Integral approach of successive collisions to linear charged-particle transport through a slab

A method is described for calculating steady-state velocity distribution functions of charged test particles streaming through a one-dimensional slab under the influence of an applied electric field, at the same time undergoing binary collisions with a background gas. By way of combining the method of characteristics with the concept of successive collisions, the partial distribution functions $f_{\nu }$, related to the particles having undergone exactly $\nu$ collisions, can be successively calculated by integrating linear first-order ordinary differential equations. Their sum is the total distribution function satisfying the linear Boltzmann equation. Boundary conditions and strongly nonequilibrium situations can be handled in a natural way, and convergence is inferred from simple physical arguments. As an example of application, a boundary-value problem involving symmetric charge transfer is solved analytically, and a previous result is recovered in the appropriate limit.