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| The '''grand-canonical ensemble''' is for "open" systems, where the number of particles, <math>N</math>, can change. It can be viewed as an ensemble of [[canonical ensemble]]s; there being a canonical ensemble for each value of <math>N</math>, and the (weighted) sum over <math>N</math> of these canonical ensembles constitutes the grand canonical ensemble. The weighting factor is <math> \exp \left[ \beta \mu \right]</math> and is known as the [[fugacity]].
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| The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.
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| == Ensemble variables == | | == Ensemble variables == |
| * [[chemical potential]], <math> \left. \mu \right. </math>
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| * volume, <math> \left. V \right. </math>
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| * [[temperature]], <math> \left. T \right. </math>
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| == Grand canonical partition function ==
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| The grand canonical partition function for a one-component system in a three-dimensional space is given by:
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| :<math> \Xi_{\mu VT} = \sum_{N=0}^{\infty} \exp \left[ \beta \mu N \right] Q_{NVT} </math>
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| where <math>Q_{NVT}</math> represents the [[Canonical ensemble#Partition Function | canonical ensemble partition function]].
| | * Chemical Potential, <math> \left. \mu \right. </math> |
| For example, for a ''classical'' system one has
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| :<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> | | * Volume, <math> \left. V \right. </math> |
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| | * Temperature, <math> \left. T \right. </math> |
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| | == Partition Function == |
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| | ''Classical'' Partition Function (one-component system) in a three-dimensional space: <math> Q_{\mu VT} </math> |
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| | :<math> 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] </math> |
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| where: | | where: |
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| * <math>N</math> is the number of particles | | * <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (depends on the temperature) |
| * <math> \left. \Lambda \right. </math> is the [[de Broglie thermal wavelength]] (which depends on the temperature)
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| * <math> \beta </math> is the [[inverse temperature]] | | * <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]] |
| * <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> U </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model) |
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| == Helmholtz energy and partition function ==
| | * <math> \left( R^*\right)^{3N} </math> represent the 3N position coordinates of the particles (reduced with the system size): i.e. <math> \int d (R^*)^{3N} = 1 </math> |
| The corresponding thermodynamic potential, the '''grand potential''', <math>\Omega</math>,
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| for the aforementioned grand canonical partition function is:
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| : <math> \Omega = \left. A - \mu N \right. </math>,
| | == Free energy and Partition Function == |
| where ''A'' is the [[Helmholtz energy function]].
| | == Free energy and Partition Function == |
| Using the relation
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| :<math>\left.U\right.=TS -pV + \mu N</math>
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| one arrives at
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| : <math> \left. \Omega \right.= -pV</math>
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| i.e.:
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| :<math> \left. p V = k_B T \ln \Xi_{\mu V T } \right. </math>
| | The [[Helmholtz energy function]] is related to the canonical partition function via: |
| ==See also==
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| *[[Grand canonical Monte Carlo]]
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| *[[Mass-stat]]
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| ==References== | | :<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math> |
| <references/> | |
| ;Related reading
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| *[http://dx.doi.org/10.1103/PhysRev.57.1160 Richard C. Tolman "On the Establishment of Grand Canonical Distributions", Physical Review '''57''' pp. 1160-1168 (1940)]
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| [[Category:Statistical mechanics]] | | [[Category:Statistical mechanics]] |