Mathematical Methods in Kinetic Theory

The motivations that suggested the publication of this book in 1969 are still largely valid. In this revised edition I have updated the literature on the subjects that were covered in the original version; I have also added an appendix to Chapter VI, explaining the technique for solving singular integral equations and systems of such equations. In addition I have tried to give an introduction to the important developments concerning the purely mathematical theory (existence and uniqueness theorems). This part of kinetic theory has reached a mature stage in the last few years and is relevant for both the physical foundations of the subject (validity of the Boltzmann equation) and the application of kinetic theory to rarefied gas dynamics (numerical and simulation methods).

The kinetic theory of gases is a part of statistical mechanics, i.e., of the statistical theory of the dynamics of mechanical systems formed by a great number of particles, such as the number of molecules contained in a lump of matter of macroscopic dimensions. The aim of statistical mechanics is to explain the macroscopic behavior of matter in terms of the mechanical behavior of the constituent molecules, i.e., in terms of motions and interactions of a large number of particles. We shall assume that classical mechanics can be applied, and, therefore, the molecules are subject to Newton's second law.