DERIVATION OF KINETIC EQUATIONS FROM 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 δg(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 δg(x→,p→,0)δg(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. © 1969 The American Physical Society.