Stochastic models for selected slow variables in large deterministic systems

Abstract

A new stochastic mode-elimination procedure is introduced for a class of deterministic systems. Under assumptions of ergodicity and mixing, the procedure gives closed-form stochastic models for the slow variables in the limit of infinite separation of timescales. The procedure is applied to the truncated Burgers–Hopf (TBH) system as a test case where the separation of timescale is only approximate. It is shown that the stochastic models reproduce exactly the statistical behaviour of the slow modes in TBH when the fast modes are artificially accelerated to enforce the separation of timescales. It is shown that this operation of acceleration only has a moderate impact on the bulk statistical properties of the slow modes in TBH. As a result, the stochastic models are sound for the original TBH system.