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# Grand canonical ensemble

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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:

$\Xi_{\mu VT} = \sum_{N=0}^{\infty} \exp \left[ \beta \mu N \right] Q_{NVT}$

where $Q_{NVT}$ represents the canonical ensemble partition function. For example, 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)
• $\beta$ is the inverse temperature
• $U$ 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. $\int d (R^*)^{3N} = 1$

## Helmholtz energy and partition function

The corresponding thermodynamic potential, the grand potential, $\Omega$, for the aforementioned grand canonical partition function is:

$\Omega = \left. A - \mu N \right.$,

where A is the Helmholtz energy function. Using the relation

$\left.U\right.=TS -pV + \mu N$

one arrives at

$\left. \Omega \right.= -pV$

i.e.:

$\left. p V = k_B T \ln \Xi_{\mu V T } \right.$