We consider the problem of describing the steady-state spreading of a
collimated particle beam as it penetrates a background material. The e
xact description for this problem is taken as the linear transport equ
ation with full six-dimensional phase space dependence. In the limit o
f very forward peaked scattering with small energy transfer, the Fokke
r-Planck scattering description is used. To obtain a simplified model
of beam transport, we assume that the beam in question has weak spatia
l gradients in the plane perpendicular to the beam direction, and that
the beam nearly maintains its collimated integrity as it passes throu
gh the material. These assumptions lead to a hierarchy of advection-di
ffusion-like approximations for the spatial distribution of the partic
le density per unit energy. In the simple case of monoenergetic transp
ort in a purely scattering homogeneous material, these equations are e
asily solved via Laplace and Fourier transformations to obtain explici
t analytical results. Comparisons with benchmark Monte Carlo calculati
ons give an indication of the accuracy of this treatment of beam sprea
ding.