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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]].
The '''grand-canonical ensemble''' is particularly well suited to simulation studies of adsorption.  
The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.  
== Ensemble variables ==
== Ensemble variables ==
* [[chemical potential]], <math> \left. \mu \right.  </math>
* volume, <math> \left. V  \right. </math>
* [[temperature]], <math> \left. T \right. </math>
== Grand canonical partition function ==
The  grand canonical partition function for a one-component system in a three-dimensional space is given by:


:<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]].
* [[Chemical potential]], <math> \left. \mu \right.  </math>
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>
* Volume, <math> \left. V  \right. </math>
 
* Temperature, <math> \left. T \right. </math>
 
== Partition Function ==
 
''Classical'' partition function (one-component system) in a three-dimensional space: <math> Q_{\mu VT} </math>
 
:<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>


where:
where:


* <math>N</math> is the number of particles
*<math> \left. N \right. </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> 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> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]]
 
* <math> \left. U \right. </math> is the potential energy, which depends on the coordinates of the particles (and on the 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>


== Helmholtz energy and partition function ==
== Helmholtz energy and partition function ==
The  corresponding thermodynamic potential, the '''grand potential''', <math>\Omega</math>,
The  corresponding thermodynamic potential, the '''grand potential''', <math>\Omega</math>,
for the aforementioned grand canonical partition function is:
for the [[Grand canonical partition function | grand canonical partition function]] is:


: <math> \Omega = \left. A - \mu N \right. </math>,  
: <math> \Omega = \left. A - \mu N \right. </math>,  
where ''A'' is the [[Helmholtz energy function]].
where ''A'' is the [[Helmholtz energy function]].
Using the relation  
Using the relation  
:<math>\left.U\right.=TS -pV + \mu N</math>
:<math>\left.U\right.=TS -PV + \mu N</math>
one arrives at  
one arrives at  
: <math> \left. \Omega \right.= -pV</math>
: <math> \left. \Omega \right.= -PV</math>
i.e.:
i.e.:


:<math> \left. p V = k_B T \ln \Xi_{\mu V T } \right. </math>
:<math> \left. p V = k_B T \log Q_{\mu V T } \right. </math>
==See also==
 
*[[Grand canonical Monte Carlo]]
*[[Mass-stat]]


==References==
<references/>
;Related reading
*[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)]
[[Category:Statistical mechanics]]
[[Category:Statistical mechanics]]
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