The Grabert formalism as developed in [1] seems to be quite general, and it is possible to restate f.e. the Mori [2] or Mazur-Oppenheim [3] formalism in Grabert's theory (i.e. one can find relevant probability densities for both cases which result in the right projector / the right generalised Langevin equation).

In chapter 4, section 2 of [1], a generalised Fokker-Planck equation is derived (following in principle [4] or [5]) and it is claimed, that the projector

\(\mathbf{P}X(\Gamma)=\int d\alpha\ \frac{tr\{\rho_\beta \psi_\alpha X\}}{p_\beta(\alpha)}\psi_\alpha(\Gamma)\)

could also be derived from Grabert's projector with a special relevant probability density \(\overline{\rho}(t)\) (see eq. 4.2.1 in [1]).

However, I'm unable to reproduce this result. The task seems simple, as it should be only a substitution into a given formula.

Could someone familiar with Grabert's work explain the method to derive this particular projector in Grabert's formalism? Thanks.

[1] H. Grabert: Projection Operator Techniques in Nonequilibrium Statistical Mechanics

[2] H. Mori: Transport, Collective Motion, and Brownian Motion; Prog. Theor. PHys. 33,p. 423

[3] P. Mazur, I. Oppenheim: A molecular theory of Brownian motion; Physica 50,2,p.241

[4] R. Zwanzig: Problems in nonlinear transport theory; in Systems Far from Equilibrium / Lecture Notes in Physics Vol. 132, p. 198

[5] Grabert et al.: Microdynamics and nonlinear stochastic processes of gross variables; J. Stat. Phys. 22,5,p.537