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.