I am trying to understand the way generalized canonical ensembles like the pressure ensemble are derived from the standard canonical ensemble.
In the derivation for the standard form, one defines a system $S$ and a reservoir $R$. With a total microcanonical Hamiltonian: $$H(X)=H_S(X)+H_R(X)$$ My question is what do we do to put in the volume exchange? What is the basic idea of going from pure energy exchange with a reservoir to different additional things like volume.
My guess is to just add it as an energy term that is not part of the Hamiltonian?
So then: $$H(X)=H_S(X)-V p+H_R(X)$$ or in general with $y$ being an intensive and $x$ being an intensive variable: $$H(X)=H_S(X)+x y +H_R(X)$$
I would be glad about a correction of my guess or a verification, of course.