Grand canonical ensemble

From SklogWiki
Jump to navigation Jump to search

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[edit]

Grand canonical partition function[edit]

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[edit]

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


See also[edit]


Related reading