Start with the Liouville operator $ℒ$ where for any Heisenberg observable $A_H(t)\in 𝔸$

and

This generalizes to both classical mechanics where $𝔸={ℝ}^{2n}\to ℝ$, a function over the positions and momentum, or quantum mechanics where $𝔸=({ℝ}^n\times {ℝ}^n)\to ℂ$ is a self Hermetian matrix. If the Hamiltonian isn't time dependent, then neither is the Heisenberg Hamiltonian, and thus the Liouville operator won't be time dependent. Then time evolution can just be simplified to

Projection Operator

We want to split the observables into "relevant" and "irrelevant" groups, then split equations of motion of the relevant observables into a part that only depends on the relevant observables and a part that depends on the irrelevant observables. We assume, with minimal assumptions, arbitrary projection operator $𝒫\in 𝔸\to 𝔸$.

Weird Identity

Let

then

and operators commute with their exponent

And fundamental theorem of calculus

Then left multiply

So

We have

Then use the operator idendity on the right one

more stuff

Projected force

Let the projected force be defined as

Then gives

Note that we didn't make any assumptions about the projection operator $𝒫$ or the Liouville operator $ℒ$, so it should work for any projection operator on any time evolution.

Zwanzig Projection Operator

The Zwanzig projection operator can be defined by

Where ${\rho }_{\text{eq}}(\mathrm{\Gamma })=\frac{1}{Z}{e}^{-\frac{H(\mathrm{\Gamma })}{k_{B}T}}$ is the cannonical equilibrium distribution, and $g\in {ℝ}^{2n}\to {ℝ}^m$ is some collection of observables. Let these be called the "relevant" observables. We want to find the time evolution of these relevant observables.

First Term

we can do integration by parts and assume $p_{\text{eq}}$ goes to zero at infinity

And $ℒp_{\text{eq}}=0$ so

Then integrate by parts again

Chain rule

Let

Second and Third Term

We probably can't get an exact expression for the second term but we can relate it to the third term.

Looking at just

Next we show that