  The Maxwell-Smoluchowski (MS) theory of gas diffusion is revisited here in
the context of gas transport in straight channels in the Knudsen regime of
large mean free path. This classical theory is based on a phenomenological
model of gas-surface interaction that posits that a fraction $\vartheta$ of
molecular collisions with the channel surface consists of diffuse collisions,
i.e., the direction of post-collision velocities is distributed according to
the Knudsen Cosine Law, and a fraction $1-\vartheta$ undergoes specular
reflection. From this assumption one obtains the value
$\mathcal{D}=\frac{2-\vartheta}{\vartheta}\mathcal{D}_K$ for the
self-diffusivity constant, where $\mathcal{D}_K$ is a reference value
corresponding to $\vartheta=1$. In this paper we show that $\vartheta$ can be
expressed in terms of micro- and macro-geometric parameters for a model
consisting of hard spheres colliding elastically against a rigid surface with
prescribed microgeometry. Our refinement of the MS theory is based on the
observation that the classical surface scattering operator associated to the
microgeometry has a canonical velocity space diffusion approximation by a
generalized Legendre differential operator whose spectral theory is known
explicitly. More specifically, starting from an explicit description of the
effective channel surface microgeometry -- a concept which incorporates both
the actual surface microgeometry and the molecular radius -- and using this
operator approximation, we show that $\vartheta$ can be resolved into easily
obtained geometric parameters.
