Grand canonical ensemble - Revision history

Carl McBride: /* See also */ Updated seation

2014-04-01T15:09:49Z

Carl McBride: /* Grand canonical partition function */ Added an internal link

2011-08-31T11:01:50Z

Grand canonical partition function: Added an internal link

← Older revision		Revision as of 13:01, 31 August 2011
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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>

	~~i.e~~. for a ''classical'' system one has		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>

Carl McBride: /* Grand canonical partition function */ added an internal link to inverse temperature

2011-08-31T10:53:42Z

Grand canonical partition function: added an internal link to inverse temperature

← Older revision		Revision as of 12:53, 31 August 2011
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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 ~~= \frac{1}{k_B T}~~ </math>~~, with <math> k_B </math> being~~ the [[~~Boltzmann constant~~]]		* <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>

Carl McBride: Slight tidy.

2011-08-31T10:30:00Z

Slight tidy.

← Older revision		Revision as of 12:30, 31 August 2011
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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>		* [[chemical potential]], <math> \left. \mu \right. </math>
	* ~~Volume~~, <math> \left. V \right. </math>		* volume, <math> \left. V \right. </math>
	* [[~~Temperature~~]], <math> \left. T \right. </math>		* [[temperature]], <math> \left. T \right. </math>
	== Grand canonical partition function ==		== Grand canonical partition function ==
	The ~~classical~~ grand canonical partition function for a one-component system in a three-dimensional space is given by:		The grand canonical partition function for a one-component system in a three-dimensional space is given by:

	:<math> Q_{\mu VT} = \sum_{N=0}^{\infty} \~~frac~~{ \exp \left[ \beta \mu N \right]}{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] 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:

	* ''N'' 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 = \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]]
	* ''U'' 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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	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 \~~log Q_~~{\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 in the grand-canonical ensemble]]		*[[Monte Carlo in the grand-canonical ensemble]]
	==References==		==References==
	#[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)]		<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]]

82.53.107.46: /* Grand canonical partition function */

2011-08-30T10:08:43Z

Grand canonical partition function

← Older revision		Revision as of 12:08, 30 August 2011
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	The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:		The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:

	:<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>		:<math> Q_{\mu VT} = \sum_{N=0}^{\infty} \frac{ \exp \left[ \beta \mu N \right]}{N! \Lambda^{3N} } \int d (R^)^{3N} \exp \left[ - \beta U \left( V, (R^)^{3N} \right) \right] </math>

	where:		where:
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	* ''U'' is the potential energy, which depends on the coordinates of the particles (and on the [[models \| interaction model]])		* ''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>,

Carl McBride: Added a reference

2008-07-30T13:02:03Z

Added a reference

← Older revision		Revision as of 15:02, 30 July 2008
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	*[[Monte Carlo in the grand-canonical ensemble]]		*[[Monte Carlo in the grand-canonical ensemble]]
	==References==		==References==
			#[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]]

Carl McBride: Added a 'see also' section.

2008-04-01T12:44:07Z

Added a 'see also' section.

← Older revision		Revision as of 14:44, 1 April 2008
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	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>		* [[Chemical potential]], <math> \left. \mu \right. </math>

	* Volume, <math> \left. V \right. </math>		* Volume, <math> \left. V \right. </math>
			* [[Temperature]], <math> \left. T \right. </math>
	* Temperature, <math> \left. T \right. </math>

	== Grand canonical partition function ==		== Grand canonical partition function ==
	The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:		The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:
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	* ''N'' is the number of particles		* ''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 = \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]]

	* ''U'' is the potential energy, which depends on the coordinates of the particles (and on the [[models \| interaction model]])		* ''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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	:<math> \left. p V = k_B T \log Q_{\mu V T } \right. </math>		:<math> \left. p V = k_B T \log Q_{\mu V T } \right. </math>
			==See also==
			*[[Monte Carlo in the grand-canonical ensemble]]
			==References==

	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

Carl McBride at 15:33, 26 June 2007

2007-06-26T15:33:17Z

← Older revision		Revision as of 17:33, 26 June 2007
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	* Temperature, <math> \left. T \right. </math>		* Temperature, <math> \left. T \right. </math>

	== ~~Partition Function~~ ==		== Grand canonical partition function ==
	The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:		The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:

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	where:		where:

	*~~<math> \left.~~ N ~~\right. </math>~~ is the number of particles		* ''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)

Carl McBride: /* Helmholtz energy and partition function */

2007-06-26T15:31:55Z

Helmholtz energy and partition function

← Older revision		Revision as of 17:31, 26 June 2007
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	== 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 ~~[[Grand canonical partition function \|~~ grand canonical partition function]] is:		for the aforementioned grand canonical partition function is:

	: <math> \Omega = \left. A - \mu N \right. </math>,		: <math> \Omega = \left. A - \mu N \right. </math>,

Carl McBride: /* Partition Function */

2007-06-26T15:30:57Z

Partition Function

← Older revision		Revision as of 17:30, 26 June 2007
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	== Partition Function ==		== Partition Function ==
			The classical grand canonical partition function for a one-component system in a three-dimensional space is given by:
	~~''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>		:<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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	* <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> \left.~~ U ~~\right. </math>~~ is the potential energy, which depends on the coordinates of the particles (and on the interaction model)		* ''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>

← Older revision		Revision as of 17:09, 1 April 2014
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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 ~~in the grand~~-~~canonical ensemble~~]]		*[[Grand canonical Monte Carlo]]
			*[[Mass-stat]]

	==References==		==References==
	<references/>		<references/>