  We study the formation and the dynamics of correlations in the velocity field
for 1D and 2D cooling granular gases with the assumption of negligible density
fluctuations (``Homogeneous Velocity-correlated Cooling State'', HVCS). It is
shown that the predictions of mean field models fail when velocity fluctuations
become important. The study of correlations is done by means of molecular
dynamics and introducing an Inelastic Lattice Maxwell Models. This lattice
model is able to reproduce all the properties of the Homogeneous Cooling State
and several features of the HVCS. Moreover it allows very precise measurements
of structure functions and other crucial statistical indicators. The study
suggests that both the 1D and the 2D dynamics of the velocity field are
compatible with a diffusive dynamics at large scale with a more complex
behavior at small scale. In 2D the issue of scale separation, which is of
interest in the context of kinetic theories, is addressed.
