Kinetic-Equation Approach to Time-Dependent Correlation Functions

We have developed and, to some extent, solved a number of kinetic equations for displaced correlation functions in a classical fluid. These functions, of which the Van Hove neutron scattering function is a special example, are one-particle distribution functions obtained from a Gibbs ensemble which is initially, at t=0, in equilibrium except for one labeled particle whose distribution W (r, v) at t=0 differs from its equilibrium value h₀ (v), where is the average fluid density, and h₀ (v) is the Maxwellian velocity distribution function. We investigate the time evolution of the (self-) distribution function of this labeled particle, fₒ (r, v, t), as well as the deviation from equilibrium, (r, v, t), of the total one-particle distribution function. The latter represents the density of fluid particles, labeled and unlabeled, at position r and velocity v. Since both fₒ and are linear functionals of W, they will satisfy exactly a linear non-Markovian kinetic equation of the form f=Bf+0^tdt^'M (t^') f (t-t^'). B is a time-in-dependent and M a time-dependent (memory) operator (nonsingular in t). Our kinetic equations (first- and higher-order) are based on neglecting or approximating M in such a way that the short-time behavior of fₒ and is described exactly. The rationale behind this scheme is that our choice of initial ensemble is precisely of the type generally assumed in the "derivation" of kinetic equations. The calculation of B is straightforward and depends in a very important way on whether the interparticle potential in the fluid is smooth or contains a hard core. In the former case, the first-order kinetic equation is of the Vlasov type with an effective potential given by the equilibrium direct correlation function, while in the latter case, B contains, in addition, a linear Enskog-type collision term. We show that this Vlasov equation (also derived previously by many authors) gives a damping linear in the wave number k for small k instead of the hydrodynamic k^2 dependence. The kinetic equation for systems with hard cores does not give correct hydrodynamic behavior. (For a one-dimensional system of hard rods, the first-order kinetic equation is exact. ) We also obtain and solve a second-order kinetic equation, which is a generalized Vlasov-Fokker-Planck-type equation, for systems with continuous potentials.