Derivation of Kinetic Equations from the Generalized Langevin Equation

Abstract

The projection operator techniques of Zwanzig and Mori are used to obtain a generalized Langevin equation describing the time evolution of the fluctuation of the microscopic phase density $\ensuremath{\delta}g(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},t)\ensuremath{\equiv}g(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},t)\ensuremath{-}〈g(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},t)〉$for a classical many-particle system. This equation is then used to develop an exact kinetic equation for the time-correlation function $\ensuremath{\delta}g(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}},\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}},0)\ensuremath{\delta}g({\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}}^{\ensuremath{'}},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}}^{\ensuremath{'}},t)$ [which is the generalization of the Van Hove time-dependent pair correlation function $G(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}},t)$]. In the lowest order of approximation, this kinetic description reduces to the Vlasov-like equation which has been used to study neutron scattering from liquids. A less restrictive approximation is obtained by utilizing weak-coupling perturbation theory to yield a generalized Fokker-Planck equation for the time-correlation function. Other possible approximation schemes are also discussed.