Equilibrium Statistical Operator for Grand Canonical Ensemble

[Petar Mali]

Canonical ensemble

equilibrium average value

$$\left\langle...\right\rangle_0=Q^{-1}Tr(e^{-\beta \hat{H}_0}...)$$

$$\hat{\rho}_0=Q^{-1}e^{-\beta \hat{H}_0}$$

Grand canonical ensemble

$$\left\langle...\right\rangle_0=Q^{-1}Tr(e^{-\beta (\hat{H}_0-\mu\hat{N})}...)$$

for example let's say

$$\hat{\mathcal{H}}_0=\hat{H}_0-\mu\hat{N}$$

Can I say that the equilibrium statistical operator

$$\hat{\rho}_0=Q^{-1}e^{-\beta (\hat{H}_0-\mu\hat{N})}=Q^{-1}e^{-\beta \hat{\mathcal{H}}_0 }$$

 

[Petar Mali]

Generally my question is about this ''$$0$$'' terms. Do I need to write this?