  We provide new concepts for understanding transport phenomena in flows of
granular materials by using a non-Fickian macroscopic model of axial diffusion
of a granular material in a finite cylindrical tumbler. The model accounts for
diffusion induced by particle collisions only in a thin surface flowing layer
due to localization of shear within the cross-section of the drum. All model
parameters are related to measurable quantities in a granular flow. It is shown
that the proposed model is a member of the general class of linear constitutive
relations with memory. An exact solution for the spreading of a finite-width
pulse initial condition under the proposed non-Fickian model is derived and
compared to the solution of a Fickian model (i.e., the ``classical'' diffusion
equation).
