Stochastic modelling: replacing fast degrees of freedom by noise

In systems with timescale separation, where the fast degrees of freedom exh ibit chaotic motion, the latter are replaced by suitable stochastic process es. A projection technique is employed to derive equations of motion for th e phase space density of the slow variables by eliminating the fast ones. T he resulting equations can be approximated in a controlled way by Fokker-Pl anck equations or equivalently by stochastic differential equations for slo w degrees of freedom. We discuss some model situations and explore the accu racy of the approximations by numerical simulations.