Grand canonical ensemble

Ensemble variables

• Chemical Potential, $\left.\mu \right.$

• Volume, $\left.V\right.$

• Temperature, $\left.T\right.$

Partition Function

Classical Partition Function (one-component system) in a three-dimensional space: $Q_{\mu VT}$

$Q_{\mu VT}=\sum _{N=0}^{\infty }{\frac {\exp \left[\beta \mu N\right]V^{N}}{N!\Lambda ^{3N}}}\int d(R^{*})^{3N}\exp \left[-\beta U\left(V,(R^{*})^{3N}\right)\right]$

where:

• $\Lambda$ is the de Broglie thermal wavelength (depends on the temperature)

• $\beta ={\frac {1}{k_{B}T}}$, with $k_{B}$ being the Boltzmann constant

• $U$ is the potential energy, which depends on the coordinates of the particles (and on the interaction model)

• $\left(R^{*}\right)^{3N}$ represent the 3N position coordinates of the particles (reduced with the system size): i.e. $\int d(R^{*})^{3N}=1$

Free energy and Partition Function

The Helmholtz energy function is related to the canonical partition function via:

$A\left(N,V,T\right)=-k_{B}T\log Q_{NVT}$