Kinetic-Equation Approach to Time-Dependent Correlation Functions

Abstract

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 $W(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ at $t=0$ differs from its equilibrium value $\ensuremath{\rho}{h}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$, where $\ensuremath{\rho}$ is the average fluid density, and ${h}_{0}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}})$ is the Maxwellian velocity distribution function. We investigate the time evolution of the (self-) distribution function of this labeled particle, ${f}_{s}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}},t)$, as well as the deviation from equilibrium, $\ensuremath{\eta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}},t)$, of the total one-particle distribution function. The latter represents the density of fluid particles, labeled and unlabeled, at position $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$ and velocity $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$. Since both ${f}_{s}$ and $\ensuremath{\eta}$ are linear functionals of $W$, they will satisfy exactly a linear non-Markovian kinetic equation of the form $f=\mathrm{B}f+\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}\mathrm{M}({t}^{\ensuremath{'}})f(t\ensuremath{-}{t}^{\ensuremath{'}})$. 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 ${f}_{s}$ and $\ensuremath{\eta}$ 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.