When a system can exchange both energy and particles with a large reservoir, equilibrium statistical mechanics describes it with the grand canonical ensemble.

The controlled macroscopic variables are temperature $T$, volume $V$ and chemical potential $\mu$, while the particle number $N$ is allowed to fluctuate.

A microstate $i$ with energy $E_i$ and particle number $N_i$ receives statistical weight

$$e^{-\beta(E_i-\mu N_i)},$$

where

$$\beta=\frac{1}{k_BT}.$$

The normalization is the grand partition function

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

so the probability of microstate $i$ is

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

The chemical-potential term matters because microstates with different particle numbers exchange particles with the reservoir. Positive or negative changes in $N_i$ therefore alter the reservoir-system thermodynamic balance as well as the microstate energy.

Mean particle number

Differentiating the grand partition function gives

$$\boxed{\langle N\rangle=\frac{1}{\beta}\left(\frac{\partial\ln\Xi}{\partial\mu}\right)_{T,V}}.$$

Thus the ensemble predicts an average particle number even though individual microstates can contain different $N$.

Grand potential

The thermodynamic potential associated with fixed $T$, $V$ and $\mu$ is the grand potential

$$\boxed{\Omega=-k_BT\ln\Xi}.$$

For a homogeneous equilibrium system under ordinary conditions,

$$\Omega=-pV.$$

The grand canonical ensemble is especially useful for open subsystems, adsorption, quantum gases and particle populations whose number is not fixed. It supplies the statistical-mechanical machinery behind chemical potential rather than defining chemical potential for the first time.