Asymptotic theory of the linear transport equation for small mean free paths. I

We consider the linear transport equation, which describes the interaction of a rarefied gas with a denser gas. We require the spatial variation of all quantities in this equation to be small over the distance of a typical mean free path, and the time variation of all quantities to be small during a typical mean free time. We also require the average velocity of the denser gas, the external electric field, and the internal sources to be small. These smallnesses are expressed analytically by means of a small parameter $\epsilon$. A solution of the transport equation, subject to the above restrictions, is constructed which is asymptotic with respect to $\epsilon$. To leading order the solution has the form $\psi \sim A_0(\overrightarrow{\mathrm{r}},t)\phi (\overrightarrow{\mathrm{r}},v,t)$, where $\phi$ is the eigenfunction of a certain operator and $A_0$ satisfies a diffusion equation containing effects due to the motion of the denser gas, inhomogeneities in the denser gas, and the external electric and magnetic fields. We discuss the physical interpretation of this equation and give expressions for the density and average velocity of the rarefied particles.