  We explore situations in which certain stochastic and high-dimensional
deterministic systems behave effectively as low-dimensional dynamical systems.
We define and study moment maps, maps on spaces of low-order moments of
evolving distributions, as a means of understanding equations-free multiscale
algorithms for these systems. We demonstrate how nonlinearity arises in these
maps and how this results in the stabilization of metastable states. Examples
are shown for a hierarchy of models, ranging from simple stochastic
differential equations to molecular dynamics simulations of a particle in
contact with a heat bath.
