Eigenvalues of collision operators: Properties and methods of computation

The linear Boltzmann equation for active atoms submerged in the much denser perturber gas contains a collision rate and a kernel. These two quantities are combined into a single entity-the collision operator. The collision operator possesses several interesting properties, the most important being that it is Hermitian. The eigenvalues are negative with the exception of one eigenvalue, which is zero and corresponds to the Maxwellian (steady-state) velocity distribution. A set of functions, closely related to the eigenfunctions of the quantum-mechanical harmonic oscillator, is postulated to approximate the true eigenfunctions. This assumption was a basis of the method of modeling various physical phenomena occurring in the gaseous mixtures, subjected to a radiation field. The eigenvalues of the collision operator were treated as free parameters. In this paper we establish a direct relationship between the eigenvalues and the collision integrals, or transport coefficients, known from the kinetic theory of gases. The generating function approach is employed to derive expressions yielding the eigenvalues. The obtained results form a bridge between kinetic theory, atomic physics, and quantum optics.