Grand canonical ensemble

Ensemble variables

• Chemical Potential, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\mu \right.}

• Volume, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.V\right.}

• Temperature, $\left.T\right.$

Partition Function

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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:

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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

(THis subsection should be checked)

The Corresponding thermodynamic potentail for the Grand Canonical Partition function is:

$\left.A-\mu N\right.$, i.e.:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. p V = k_B T \log Q_{\mu V T } \right. }