Dynamics and kinetic theory of hard spheres under strong confinement

TL;DR

This paper develops a rigorous kinetic theory for dilute hard sphere gases confined between two plates, deriving equations that align with equilibrium statistical mechanics and revealing complex density profiles under external fields.

Contribution

It provides a systematic derivation of kinetic equations for strongly confined gases from first principles, improving upon heuristic models.

Findings

Stationary solutions consistent with equilibrium statistical mechanics.

Complex density profiles due to confinement and external fields.

A solid theoretical basis for studying strongly confined dilute gases.

Abstract

The kinetic theory description of a low density gas of hard spheres or disks, confined between two parallel plates separated a distance smaller than twice the diameter of the particles, is addressed starting from the Liouville equation of the system. The associated BBGKY 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, both in 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.