Mathematics Colloquium: Simple low-dimensional examples to illustrate mode-reduction and probabilistic averaging
4:10 pm Neill 5W
Abstract: In the second part of my talk I will discuss a novel application of the stochastic mode-reduction procedure for a class of deterministic systems. Under assumptions of mixing and ergodicity, the procedure gives closed-form stochastic models for the slow variables in the limit of infinite separation of time-scales. We show that under these assumptions the ad-hoc modification of the nonlinear self-interactions of the fast degrees of freedom can be avoided. 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 behavior of the slow modes in TBH when the fast modes are artificially accelerated to enforce the separation of time-scales. 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.