note: I expect that a microscopic treatment of the rate of heat transfer between gases and solid surfaces will likely need to address this as well and so I'm looking in to that today, but if someone beats me to it that would be great!
For example Singh et al. (2009) J. App. Phys. 106, 024314 Modeling of subcontinuum thermal transport across semiconductor-gasinterfaces seems to address this using a Boltzman distribution for the gas atoms in the vauum and for phonons rather than individual atoms of the solid. However I can’t see how to apply that microscopically on a per-collision basis without estimating how many phonons are involved; is it one phonon on average? More? Less?
Figures at the end of The Interaction of Molecular Beams with Solid Surfaces show that the spread in angles of atoms incident at 45 degrees does get wider with increased surface temperature, but there is no information on changes in kinetic energy, but Wikipedia's Helium atom scattering; Inelastic measurements suggests that interactions with phonons can lead to a much more complicated picture.
I'm writing a simple Monte-Carlo simulation for an ultra-high vacuum chamber. I'll initially distribute atoms randomly in position and direction and with speeds derived from a Maxwell-Boltzmann distribution. At one end there is a tube of variable length; once a particle passes the far end of it it will be considered "pumped" and no longer tracked.
The idea is to address the effect of a long narrow tube between the chamber and the pump on the effective pumping speed.
I'll assume a monatomic gas with no internal degrees of freedom.
I can introduce the effect of wall roughness by randomly varying the microscopic surface normal from the macroscopic normal at each collision, but I can't think of a way to handle the random momentum transfer effects due to thermal vibrations of the atoms in the wall. These can both change the kinetic energy and the direction of scattering.
What would be a simple, first principles way to introduce it?
I can imagine treating the atom in the wall as a free particle with a similar thermal distribution of speed and direction, but the atoms of the wall are not free, they're constrained by bonds to adjacent atoms.