Grand canonical ensemble

The grand-canonical ensemble is for "open" systems, where the number of particles, $N$ , can change. It can be viewed as an ensemble of canonical ensembles; there being a canonical ensemble for each value of $N$ , and the (weighted) sum over $N$ of these canonical ensembles constitutes the grand canonical ensemble. The weighting factor is $\exp \left[\beta \mu \right]$ and is known as the fugacity. The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.

Ensemble variables

chemical potential, $\left.\mu \right.$
volume, $\left.V\right.$
temperature, $\left.T\right.$

Grand canonical partition function

The grand canonical partition function for a one-component system in a three-dimensional space is given by:

\Xi _{\mu VT}=\sum _{N=0}^{\infty }\exp \left[\beta \mu N\right]Q_{NVT}

i.e. for a classical system one has

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

where:

$N$ is the number of particles
$\left.\Lambda \right.$ is the de Broglie thermal wavelength (which depends on the temperature)
$\beta ={\frac {1}{k_{B}T}}$ , with $k_{B}$ being the Boltzmann constant
$U$ 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. $\int d(R^{*})^{3N}=1$