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# Grand canonical ensemble

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

## Contents

## Ensemble variables

- chemical potential,
- volume,
- temperature,

## Grand canonical partition function

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

where represents the canonical ensemble partition function.
For example, for a *classical* system one has

where:

- is the number of particles
- is the de Broglie thermal wavelength (which depends on the temperature)
- is the inverse temperature
- is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- represent the position coordinates of the particles (reduced with the system size): i.e.

## Helmholtz energy and partition function

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

- ,

where *A* is the Helmholtz energy function.
Using the relation

one arrives at

i.e.:

## See also

## References

- Related reading