Validity of RPA for the initial state in the time evolution of a fluid

Zwanzig introduced a projection operator P-circumflexz such that P-circumflexzf, where f solves the Liouville equation, obeys a closed kinetic equation. On multiplying P-circumflexzf by a finite set of phase functions {A-circumflexi(x)} and integrating over phase space, we get values {αi} agreeing with those calculated from f. Furthermore, one can find P-circumflex such that P-circumflex f = fr has a specified form, e.g. fT, the linearization of the Jaynes distribution fJ, and also satisfies the integral conditions imposed on P-circumflexzf. The equation for fr = fT given by Los linearizes the Robertson kinetic equation for fJ. Multiplying the Los equation by A-circumflexi(x) and integrating over x, we get an equation for α-doti describing the irreversible approach to equilibrium. This agrees with existing phenomenology if the initial state term involving fT(0, x) vanishes and relaxation times are >0. Here we show that the initial state term vanishes if t = 0 is far enough along in system evolution and C(t) decays exponentially at long times. This work addresses problems left hitherto incomplete in the projection operator approach to irreversibility