Grand canonical ensemble

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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.

Ensemble variables

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