# Isothermal-isobaric and grand canonical ensembles

Also useful are the isothermal-isobaric and grand canonical ensembles, which are defined just as they are for the classical cases:

• isothermal-isobaric:

$\Delta(N,P,T) = \int_0^{\infty}dV e^{-\beta PV} Q(N,V,T) =\int_0^{\infty} dV {\rm Tr}(e^{-\beta (H+PV)})$

• grand canonical ensembles

${\cal Z}(\mu,V,T) = \sum_{N=0}^{\infty}e^{\beta \mu N}Q(N,V,T) = \sum_{N=0}^{\infty} {\rm Tr}(e^{-\beta (H-\mu N)})$