Grand Canonical Ensemble

Both energy AND particles are exchanged

$P(N,\epsilon )=\frac{e^{\beta (\mu N-\epsilon )}}{\mathbb{Z}}$

• Boltzmann distribution for grand canonincal ensemble
• The probability of a system of $N$ particles at energy $\epsilon$
• $\mathbb{Z}$ is the grand partition function

$⟨N⟩=\lambda \frac{\partial }{\partial \lambda }\ln \mathbb{Z}=\frac{\lambda }{\mathbb{Z}}\frac{\partial \mathbb{Z}}{\partial \lambda }=\beta \frac{\partial }{\partial \mu }\ln \mathbb{Z}$

• Thermal average number of particles
• $\lambda$ is the absolute activity
• $\mu$ is the [chemical potential]