Grand canonical ensemble
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The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.
Contents |
[edit] Ensemble variables
- Chemical potential,
- Volume,
- Temperature,
[edit] Grand canonical partition function
The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:
![Q_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right] V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right]](/SklogWiki/images/math/0/a/f/0afbcd39208ab221c143e9579ab71d51.png)
where:
- N is the number of particles
-
is the de Broglie thermal wavelength (which depends on the temperature)
-
, with kB being the Boltzmann constant
- U is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
-
represent the 3N position coordinates of the particles (reduced with the system size): i.e. 
[edit] Helmholtz energy and partition function
The corresponding thermodynamic potential, the grand potential, Ω, for the aforementioned grand canonical partition function is:
-
,
where A is the Helmholtz energy function. Using the relation
one arrives at
i.e.:
