The Trend to Equilibrium According to the Kinetic Theory of Gases

Abstract

When we study the Maxwell-Boltzmann equation,

$${\partial _t}F + {\text{v}} \cdot {\partial _x}F + b \cdot {\partial _v}F = \mathbb{C}\left[ F \right],$$

(8.15)

$$mathbb{C}\left[ F \right] = \iint {w\left( {F'F*' - FF*} \right)dS dv}*,$$

(18.17)

we distinguish between two kinds of solutions. The general solutions correspond to an initial-value problem. In principle, at least, they trace the evolution of the gas as some initial molecular-density function F(x, v, 0) is converted, according to the Maxwell-Boltzmann equation, into the present molecular-density function F(x, v, t). The various gross quantities defined from these solutions obey the field equations (8.22) of continuum mechanics; the kinetic equilibrium solutions lead to the same gross equations as those of Eulerian hydrostatics; but in general gas flows there is no reason to think that either the irreversibility or any constitutive equation of continuum thermomechanics is satisfied. Solutions of a second and special kind, called dominant solutions, have been conjectured to exist, and formal procedures have been devised so as to find successive approximations to properties enjoyed by them. These solutions may or may not conform with the irreversibility of continuum thermomechanics, but certainly they do obey specific constitutive relations appropriate to a certain kind of non-simple material; indeed, the main aim of the classical approach to the kinetic theory is to calculate the response functions.