Fluctuating hydrodynamics and correlation lengths in a driven granular fluid

Static and dynamical structure factors for shear and longitudinal modes of the velocity and density fields are computed for a granular system fluidized by a stochastic bath with friction. Analytical expressions are obtained through fluctuating hydrodynamics and are successfully compared with numerical simulations up to a volume fraction similar to 50%. The hydrodynamic noise is the sum of the external noise due to the bath and the internal one due to collisions. Only the latter is assumed to satisfy the fluctuation-dissipation relation with the average granular temperature. The static velocity structure factors S-perpendicular to(k) and S-parallel to(k) display a general non-constant behavior with two plateaux at large and small k, representing the granular temperature T-g and the bath temperature T-b > T-g respectively. From this behavior, two different velocity correlation lengths are measured, both increasing as the packing fraction is raised. This growth of spatial order is in agreement with the behavior of dynamical structure factors, the decay of which becomes slower and slower at increasing density.