  For a system of weakly interacting anharmonic oscillators, perturbed by an
energy preserving stochastic dynamics, we prove an autonomous (stochastic)
evolution for the energies at large time scale (with respect to the coupling
parameter). It turn out that this macroscopic evolution is given by the so
called conservative (non-gradient) Ginzburg-Landau system of stochastic
differential equations. The proof exploits hypocoercivity and hypoellipticity
properties of the uncoupled dynamics.
