Thermal relaxation of light dilute particles in a heat bath: a model problem

Abstract

The relaxation of a very dilute test particle species undergoing elastic collisions with a distinct, thermally distributed second particle species is considered. The Boltzmann equation is set up for the component of the test particle distribution function isotropic in velocity space; external forces and spatial dependence are ignored. For small test particle: heat-bath particle mass ratio, collisions are adequately described by a differential operator that is second-or der with respect to collision speed, which depends on the functional form of the collision frequency in terms of speed. The evolution equation is transformed to the form of the time-dependent Schrödinger equation, and the equivalent Schrödinger potential evaluated in terms of the collision frequency-speed relation. The relaxation problem may be solved by specifying the Green's propagator, describing the relaxation of an initial monoenergetic distribution. The case of constant Schrödinger potential is studied: the corresponding form of the collision frequency is found by an inversion process, and the evolution equation studied in Schrödinger form. The transformed speed variable has a finite range as particle speeds vary from 0 to ∞, and it is necessary to specify that particles are not lost at the finite values corresponding to small and large speeds. The Green's propagator is continued beyond this physical range, rather as in the ‘method of images’ of electrostatics. The particle conservation requirements induce a Laplace transform problem for the initial value of the Green's propagator, which is reduced to a polynomial recurrence relation and solved by finding the generating function. The Green's propagator is itself propagated forward from its initial value by a ‘primitive’ propagator which is the solution of the problem for initial delta function not only in the physical range, but anywhere in the space of the transformed variable. This primitive propagator is readily found, and the propagation quadrature performed to yield the true propagator in terms of parabolic cylinder functions. A check is made with a special case, for which relaxation to thermal form is explicitly demonstrated. Differences and similarities between this analysis and those corresponding to more general collision frequencies and potentials are considered.