Grand canonical ensemble

Ensemble variables

• Chemical Potential, $\mu$

• Volume, $V$

• Temperature, $T$

Partition Function

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

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

where:

• $\mathrm{\Lambda }$ 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)

• ${(R^{*})}^{3N}$ 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:

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