Grand canonical ensemble: Difference between revisions
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:<math> \Xi_{\mu VT} = \sum_{N=0}^{\infty} \exp \left[ \beta \mu N \right] Q_{NVT} </math> | :<math> \Xi_{\mu VT} = \sum_{N=0}^{\infty} \exp \left[ \beta \mu N \right] Q_{NVT} </math> | ||
where <math>Q_{NVT}</math> represents the [[Canonical ensemble#Partition Function | canonical ensemble partition function]]. | |||
For example, for a ''classical'' system one has | |||
:<math> \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] </math> | :<math> \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] </math> | ||
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* <math>N</math> is the number of particles | * <math>N</math> is the number of particles | ||
* <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (which depends on the temperature) | * <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (which depends on the temperature) | ||
* <math> \beta | * <math> \beta </math> is the [[inverse temperature]] | ||
* <math>U</math> is the potential energy, which depends on the coordinates of the particles (and on the [[models | interaction model]]) | * <math>U</math> is the potential energy, which depends on the coordinates of the particles (and on the [[models | interaction model]]) | ||
* <math> \left( R^*\right)^{3N} </math> represent the <math>3N</math> position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math> | * <math> \left( R^*\right)^{3N} </math> represent the <math>3N</math> position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math> | ||
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:<math> \left. p V = k_B T \ln \Xi_{\mu V T } \right. </math> | :<math> \left. p V = k_B T \ln \Xi_{\mu V T } \right. </math> | ||
==See also== | ==See also== | ||
*[[Monte Carlo | *[[Grand canonical Monte Carlo]] | ||
*[[Mass-stat]] | |||
==References== | ==References== | ||
<references/> | <references/> | ||
Latest revision as of 17:09, 1 April 2014
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.
![{\displaystyle \exp \left[\beta \mu \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee894e58e3416f71930628e1b50c7ac1aa832e3)
Ensemble variables[edit]
- chemical potential,
- volume,
- temperature,
Grand canonical partition function[edit]
The grand canonical partition function for a one-component system in a three-dimensional space is given by:
![{\displaystyle \Xi _{\mu VT}=\sum _{N=0}^{\infty }\exp \left[\beta \mu N\right]Q_{NVT}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef904e42cf93b2f11bcec2757385884620f8c7a4)
where represents the canonical ensemble partition function. For example, for a classical system one has
![{\displaystyle \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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2918a47c2ede5c1cc4f906029ca17adbff6daac9)
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[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
i.e.:
See also[edit]
References[edit]
- Related reading