Boltzmann kinetic equation for a strongly confined gas of hard spheres

A Boltzmann-like kinetic equation for a quasi-two-dimensional gas of hard spheres is derived. The system is confined between two parallel hard plates separated a distance between one and two particle diameters. An entropy Lyapunov function for the equation is identified. In addition to the usual Boltzmann expression, it contains a contribution associated to the confinement of the particles. The steady properties of the system agree with equilibrium statistical mechanics results. Equations describing the energy transfer between the degrees of freedom parallel and perpendicular to the confining plates are obtained for some simple initial configurations. The theoretical predictions are compared with molecular dynamics simulation data and a good agreement is found.

Figure 1

Sketch of the collision between two particles in the quasi-two-dimensional system of hard spheres described in the main text. Particles are confined by two hard parallel walls, separated a distance $h$ between one and two particle diameters. Given the coordinate $z$ of one of the particles, there is a biunivocal correspondence between the coordinate $z_1$ of the other particle involved in the collision and the polar angle $\theta$.

Figure 2

Decay of the averaged vertical, ${\overline{T}}_z$, and horizontal, ${\overline{T}}_{xy}$, temperature parameters toward their equal steady values in a confined system that is homogeneous in the horizontal $x-y$ plane. Time is measured in the dimensionless units indicated in the label, with $v_0\equiv \sqrt{2k_{B}T\left(0\right)/m}$. The solid lines are MD simulation results while the dashed lines are the theoretical predictions given by Eqs. (75) and (76). Moreover, the two (red) upper lines correspond to the vertical temperature parameter and the two (black) lower lines to the horizontal temperature parameter.