  We develop an analytical framework and numerical approach to obtain the
coefficient of self-diffusivity for the transport of a rarefied gas in channels
in the limit of large Knudsen number. This framework provides a method for
determining the influence of channel surface microstructure on the value of
diffusivity that is particularly effective when the microstructure exhibits
relatively low roughness. This method is based on the observation that the
Markov transition (scattering) operator determined by the microstructure, under
the condition of weak surface scattering, has a universal form given, up to a
multiplicative constant, by the classical Legendre differential operator. We
also show how characteristic numbers of the system -- namely geometric
parameters of the microstructure, the spectral gap of a Markov operator, and
the tangential momentum accommodation coefficient of a commonly used model of
surface scattering -- are all related. Examples of microstructures are
investigated to illustrate the relation of these quantities numerically and
analytically.
