  The dynamics of a system composed of inelastic hard spheres or disks that are
confined between two parallel vertically vibrating walls is studied (the
vertical direction is defined as the direction perpendicular to the walls). The
distance between the two walls is supposed to be larger than twice the diameter
of the particles so that the particles can pass over each other, but still much
smaller than the dimensions of the walls. Hence, the system can be considered
to be quasi-two-dimensional (quasi-one-dimensional) in the hard spheres (disks)
case. For dilute systems, a closed evolution equation for the one-particle
distribution function is formulated that takes into account the effects of the
confinement. Assuming the system is spatially homogeneous, the kinetic equation
is solved approximating the distribution function by a two-temperatures
(horizontal and vertical) gaussian distribution. The obtained evolution
equations for the partial temperatures are solved, finding a very good
agreement with Molecular Dynamics simulation results for a wide range of the
parameters (inelasticity, height and density) for states whose projection over
a plane parallel to the walls is homogeneous. In the stationary state, where
the energy lost in collisions is compensated by the energy injected by the
walls, the pressure tensor in the horizontal direction is analyzed and its
relation with an instability of the homogenous state observed in the
simulations is discussed.
