Dynamics and kinetic theory of hard spheres under strong confinement

The kinetic theory description of a low-density gas of hard spheres or disks, confined between two parallel plates separated at a distance smaller than twice the diameter of the particles, is addressed starting from the Liouville equation of the system. The associated Bogoliubov, Born, Green, Kirkwood, and Yvon hierarchy of equations for the reduced distribution functions is expanded in powers of a parameter measuring the density of the system in the appropriate dimensionless units. The Boltzmann level of description is obtained by keeping only the two lowest orders in the parameter. In particular, the one-particle distribution function obeys a couple of equations. Contrary to what happens with a Boltzmann-like kinetic equation that has been proposed for the same system on a heuristic basis, the kinetic theory formulated here admits stationary solutions that are consistent with equilibrium statistical mechanics, in both the absence and presence of external fields. In the latter case, the density profile is rather complex due to the coupling between the inhomogeneities generated by the confinement and by the external fields. The general theory formulated provides a solid basis for the study of the properties of strongly confined dilute gases.