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

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\Lambda \right.} is the de Broglie thermal wavelength (which depends on the temperature)

• $\beta ={\frac {1}{k_{B}T}}$, 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 (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 \int d (R^*)^{3N} = 1 }

Helmholtz energy and partition function

The corresponding thermodynamic potential for the grand canonical partition function is:

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. 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. }

where A is the Helmholtz energy function.