Question

We have seen the microcanonical ensemble, Ω(U,V,N), the canonical ensemble, Z(T,V,N), and the grand canonical ensemble, Z(T,V,μ). Why do we not create a hyper-grand canonical ensemble, Ξ(T, P, μ)?

Answer 1

We cannot do so for a one species system, i.e., where only one type of particles ( described by a single chemical potential $\mu$ ) is there. The reason is that the corresponding thermodynamic potential $G(T,P,\mu)$ , say, is identically zero:

$(U,V,N)\rightarrow( T,P,\mu) \implies G(T,P,\mu)=U-TS+PV-\mu N=0$

Other handwaving arguments may be given such as, all of the control parameters would be intensive, which leaves all the extensive conjugated parameters unbounded, and you can't write your sums and integrals anymore.