Fixed point properties of gas-surface scattering operators

The calculation of the particle flux in stationary free-molecular flow of monatomic gases enclosed by solid surfaces in a given geometry is considered as a fixed-point problem. Physically plausible properties of the boundary conditions are formulated, which insure the existence of a unique fixed point of the corresponding operator in flux space. Sufficient conditions are given for any trial function to be transformed into this fixed point in the limit of infinitely many iterations of the boundary operator. The general theory applies to an exactly solvable model of gas-surface scattering