Grand canonical ensemble
The grand-canonical ensemble is for "open" systems, where the number of particles, \(N\), can change. It can be viewed as an ensemble of canonical ensembles; there being a canonical ensemble for each value of \(N\), and the (weighted) sum over \(N\) of these canonical ensembles constitutes the grand canonical ensemble. The weighting factor is \( \exp \left[ \beta \mu \right]\) and is known as the fugacity. The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.
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[edit] Ensemble variables
- chemical potential, \( \left. \mu \right. \)
- volume, \( \left. V \right. \)
- temperature, \( \left. T \right. \)
[edit] Grand canonical partition function
The grand canonical partition function for a one-component system in a three-dimensional space is given by:
\[ \Xi_{\mu VT} = \sum_{N=0}^{\infty} \exp \left[ \beta \mu N \right] Q_{NVT} \]
where \(Q_{NVT}\) represents the canonical ensemble partition function. For example, for a classical system one has
\[ \Xi_{\mu VT} = \sum_{N=0}^{\infty} \exp \left[ \beta \mu N \right] \frac{ V^N}{N! \Lambda^{3N} } \int d (R^*)^{3N} \exp \left[ - \beta U \left( V, (R^*)^{3N} \right) \right] \]
where:
- \(N\) is the number of particles
- \( \left. \Lambda \right. \) is the de Broglie thermal wavelength (which depends on the temperature)
- \( \beta \) is the inverse temperature
- \(U\) is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- \( \left( R^*\right)^{3N} \) represent the \(3N\) position coordinates of the particles (reduced with the system size): i.e. \( \int d (R^*)^{3N} = 1 \)
[edit] Helmholtz energy and partition function
The corresponding thermodynamic potential, the grand potential, \(\Omega\), for the aforementioned grand canonical partition function is:
- \( \Omega = \left. A - \mu N \right. \),
where A is the Helmholtz energy function. Using the relation \[\left.U\right.=TS -pV + \mu N\] one arrives at
- \( \left. \Omega \right.= -pV\)
i.e.:
\[ \left. p V = k_B T \ln \Xi_{\mu V T } \right. \]
[edit] See also
[edit] References
- Related reading
