The discrete model of the Boltzmann equation

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Abstract
In this paper we study the general model of a gas with a discrete distribution of velocities. The medium is composed of identical particles which can only have velocity vectors belonging to a finite set of p vectors. When the medium is sufficiently rarefied, only binary collisions are considered. The original Boltzmann equation is replaced by a system of p first order partial differential equations. Each equation in the system is linear with respect to the derivatives of the unknown functions and quadratic with respect to the functions themselves. As in classical kinetic theory, by introducing a properly defined H-Boltzmann function, we can prove that for a gas in a uniform state, the distribution of velocities tends to a distribution, called Maxwellian, in which each collision brings no contribution to the evolution of densities. Among all distributions of velocities which correspond to given state variables, one and only one is Maxwellian, and the corresponding H-function is minimal. When the collision cross-section is infinite, the velocity distribution is Maxwellian, and the evolution of the gas is governed by the Euler equations (which form a hyperbolic system) and by the associated shock equations.