Re-normalized quasi-linear approximation of plasma turbulence: Part 2. Validity of short-time expansion and Markovian truncation of the propagator

In part 1 (Misguich & Balescu 1975) the Dupree description of plasma turbulence (i.e. the weak-coupling approximation of Weinstock) was derived and completed in the framework of the re-normalized quasi-linear approximation. The time behaviour of the frequency shift and of the Dupree damping was proved to be modified by time-ordering effects, usually neglected. In part 2 we show that a Markovian truncation must be applied to the basic average propagator Ū, to recover the short-time expansion of Birmingham & Bornatici. Such an expansion is shown to be generally non-convergent. The corresponding short-time results are invalid.