Grand Canonical Ensemble

#Physics
Both energy AND particles are exchanged

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

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

$\displaystyle {\left\langle{N}\right\rangle}=\lambda \frac{ \partial }{ \partial \lambda }\ln \mathbb{Z}=\frac{\lambda}{\mathbb{Z}}\frac{ \partial \mathbb{Z}}{ \partial \lambda }=\beta \frac{\partial }{\partial \mu}\ln \mathbb{Z}$

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