Self-structure factor of hard-sphere gases for arbitrary ratio of bath to test particle masses

Abstract

The self-structure factor Ss(k,ω) for test particles of mass different from the mass of the bath particles is considered. For a dilute hard-sphere gas the self-Enskog kinetic equation is solved to high accuracy for Ss(k,ω) by the Gross–Jackson (GJ) modeling procedure. When the variables in the self-Enskog kinetic equation are scaled to customary dimensionless parameters, the peak height and width of Ss(k,ω) vary only about 10% within a large mass ratio range. The GJ solution approaches smoothly the Ss(k,ω) of the Fokker–Planck equation (small bath particle mass). On the other hand, large variation of Ss(k,ω) occurs near the Lorentz limit (large bath particle mass). The optimized Bhatnagar–Gross–Krook (BGK) model of Groome, Dufty, and Lindenfeld shows better quantitative agreement than the usual BGK model but does not predict some qualitative features. Hydrodynamic limits are also discussed in some detail. It is demonstrated that the hydrodynamic limit of Ss(k,ω) for the Lorentz equation does not have the usual Lorentzian line shape.