Kinetic Theory of Dilute Gases

Abstract

Kinetic theory of gases can be considered as a branch of statistical mechanics, although it preceded it from a historical point of view (Brush, 1965; Brush, 1976; Cercignani, 1998). The aim of kinetic theory is to explain, by using a statistical approach, the macroscopic properties of gases out of equilibrium from the knowledge of the laws governing the dynamics of the microscopic constituents (atoms or molecules). The first serious advances in kinetic theory began in the 18th Century with the work of Daniel Bernoulli (1700–1782), continued along the 19th Century with John Herapath (1790–1868), John J. Waterston (1811–1883), James P. Joule (1818–1889), August K. Kronig (1822–1879), and, especially, Rudolf Clausius (1822–1888). The theory reached its maturity with James C. Maxwell (1831–1879) and Ludwig Boltzmann (1844–1906). The former derived the velocity distribution function of a gas in equilibrium, obtained the transfer equations for mass, momentum, and energy in a dilute gas, and showed that the coefficients of shear viscosity and thermal conductivity of a dilute gas are independent of the density. Maxwell also analyzed a class of interaction models where the particles repel each other with a potential inversely proportional to the ω-th power of the distance and found that the collisional rate of change of a given quantity involved the relative velocity raised to the power (ω − 4)/ω; in the special case of ω = 4 the relative velocity drops out and the velocity integration can be carried out without the explicit knowledge of the velocity distribution function. This simple interaction model with ω = 4 defines the so-called Maxwell molecules.