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.
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
- 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
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