Grand canonical ensemble: Difference between revisions
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The '''grand-canonical ensemble''' is particularly well suited to simulation studies of adsorption. | 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. | |||
== 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 grand canonical partition function for a one-component system in a three-dimensional space is given by: | ||
:<math> | :<math> \Xi_{\mu VT} = \sum_{N=0}^{\infty} \exp \left[ \beta \mu N \right] Q_{NVT} </math> | ||
i.e. 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> | |||
where: | where: | ||
* | * <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 = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]] | * <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> \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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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 \ln \Xi_{\mu V T } \right. </math> | ||
==See also== | ==See also== | ||
*[[Monte Carlo in the grand-canonical ensemble]] | *[[Monte Carlo in the grand-canonical ensemble]] | ||
==References== | ==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]] | ||
Revision as of 11:30, 31 August 2011
The grand-canonical ensemble is for "open" systems, where the number of particles, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} , can change. It can be viewed as an ensemble of canonical ensembles; there being a canonical ensemble for each value of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} , and the (weighted) sum over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} of these canonical ensembles constitutes the grand canonical ensemble. The weighting factor is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \exp \left[\beta \mu \right]} and is known as the fugacity. The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.
Ensemble variables
- chemical potential, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \mu \right. }
- volume, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. V \right. }
- temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. T \right. }
Grand canonical partition function
The grand canonical partition function for a one-component system in a three-dimensional space is given by:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Xi_{\mu VT} = \sum_{N=0}^{\infty} \exp \left[ \beta \mu N \right] Q_{NVT} }
i.e. for a classical system one has
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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]}
where:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the number of particles
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \Lambda \right. } is the de Broglie thermal wavelength (which depends on the temperature)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta = \frac{1}{k_B T} } , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B } being the Boltzmann constant
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( R^*\right)^{3N} } represent the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3N} position coordinates of the particles (reduced with the system size): i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int d (R^*)^{3N} = 1 }
Helmholtz energy and partition function
The corresponding thermodynamic potential, the grand potential, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} , for the aforementioned grand canonical partition function is:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = \left. A - \mu N \right. } ,
where A is the Helmholtz energy function. Using the relation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.U\right.=TS -pV + \mu N}
one arrives at
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\Omega \right.=-pV}
i.e.:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. p V = k_B T \ln \Xi_{\mu V T } \right. }
See also
References
- Related reading