Kinetic equation for one-dimensional motion of spheres

A dynamic system of identical spheres in a vessel is considered as a gas model. The spheres and the walls of the vessel are assumed to be absolutely rigid and elastic. For three-dimensional motion, the chain of Bogolyubov equations is derived in a form not available in the literature. It is shown that the reason for the noninvertibility of the Boltzmann kinetic equation is an approximate description of the dynamic system. For the one-dimensional motion of the spheres along the straight-line segment between the walls, the Bogolyubov chain is closed in the class of multiplicative distributions by a limiting transition in which the number of spheres tends to infinity and the sum of their diameters remains constant. The obtained kinetic equation differs substantially in structure from the Boltzmann equation. For example, it is invertible. It implies the equations of multivelocity hydrodynamics. The existence of the solution in the large in time is established. It is shown that a closed kinetic equation for the one-particle projection does not exist in the class of arbitrary distributions.