The Source of Examples

Abstract

The Boltzmann equation is the first and the most celebrated nonlinear kinetic equation introduced by the great Austrian scientist Ludwig Boltzmann in 1872 [111]. This equation describes the dynamics of a moderately rarefied gas, taking into account two processes: the free flight of the particles, and their collisions. In its original version, the Boltzmann equation has been formulated for particles represented by hard spheres. The physical condition of rarefaction means that only pair collisions are taken into account, a mathematical specification of which is given by the it Grad-Boltzmann limit [200]: If N is the number of particles, and σ is the diameter of the hard sphere, then the Boltzmann equation is expected to hold when N tends to infinity, σ tends to zero, Nσ³ (the volume occupied by the particles) tends to zero, while Nσ² (the total collision cross section) remains constant. The microscopic state of the gas at time t is described by the one-body distribution function Px,v,t, where x is the position of the center of the particle, and v is the velocity of the particle. The distribution function is the probability density of finding the particle at time t within the infinitesimal phase space volume centered at the phase point (x,v). The collision mechanism of two hard spheres is presented by a relation between the velocities of the particles before [v and w] and after [v’ and W’] their impact: