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 ρh0(v→), where ρ 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, fs(r→,v→,t), as well as the deviation from equilibrium, η(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 fs and η are linear functionals of W, they will satisfy exactly a linear non-Markovian kinetic equation of the form f=Bf+0tdt′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 fs and η 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 k2 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.