Connection between Grabert formalism and the Zwanzig (1980)/Grabert-Hänggi-Talkner projector

Question

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
 

Answer 1

Prof. Grabert was so kind to point out the derivation.

The relevant probability density of eq. 4.1.10

\(\bar{\rho}(\Gamma,t)=\rho_\beta(\Gamma)\int d\alpha\ \frac{p(\alpha)}{p_\beta(\alpha)}\Psi_\alpha(\Gamma)\)

is linear in the "mean values" \(p(\alpha)\).

Going back to the original definition of the Grabert projection operator

\(\mathbf{P}[a]F=\rm{tr}\{\bar{\rho}F\}+\sum_i(A_i-a_i)\rm{tr}\{(\partial_i\bar{\rho})F\}\),

the projection operator simplifies to

\(\mathbf{P}[a]F=\sum_iA_i\rm{tr}\{\frac{\partial\bar{\rho}}{\partial a_i}F\}\),

because the relevant probability density satisfies

\(\bar{\rho}=\sum_ia_i\frac{\partial\bar{\rho}}{\partial a_i}\)

due to its linearity in the mean values.

\(\frac{\partial\bar{\rho}}{\partial a_i}\) is in our case the variational derivative with respect to \(p(\alpha)\) and therefore equal to

\(\rho_\beta(\Gamma)\frac{1}{p_\beta(\alpha)}\Psi_\alpha(\Gamma)\).

It remains to convert the particular projection operator above to the relevant variables used (i.e. \(A_i\rightarrow \Psi_\alpha(\Gamma), a_i\rightarrow p(\alpha),\sum_i \rightarrow\int d\alpha,\) and the Zwanzig/GHT-projector follows.