The theory of thermal relaxation of light dilute particles in a heat bath: Integral and differential elastic collision operators

Abstract

The field-free relaxation in velocity space of an infinitely dilute test particle species colliding elastically with a thermally distributed host particle species is considered. Spatial homogeneity is assumed. The test particle distribution function evolves in velocity space according to the Boltzmann equation, and its spherical harmonic components are decoupled. The lth harmonic ƒ l evolves according to ∂ƒ l/∂t = C lƒ l where the linear operator C l depends on the differential collision cross-section and the test particle: host particle mass ratio. C l is an integral operator, but may be expanded in ascending powers of mass ratio: the coefficient of the nth power is a 2 nth order differential operator in | v |. A discussion of the evolution of the harmonics ƒ l is presented both to leading order in mass ratio and with the first correction included. For anisotropic harmonics the leading order and first correction terms are n = 0 and n = 1, while for the isotropic component the n = 0 term vanishes and the leading order and first correction terms correspond to n = 1 (the Davydov operator) and n = 2. Properties of the evolution equation and its solution are studied. The best method of solution is to find the Green's propagator (initially a delta function in velocity space) by Laplace transformation with respect to time and, when possible, by eigenfunction expansion in the variable | v |. If this eigen problem is irregular, other methods are used. Most of the work is a study of the Davydov operator; it is useful to transform its relaxation problem to the form of the time-dependent Schrödinger equation, and an important model problem is solved in this representation. Several physical properties of the exact solution are retained to leading order in mass ratio, but break down thereafter. All of the results derived are useful in swarm analysis.