  We present recent results on coarse-graining techniques for thermodynamic
quantities (canonical averages) and dynamical quantities (averages of path
functionals over solutions of overdamped Langevin equations). The question is
how to obtain reduced models to compute such quantities, in the specific case
when the functional to be averaged only depends on a few degrees of freedom. We
mainly review, numerically illustrate and extend results from [3,18],
concerning the computation of the stress-strain relation for one-dimensional
chains of atoms, and the construction of an effective dynamics for a scalar
coarse-grained variable when the complete system evolves according to the
overdamped Langevin equation.
