Small and Large Mean Free Paths

Abstract

It was pointed out in Chapter IV, Section 1, that if we want to solve the Boltzmann equation for realistic nonequilibrium situations, we must rely upon approximation methods, in particular, perturbation procedures. In order to do this, we have to look for a parameter ɛ which can be considered to be small in some situations. In Chapter IV, Section 2, ɛ was assumed not to appear directly in the Boltzmann equation. This led us to considering the linearized Boltzmann equation, which turns out to be useful for describing situations in which deviations of velocity and temperature from their average values are small. If we look for different expansions, a first step consists in investigating the order of magnitude of the various terms appearing in the Boltzmann equation. If we denote by τ a typical time scale, by d a typical length scale and by ξ a typical molecular velocity, then [see, for example, Eq. (II.3.15)]: where ∼ denotes that two quantities are of the same order of magnitude, n = ρ/m is the number density of molecules and σ the molecular diameter (or range of the interaction potential).