Partition function - Revision history

Vql at 17:34, 16 January 2008

2008-01-16T17:34:21Z

← Older revision		Revision as of 19:34, 16 January 2008
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	where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math>		where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math>
	the [[Boltzmann constant]].		the [[Boltzmann constant]].

			In classical statistical mechanics, there is a close connection
			between the partition function and the
			[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configuration integral],
			which has played an important role in many applications
			(e.g., drug design).

	==Helmholtz energy function==		==Helmholtz energy function==

Carl McBride at 14:32, 24 September 2007

2007-09-24T14:32:49Z

← Older revision		Revision as of 16:32, 24 September 2007
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	The ~~[[canonical ensemble]]~~ '''partition function''' of a system ~~in contact with a thermal bath~~		The '''partition function''' of a system is given by
	~~at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]~~
	~~function, and therefore its expression~~ is given by

	:<math> \left. Z \right.= {\mathrm {Tr}} \{ e^{-\beta H} \}</math>		:<math> \left. Z \right.= {\mathrm {Tr}} \{ e^{-\beta H} \}</math>

	where ''H'' is the [[Hamiltonian]], ~~or as~~		where ''H'' is the [[Hamiltonian]]. The symbol ''Z'' is from the German ''Zustandssumme'' meaning "sum over states". The [[canonical ensemble]] partition function of a system in contact with a thermal bath
			at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]
			function, and therefore its expression is given by

	:<math>Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE</math>,		:<math>Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE</math>,

	where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math>		where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math>
	the [[Boltzmann constant]]~~. The symbol ''Z'' is from the German ''Zustandssumme'' meaning "sum over states"~~.		the [[Boltzmann constant]].

	==Helmholtz energy function==		==Helmholtz energy function==

Carl McBride at 14:21, 24 September 2007

2007-09-24T14:21:23Z

← Older revision		Revision as of 16:21, 24 September 2007
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	at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]		at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]
	function, and therefore its expression is given by		function, and therefore its expression is given by

			:<math> \left. Z \right.= {\mathrm {Tr}} \{ e^{-\beta H} \}</math>

			where ''H'' is the [[Hamiltonian]], or as

	:<math>Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE</math>,		:<math>Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE</math>,

Carl McBride at 15:24, 26 June 2007

2007-06-26T15:24:25Z

← Older revision		Revision as of 17:24, 26 June 2007
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	The '''partition function''' of a system in contact with a thermal bath		The [[canonical ensemble]] '''partition function''' of a system in contact with a thermal bath
	at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]		at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]
	function, and therefore its expression is given by		function, and therefore its expression is given by
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	where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math>		where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math>
	the [[Boltzmann constant]].		the [[Boltzmann constant]]. The symbol ''Z'' is from the German ''Zustandssumme'' meaning "sum over states".

			==Helmholtz energy function==
	The partition function of a system is related to the [[Helmholtz energy function]] through the formula		The partition function of a system is related to the [[Helmholtz energy function]] through the formula

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	This connection can be derived from the fact that <math>k_B\log\Omega(E)</math> is the		This connection can be derived from the fact that <math>k_B\log\Omega(E)</math> is the
	[[entropy]] of a system with total energy <math>E</math>. This is an [[extensive magnitude]] in the		[[entropy]] of a system with total energy <math>E</math>. This is an [[Extensive properties \| extensive magnitude]] in the
	sense that, for large systems (i.e. in the [[thermodynamic limit]], when the number of		sense that, for large systems (i.e. in the [[thermodynamic limit]], when the number of
	particles <math>N\to\infty</math>		particles <math>N\to\infty</math>
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	These equations provide a link between [[Classical thermodynamics \| classical thermodynamics]] and		These equations provide a link between [[Classical thermodynamics \| classical thermodynamics]] and
	[[Statistical mechanics \| statistical mechanics]]		[[Statistical mechanics \| statistical mechanics]]
			==See also==
			*[[Ideal gas partition function]]
	[[category:classical thermodynamics]]		[[category:classical thermodynamics]]
	[[category:statistical mechanics]]		[[category:statistical mechanics]]

Carl McBride at 16:32, 7 June 2007

2007-06-07T16:32:55Z

← Older revision		Revision as of 18:32, 7 June 2007
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	:<math>\left.A\right.=-k_BT\log Z(T)</math>		:<math>\left.A\right.=-k_BT\log Z(T)</math>

	we also have the [[~~First law of thermodynamics \|~~ internal energy]], which is given by		we also have the [[internal energy]], which is given by

