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:

• $\left.N\right.$ is the number of particles

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

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \beta ={\frac {1}{k_{B}T}}} , with $k_{B}$ being the Boltzmann constant

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

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left(R^{*}\right)^{3N}} represent the 3N position coordinates of the particles (reduced with the system size): i.e. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int d(R^{*})^{3N}=1}

Free energy and Partition Function

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

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