0 0.1 0.2 0.3 0 0.4 0.8 1.2 T$_{eff}$/T$_{P}$ FIG. 8: Ratio T$_{eff}$/T$_{P}$ for large values of M/m as a function of 1 − α 2 for the same systems as in Fig. 5. The symbols are simulation results and the dashed line a linear fit of them. the figure by the solid straight line. Nevertheless, it is important to realize that the interval of values of M/m for which the above dependence is identified is rather narrow, just one order of magnitude, so that its range of validity can be limited. The aim here has been to investigate the volume fluctuations of a vibrated low density gas of inelastic hard disks in presence of gravity, and confined by a movable piston on the top. The study has been restricted to the parameter region in which the system reaches a steady state with gradients only in the direction of the external gravitational field, i.e. perpendicular to the movable piston. In practice, this has limited the values of the co- efficient of normal restitution of the gas particles to the interval 0 . 85 ≤ α < 1. Due to the coupling between inelasticity and hydrodynamic gradients, which is peculiar of steady states of granular systems, the above limitation also implies restriction to small gradients. IV. DISCUSSION AND SUMMARY 17