	:<math>U=k_B T^{2} \left. \frac{\partial \log Z(T)}{\partial T} \right\vert_{N,V}</math>		:<math>U=k_B T^{2} \left. \frac{\partial \log Z(T)}{\partial T} \right\vert_{N,V}</math>

Carl McBride at 16:18, 7 June 2007

2007-06-07T16:18:45Z

← Older revision		Revision as of 18:18, 7 June 2007
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	:<math>\left.A\right.=-k_BT\log Z(T)</math>		:<math>\left.A\right.=-k_BT\log Z(T)</math>

	we also have the [[internal energy]], which is given by		we also have the [[First law of thermodynamics \| internal energy]], which is given by

	:<math>U=k_B T^{2} \left. \frac{\partial \log Z(T)}{\partial T} \right\vert_{N,V}</math>		:<math>U=k_B T^{2} \left. \frac{\partial \log Z(T)}{\partial T} \right\vert_{N,V}</math>

Nice and Tidy: /* Connection with thermodynamics */

2007-05-24T14:40:23Z

Connection with thermodynamics

← Older revision		Revision as of 16:40, 24 May 2007
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	:<math>p=k_B T \left. \frac{\partial \log Z(T)}{\partial V} \right\vert_{N,T}</math>.		:<math>p=k_B T \left. \frac{\partial \log Z(T)}{\partial V} \right\vert_{N,T}</math>.

	These equations ~~provides~~ a link between [[Classical thermodynamics \| classical thermodynamics]] and		These equations provide a link between [[Classical thermodynamics \| classical thermodynamics]] and
	[[Statistical mechanics \| statistical mechanics]]		[[Statistical mechanics \| statistical mechanics]]

	[[category:classical thermodynamics]]		[[category:classical thermodynamics]]
	[[category:statistical mechanics]]		[[category:statistical mechanics]]

Carl McBride at 13:13, 24 May 2007

2007-05-24T13:13:57Z

← Older revision		Revision as of 15:13, 24 May 2007
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	the thermodynamic definition of the [[Helmholtz energy function]]. Thus, when <math>N</math> is large,		the thermodynamic definition of the [[Helmholtz energy function]]. Thus, when <math>N</math> is large,

	:<math>\left.A\right.=-k_BT\log Z.</math>		:<math>\left.A\right.=-k_BT\log Z(T).</math>
			==Connection with thermodynamics==
			We have the aforementioned [[Helmholtz energy function]],

	~~The internal energy is given by~~		:<math>\left.A\right.=-k_BT\log Z(T)</math>

	:<math>U=k_B T^{2} \frac{\partial \log Z(T)}{\partial T}</math>		we also have the [[internal energy]], which is given by

			:<math>U=k_B T^{2} \left. \frac{\partial \log Z(T)}{\partial T} \right\vert_{N,V}</math>

			and the pressure, which is given by

			:<math>p=k_B T \left. \frac{\partial \log Z(T)}{\partial V} \right\vert_{N,T}</math>.

	These equations provides a link between [[Classical thermodynamics \| classical thermodynamics]] and		These equations provides a link between [[Classical thermodynamics \| classical thermodynamics]] and
	[[Statistical mechanics \| statistical mechanics]]		[[Statistical mechanics \| statistical mechanics]]

			[[category:classical thermodynamics]]
			[[category:statistical mechanics]]

80.38.96.35 at 10:41, 24 May 2007

2007-05-24T10:41:33Z

← Older revision		Revision as of 12:41, 24 May 2007
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	The internal energy is given by		The internal energy is given by

	:<math>U=k_B T^{2} \frac{\partial Z(T)}{\partial T}</math>		:<math>U=k_B T^{2} \frac{\partial \log Z(T)}{\partial T}</math>

	These equations provides a link between [[Classical thermodynamics \| classical thermodynamics]] and		These equations provides a link between [[Classical thermodynamics \| classical thermodynamics]] and
	[[Statistical mechanics \| statistical mechanics]]		[[Statistical mechanics \| statistical mechanics]]

Carl McBride at 16:43, 21 May 2007

2007-05-21T16:43:54Z

← Older revision		Revision as of 18:43, 21 May 2007
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	:<math>U=k_B T^{2} \frac{\partial Z(T)}{\partial T}</math>		:<math>U=k_B T^{2} \frac{\partial Z(T)}{\partial T}</math>

	~~This equation~~ provides a link between [[Classical thermodynamics \| classical thermodynamics]] and		These equations provides a link between [[Classical thermodynamics \| classical thermodynamics]] and
	[[Statistical mechanics \| statistical mechanics]]		[[Statistical mechanics \| statistical mechanics]]