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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

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]


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


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

See also


Related reading