Optimal prediction of stiff oscillatory systems

We consider some large systems of differential equations that have been introduced as model many-body problems. These systems have solutions that oscillate on a wide range of time scales. We apply the formalism of optimal prediction to these systems, using conditional expectations of the equations of motion to construct effective equations for the most slowly-varying quantities. We verify the accuracy of the effective equations in examples, comparing solutions of the original and new systems, and we show that the new equations give accurate answers for slow variables with relatively little computational effort