Stochastic modelling: replacing fast degrees of freedom by noise

In systems with timescale separation, where the fast degrees of freedom exhibit chaotic motion, the latter are replaced by suitable stochastic processes. A projection technique is employed to derive equations of motion for the phase space density of the slow variables by eliminating the fast ones. The resulting equations can be approximated in a controlled way by Fokker-Planck equations or equivalently by stochastic differential equations for slow degrees of freedom. We discuss some model situations and explore the accuracy of the approximations by numerical simulations.