[en] We introduce a model described in terms of a scalar velocity field on a 1D lattice, evolving through collisions that conserve momentum but do not conserve energy. Such a system possesses some of the main ingredients of fluidized granular media and naturally models them. We deduce non-linear fluctuating hydrodynamics equations for the macroscopic velocity and temperature fields, which replicate the hydrodynamics of shear modes in a granular fluid. Moreover, this Landau-like fluctuating hydrodynamics predicts an essential part of the peculiar behaviour of granular fluids, like the instability of homogeneous cooling state at large size or inelasticity. We also compute the exact shape of long range spatial correlations which, even far from the instability, have the physical consequence of noticeably modifying the cooling rate. This effect, which stems from momentum conservation, has not been previously reported in the realm of granular fluids.
