there is no friction between the piston and the lateral walls of the vessel containing the gas. The system of particles is kept fluidized and at low density by injecting energy through the bottom wall, which is vibrating. Inelasticity in collisions between particles is modeled by means of a constant coefficient of normal restitution, α , defined in the interval 0 < α < 1. Thus when two particles i and j collide their velocities change instantaneously from the initial values v$_{i}$ , v$_{j}$ to the post-collisional ones given by where v$_{ij}$ ≡ v$_{i}$ − v$_{j}$ is the relative velocity and ̂ σ is the unit vector joining the center of the two particles at contact. The z axis will be taken in the direction of the gravitational field, so that the particles are submitted to an external force of the form f = − mg 0 ̂ e$_{z}$ , g 0 being a positive constant and ̂ e$_{z}$ the positive unit vector along the z axis. Collisions of particles with the movable piston on the top are also considered as smooth and inelastic, α$_{P}$ being the coefficient of normal restitution for them. Therefore, in a collision between particle i and the piston, the component v$_{i,x}$ of the velocity of the particle perpendicular to the z axis remains unchanged, while the component v$_{i,z}$ of the velocity of the particle and the velocity V$_{z}$ of the piston are instantaneously modified accordingly with and respectively. When the transversal section of the system, i.e. the size of the piston W , is smaller than a critical value [15, 16], the gas reaches, after a transient time interval, a stationary state with gradients only in the direction of the gravitational field. If, in addition, the inelasticity of the system is small, the inelastic hydrodynamic Navier-Stokes equations with the appropriate boundary conditions [17, 18] provide an accurate description of the stationary state [19]. As the coefficient of normal restitution of the gas α decreases, signif- icant deviations from the predictions following from the Navier-Stokes equations show up. v ′ i,x = v$_{i,x}$, (3) v ′ i,z = v$_{i,z}$ − M m + M (1 + α$_{P}$ )( v$_{i,z}$ − V$_{z}$ ) (4) V ′ z = V$_{z}$ + m m + M (1 + α$_{P}$ )( v$_{i,z}$ − V$_{z}$ ) , (5) 4