Fluid Dynamic Limits of the Boltzmann Equation

Abstract

In 1866, James Clerk Maxwell (1831–1879) developed a fundamental theoretical basis for the kinetics theory of gases. Maxwell’s theory is based on the idea of Daniel Bernoulli (1738), which gave birth to the kinetic theory of gases, that gases are formed of electric molecules rushing hither and thither at high speeds, colliding and rebounding according to the laws of elementary mechanics (see, Cercignani, Illner, and Pulvirenti 1994, pp. 8–12). In fact, Maxwell developed, first, a theory of transport processes and gave a heuristic derivation of the velocity distribution function that bears his name. Next, he developed a much more accurate model (Maxwell 1867), based on transfer equations, in fact, a model, according to which the molecules interact with a force inversely proportional to the fifth power of the distance between them (now commonly called Maxwellian molecules). With these transfer equations, Maxwell came very close to an evolution equation for the distribution, but this step (1872) must be credited to Ludwig Boltzmann (1844–1906). The equation under consideration is usually called the Boltzmann equation.