Moment series expansions for a model kinetic equation

Abstract

A steady-state, one-dimensional model kinetic equation with a simple external force and a Lorentz collision term is solved by the approximation methods of moment expansions, by a neoclassical type of asymptotic expansion, and by other simple expansions. The very high-order moment expansions are assumed to give the correct solution, and a comparison is given with low-order moment expansions, the neoclassical expansion, and a small external force expansion. All the calculations and expansions are given for the cases in which the three fundamental dimensional physical parameters that characterize the problem are small. These parameters are the collisionality, the mean velocity to which the system is driven, and the magnitude of the spatially varying external force. Each of the expansions has regions of applicability and utility. Even in the very low collisionality case, low-order moment expansions give the correct order of magnitude of the solution, the mean flow velocity within 30%, the temperature quite accurately, the average value of the heat flow poorly, but give the spatially varying part of the heat flow very well. For the neoclassical expansion to be accurate, not only must the collisionality be small, but it also must be sufficiently smaller than other small physical parameters in the system. For parameter values somewhat less than one moment expansions of order five are quite accurate. For large values of the collisionality, the Hilbert–Chapman–Enskog expansion is recovered.