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The '''grand-canonical ensemble''' | 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 == | == Grand canonical partition function == | ||
The | The classical grand canonical partition function for a one-component system in a three-dimensional space is given by: | ||
:<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: | ||
* | * ''N'' 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> | * <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]] | ||
* | * ''U'' 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> | ||
== 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>, | ||
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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 - | :<math>\left.U\right.=TS -PV + \mu N</math> | ||
one arrives at | one arrives at | ||
: <math> \left. \Omega \right.= - | : <math> \left. \Omega \right.= -PV</math> | ||
i.e.: | i.e.: | ||
:<math> \left. p V = k_B T \ | :<math> \left. p V = k_B T \log Q_{\mu V T } \right. </math> | ||
==See also== | ==See also== | ||
*[[ | *[[Monte Carlo in the grand-canonical ensemble]] | ||
==References== | |||
[[Category:Statistical mechanics]] | [[Category:Statistical mechanics]] |