The Boltzmann equation for dilute gases

Abstract The Boltzmann equation, which is the first and best known kinetic equation, describes the dynamics of classical dilute gases. For its derivation, the motion of the atoms and molecules is separated in free streaming and binary collisions. Notably, the kinetic equation that is obtained turns out to be irreversible despite the use of concepts of classical reversible mechanics. The origin of the irreversibility, quantified by the H-theorem, is explained and justified. The irreversibility manifests in that after a few collisions, the gases reach local thermal equilibrium described by Maxwellian distributions. For long times, it is shown that the system evolves via hydrodynamic equations and the transport coefficients, viscosity, and thermal conductivity, are computed in terms of the collision properties. The Boltzmann equation is extended to describe dense gases and granular media. Finally, the concepts presented in the chapter are used to explain the cooling of particles in the expanding universe.