Knudsen diffusivity in random billiards: Spectrum, geometry, and computation

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.