  A recently introduced model describing -on a 1d lattice- the velocity field
of a granular fluid is discussed in detail. The dynamics of the velocity field
occurs through next-neighbours inelastic collisions which conserve momentum but
dissipate energy. The dynamics can be described by a stochastic equation in
full phase space, or through the corresponding Master Equation for the time
evolution of the probability distribution. In the hydrodynamic limit, equations
for the average velocity and temperature fields with fluctuating currents are
derived, which are analogous to those of granular fluids when restricted to the
shear modes. Therefore, the homogeneous cooling state, with its linear
instability, and other relevant regimes such as the uniform shear flow and the
Couette flow states are described. The evolution in time and space of the
single particle probability distribution, in all those regimes, is also
discussed, showing that the local equilibrium is not valid in general. The
noise for the momentum and energy currents, which are correlated, are white and
Gaussian. The same is true for the noise of the energy sink, which is usually
negligible.
