Two-fluid theory of coupled kinetic equations for a test particle and a background fluid

Abstract

The generalized Langevin equation formalism of Mori and Zwanzig and Akcasu and Dudenstadt is used to derive coupled kinetic equations for the distribution functions of a tagged test particle and of the untagged members of the surrounding background fluid. The interaction between any two of these particles is taken to be the sum of a long-ranged continuous potential and a short-ranged, hard-core contribution. The distribution function of the background fluid is chosen to depend on the location of a solvent particle measured relative to the instantaneous position of the test particle. Because of this choice, the frequency matrix associated with the pair of distribution functions incorporates much of the coupling between the tagged and untagged particles which otherwise would be imbedded within their separate memory functions. Consequently, the mean-field approximation obtained by neglecting the memory function of this two-fluid theory is expected to be quite accurate. Explicit forms are derived for the coupled mean-field kinetic equations gotten in this way. In regions remote from the test particle, the kinetic equation for a member of the solvent reduces to previously derived kinetic equation of the type called revised or generalized Enskog equations. Finally, it is shown that the kinetic equations of this two-fluid, mean-field theory also can be gotten by truncating the appropriate BBGKY hierarchy of equations, using a plausible, easily interpretable connection between pair-space distribution functions and the singlet distribution functions specific to the tagged and untagged particles.