smalldiscretization time step in order to resolve the fast components of dynamics.In this work we propose a simple method of determining the closed model forslow variables alone, which requires only a single computation of appropriatestatistics for the fast dynamics with a certain fixed state of the slowvariables. The method is based on the first-order Taylor expansion of theaveraged coupling term with respect to the slow variables, which can becomputed using the linear fluctuation-dissipation theorem. We show that, withsimple linear coupling in both slow and fast variables, this method producesquite comparable statistics to what is exhibited by a complete two-scale model.The main advantage of the method is that it applies even when the statistics ofthe full multiscale model cannot be simulated due to computational complexity,which makes it practical for real-world large scale applications.

A simple linear response closure approximation for slow dynamics of a multiscale system with linear coupling