Derivation of Kinetic Equations from the Generalized Langevin Equation

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 ${δ}g({{→}}{x},{{→}}{p},t){≡}g({{→}}{x},{{→}}{p},t){-}〈g({{→}}{x},{{→}}{p},t)〉$for a classical many-particle system. This equation is then used to develop an exact kinetic equation for the time-correlation function ${δ}g({{→}}{x},{{→}}{p},0){δ}g({{{→}}{x}}^{{'}},{{{→}}{p}}^{{'}},t)$ [which is the generalization of the Van Hove time-dependent pair correlation function $G({{→}}{r},t)$]. 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.