Grand Canonical Ensemble

Weights

For fixed $T,V,\mu$, a microstate with energy $E_i$ and particle number $N_i$ has probability proportional to

$$\exp[-\beta(E_i-\mu N_i)].$$

The grand partition function is

$$\Xi=\sum_i e^{-\beta(E_i-\mu N_i)}.$$

Thermodynamic Potential

The grand potential is

$$\Omega=-k_BT\ln \Xi.$$

It is useful for quantum gases, adsorption, and any system where particle number fluctuates.