Eigenmodes of a dilute simple gas in equilibrium

The eigenmodes of a dilute simple gas described by the spatially inhomogeneous linearized Boltzmann operator are studied. These eigenmodes can be obtained approximately using a finite set of polynomia in the velocity of a molecule. Three approximation methods are compared: the Chapman-Enskog method, the moment method and the Bhatnagar-Gross-Krook method. The implications for the dynamic structure factor for light and neutron scattering are discussed. This factor is dominated by the hydrodynamic modes and their extensions to smaller wave lengths, λ, for λ ≧ 7.5l0 and by a Gaussian and its first correction for λ ≦ 1.5l0, where l0 is the mean free path.