Canonical ensemble - Revision history

Carl McBride: /* Partition Function */

2011-08-31T11:16:16Z

Partition Function

← Older revision		Revision as of 13:16, 31 August 2011
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	== Partition Function ==		== Partition Function ==
			The [[partition function]], <math>Q</math>,
			for a system of <math>N</math> identical particles each of mass <math>m</math> is given by

	~~The ''classical'~~' [[~~partition function~~]] ~~for~~ a one-component system in a three-dimensional space, <math> Q_{NVT} </math>,		:<math>Q_{NVT}=\frac{1}{N!h^{3N}}\iint d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]</math>

			where <math>h</math> is [[Planck constant \|Planck's constant]], <math>T</math> is the [[temperature]], <math>k_B</math> is the [[Boltzmann constant]] and <math>H(p^N, r^N)</math> is the [[Hamiltonian]]
			corresponding to the total energy of the system.
			For a classical one-component system in a three-dimensional space, <math> Q_{NVT} </math>,
	is given by:		is given by:

Carl McBride: Added a see also section

2011-08-31T11:08:11Z

Added a see also section

← Older revision		Revision as of 13:08, 31 August 2011
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	* <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>		* <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>

			==See also==
			*[[Ideal gas partition function]]
	==References==		==References==
	<references/>		<references/>

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

Carl McBride: /* Partition Function */ Added classical criteria

2011-08-31T10:49:48Z

Partition Function: Added classical criteria

← Older revision		Revision as of 12:49, 31 August 2011
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	is given by:		is given by:

	:<math> Q_{NVT} = \frac{V^N}{N! \Lambda^{3N} } \int d (R^)^{3N} \exp \left[ - \beta U \left( V, (R^)^{3N} \right) \right] </math>		:<math> Q_{NVT} = \frac{V^N}{N! \Lambda^{3N} } \int d (R^)^{3N} \exp \left[ - \beta U \left( V, (R^)^{3N} \right) \right] ~~~~~~~~~~ \left( \frac{V}{N\Lambda^3} \gg 1 \right) </math>

	where:		where:
Line 24:		Line 24:
	* <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>		* <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>

			==References==
			<references/>

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

Carl McBride: Slight tidy.

2008-02-18T12:02:34Z

Slight tidy.

← Older revision		Revision as of 14:02, 18 February 2008
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	* Volume, <math> V </math>		* Volume, <math> V </math>

	* Temperature, <math> T </math>		* [[Temperature]], <math> T </math>

	== Partition Function ==		== Partition Function ==
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	* <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (depends on the temperature)		* <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (depends on the temperature)

	* <math> \beta = \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]], and ''T'' the [[temperature]].		* <math> \beta := \frac{1}{k_B T} </math>, with <math> k_B </math> being the [[Boltzmann constant]], and ''T'' the [[temperature]].

	* <math> U </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model)		* <math> U </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model)

Carl McBride at 09:48, 25 June 2007

2007-06-25T09:48:32Z

← Older revision		Revision as of 11:48, 25 June 2007
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	== Partition Function ==		== Partition Function ==

	''~~Classical~~'' ~~Partition Function (~~one-component system) in a three-dimensional space: <math> Q_{NVT} </math>		The ''classical'' [[partition function]] for a one-component system in a three-dimensional space, <math> Q_{NVT} </math>,
			is given by:

	:<math> Q_{NVT} = \frac{V^N}{N! \Lambda^{3N} } \int d (R^)^{3N} \exp \left[ - \beta U \left( V, (R^)^{3N} \right) \right] </math>		:<math> Q_{NVT} = \frac{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> \Lambda </math> is the [[de Broglie thermal wavelength]] (depends on the temperature)		* <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (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]], and ''T'' the [[temperature]].

	* <math> U </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model)		* <math> U </math> is the potential energy, which depends on the coordinates of the particles (and on the interaction model)

Carl McBride at 15:21, 8 June 2007

2007-06-08T15:21:00Z

← Older revision		Revision as of 17:21, 8 June 2007
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	* <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>		* <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>

	~~== Helmholtz energy and partition function ==~~

	~~The [[Helmholtz energy function]] is related to the canonical [[partition function]] via:~~

	~~:<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math>~~
	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

Carl McBride: /* Free energy and Partition Function */

2007-03-05T15:25:52Z

Free energy and Partition Function

← Older revision		Revision as of 17:25, 5 March 2007
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	* <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>		* <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>

	== ~~Free~~ energy and ~~Partition Function~~ ==		== Helmholtz energy and partition function ==

	The [[Helmholtz energy function]] is related to the canonical partition function via:		The [[Helmholtz energy function]] is related to the canonical [[partition function]] via:

	:<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math>		:<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math>
	[[Category:Statistical mechanics]]		[[Category:Statistical mechanics]]

Nice and Tidy at 11:44, 27 February 2007

2007-02-27T11:44:33Z

← Older revision		Revision as of 13:44, 27 February 2007
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	where:		where:

	* <math> \Lambda </math> is the [[de Broglie wavelength]] (depends on the temperature)		* <math> \Lambda </math> is the [[de Broglie thermal wavelength]] (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]]
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	:<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math>		:<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math>
			[[Category:Statistical mechanics]]

Nice and Tidy at 17:18, 26 February 2007

2007-02-26T17:18:30Z

← Older revision		Revision as of 19:18, 26 February 2007
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	''Classical'' Partition Function (one-component system) in a three-dimensional space: <math> Q_{NVT} </math>		''Classical'' Partition Function (one-component system) in a three-dimensional space: <math> Q_{NVT} </math>

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

	where:		where:
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	The [[Helmholtz energy function]] is related to the canonical partition function via:		The [[Helmholtz energy function]] is related to the canonical partition function via:

	<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math>		:<math> A\left(N,V,T \right) = - k_B T \log Q_{NVT} </math>

Nice and Tidy at 16:44, 26 February 2007

2007-02-26T16:44:33Z

← Older revision		Revision as of 18:44, 26 February 2007
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	~~Canonical Ensemble:~~

	Variables:		Variables: