Widom test-particle method - Revision history

Carl McBride: Added a recent publication

2016-12-12T15:44:01Z

Added a recent publication

← Older revision		Revision as of 17:44, 12 December 2016
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	{{stub-general}}		{{stub-general}}
	[[Benjamin Widom]] ~~proposed~~ an elegant, general [[Computer simulation techniques \|simulation technique]] to obtain the excess [[chemical potential]] of a system. A so-called ''test particle'' is introduced in a [[Random numbers \|random]] location, and <math>\Delta\Phi</math>, the difference in [[internal energy]] before and after the insertion, is computed. For [[Intermolecular pair potential \|pairwise interactions]], this would become be the interaction potential energy between the randomly placed test particle and the <math>N</math> particles that the system is comprised of. The particle is not actually inserted, at variance with [[Monte Carlo in the grand-canonical ensemble\|grand canonical Monte Carlo]].		The '''Widom test-particle method''', proposed by [[Benjamin Widom]] <ref>[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]</ref><ref>[http://dx.doi.org/10.1021/j100395a005 B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry '''86''' pp. 869-872 (1982)]</ref>, is an elegant, general [[Computer simulation techniques \|simulation technique]] to obtain the excess [[chemical potential]] of a system. A so-called ''test particle'' is introduced in a [[Random numbers \|random]] location, and <math>\Delta\Phi</math>, the difference in [[internal energy]] before and after the insertion, is computed. For [[Intermolecular pair potential \|pairwise interactions]], this would become be the interaction potential energy between the randomly placed test particle and the <math>N</math> particles that the system is comprised of. The particle is not actually inserted, at variance with [[Monte Carlo in the grand-canonical ensemble\|grand canonical Monte Carlo]].

	The excess chemical potential is given by		The excess chemical potential is given by
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	where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]].		where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]].
	==References==		==References==
	#[http://dx.doi.org/10.~~1063~~/~~1.1734110 B~~. Widom ~~"Some Topics in~~ the ~~Theory of Fluids~~", ~~Journal of Chemical~~ Physics '''39''' pp. ~~2808~~-~~2812~~ (~~1963~~)]		<references/>
	#[http://dx.doi.org/10.~~1021~~/~~j100395a005 B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry '''86''' pp~~. ~~869 - 872 (1982)]~~		;Related reading
	~~#[http://dx.doi.org/10.1080/002689798169104~~ David S. ~~Corti~~ "~~Alternative derivation~~ of ~~Widom's~~ test ~~particle insertion method using the small system grand canonical ensemble~~", ~~Molecular~~ Physics '''93''' ~~pp. 417-420~~ (~~1998~~)]		*[http://dx.doi.org/10.1080/002689798169104 David S. Corti "Alternative derivation of Widom's test particle insertion method using the small system grand canonical ensemble", Molecular Physics '''93''' pp. 417-420 (1998)]
			*[http://dx.doi.org/10.1063/1.4968039 David M. Heyes and Andrés Santos "Chemical potential of a test hard sphere of variable size in a hard-sphere fluid", Journal of Chemical Physics '''145''' 214504 (2016)]

	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Carl McBride: Slight tidy.

2008-02-18T14:11:50Z

Slight tidy.

← Older revision		Revision as of 16:11, 18 February 2008
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	{{stub-general}}		{{stub-general}}
	[[Benjamin Widom]] proposed an elegant, general [[Computer simulation techniques \|simulation technique]] to obtain		[[Benjamin Widom]] proposed an elegant, general [[Computer simulation techniques \|simulation technique]] to obtain the excess [[chemical potential]] of a system. A so-called ''test particle'' is introduced in a [[Random numbers \|random]] location, and <math>\Delta\Phi</math>, the difference in [[internal energy]] before and after the insertion, is computed. For [[Intermolecular pair potential \|pairwise interactions]], this would become be the interaction potential energy between the randomly placed test particle and the <math>N</math> particles that the system is comprised of. The particle is not actually inserted, at variance with [[Monte Carlo in the grand-canonical ensemble\|grand canonical Monte Carlo]].
	the excess [[chemical potential]] of a system. A so-called ''test particle'' is introduced in a [[Random numbers \|random]]
	location, and <math>\Delta\Phi</math>, the difference
	in [[internal energy]] before and after the insertion,
	is computed. For [[Intermolecular pair potential \|pairwise interactions]], this would
	become be the interaction potential energy between the randomly
	placed test particle and the <math>N</math> particles that the system is comprised of.
	The particle is not actually inserted, at variance with
	[[Monte Carlo in the grand-canonical ensemble\|grand canonical Monte Carlo]].

	The excess chemical potential is given by		The excess chemical potential is given by

Nice and Tidy at 10:04, 30 January 2008

2008-01-30T10:04:36Z

← Older revision		Revision as of 12:04, 30 January 2008
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	The excess chemical potential is given by		The excess chemical potential is given by

	:<math>\mu^{ex} = -k_BT \log \langle e^{-\Delta\Phi/~~k_bT~~}\rangle_N ,</math>		:<math>\mu^{excess} = \mu -\mu^{ideal} = -k_BT \log \langle e^{-\Delta\Phi/k_BT}\rangle_N ,</math>

	where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]].		where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]].

Nice and Tidy at 09:33, 30 January 2008

2008-01-30T09:33:46Z

← Older revision		Revision as of 11:33, 30 January 2008
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	{{stub-general}}		{{stub-general}}
	[[Benjamin Widom]] proposed an elegant, general ~~method~~ to obtain		[[Benjamin Widom]] proposed an elegant, general [[Computer simulation techniques \|simulation technique]] to obtain
	the excess [[chemical potential]] of a system ~~that is being~~		the excess [[chemical potential]] of a system. A so-called ''test particle'' is introduced in a [[Random numbers \|random]]
	~~simulated~~. A ''test particle'' is introduced in a random
	location, and <math>\Delta\Phi</math>, the difference		location, and <math>\Delta\Phi</math>, the difference
	in [[internal energy]] before and after the insertion,		in [[internal energy]] before and after the insertion,
	is computed. (For pairwise interactions, this would		is computed. For [[Intermolecular pair potential \|pairwise interactions]], this would
	~~just~~ be the interaction potential energy between the randomly		become be the interaction potential energy between the randomly
	placed test particle and the ''N'' particles that the system is comprised of.)		placed test particle and the <math>N</math> particles that the system is comprised of.
	The particle is not actually inserted (at variance with [[Monte Carlo in the grand-canonical ensemble\|grand canonical MC ]]).		The particle is not actually inserted, at variance with
			[[Monte Carlo in the grand-canonical ensemble\|grand canonical Monte Carlo]].

	The excess chemical potential is given by		The excess chemical potential is given by
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	:<math>\mu^{ex} = -k_BT \log \langle e^{-\Delta\Phi/k_bT}\rangle_N ,</math>		:<math>\mu^{ex} = -k_BT \log \langle e^{-\Delta\Phi/k_bT}\rangle_N ,</math>

	where <math>k_B</math> is the [[Boltzmann constant]] and ''T'' is the [[temperature]].		where <math>k_B</math> is the [[Boltzmann constant]] and <math>T</math> is the [[temperature]].
	==References==		==References==
	#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]		#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]

Dduque at 21:04, 29 January 2008

2008-01-29T21:04:46Z

← Older revision		Revision as of 23:04, 29 January 2008
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	{{stub-general}}		{{stub-general}}
	~~The~~ excess [[chemical potential]] is ~~given by~~		[[Benjamin Widom]] proposed an elegant, general method to obtain
			the excess [[chemical potential]] of a system that is being
			simulated. A ''test particle'' is introduced in a random
			location, and <math>\Delta\Phi</math>, the difference
			in [[internal energy]] before and after the insertion,
			is computed. (For pairwise interactions, this would
			just be the interaction potential energy between the randomly
			placed test particle and the ''N'' particles that the system is comprised of.)
			The particle is not actually inserted (at variance with [[Monte Carlo in the grand-canonical ensemble\|grand canonical MC ]]).

	~~:<math>\mu^{ex} = -k_BT \ln \langle e^{-\Phi/k_bT}\rangle_N</math>~~		The excess chemical potential is given by

	where <math>k_B</math> is the [[Boltzmann constant]], ''T'' is the [[temperature]], ''N'' is the number of particles within the system and <math>\Phi</math> is the interaction potential energy between the randomly placed test particle and the ''N'' particles that the system is comprised of.		:<math>\mu^{ex} = -k_BT \log \langle e^{-\Delta\Phi/k_bT}\rangle_N ,</math>

			where <math>k_B</math> is the [[Boltzmann constant]] and ''T'' is the [[temperature]].
	==References==		==References==
	#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]		#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]

Carl McBride at 16:43, 29 January 2008

2008-01-29T16:43:08Z

← Older revision		Revision as of 18:43, 29 January 2008
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			{{stub-general}}
	The excess [[chemical potential]] is given by		The excess [[chemical potential]] is given by

Carl McBride at 10:24, 26 September 2007

2007-09-26T10:24:24Z

← Older revision		Revision as of 12:24, 26 September 2007
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	:<math>\mu^{ex} = -k_BT \ln \langle e^{-\Phi/k_bT}\rangle_N</math>		:<math>\mu^{ex} = -k_BT \ln \langle e^{-\Phi/k_bT}\rangle_N</math>

	where <math>k_B</math> is the [[Boltzmann constant]], ''T'' is the [[temperature]], ''N'' is the number of particles within the system and <math>\Phi</math> is the ~~[[intermolecular pair~~ potential]].		where <math>k_B</math> is the [[Boltzmann constant]], ''T'' is the [[temperature]], ''N'' is the number of particles within the system and <math>\Phi</math> is the interaction potential energy between the randomly placed test particle and the ''N'' particles that the system is comprised of.
	==References==		==References==
	#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]		#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]

Carl McBride at 17:01, 22 September 2007

2007-09-22T17:01:15Z

← Older revision		Revision as of 19:01, 22 September 2007
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	#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]		#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]
	#[http://dx.doi.org/10.1021/j100395a005 B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry '''86''' pp. 869 - 872 (1982)]		#[http://dx.doi.org/10.1021/j100395a005 B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry '''86''' pp. 869 - 872 (1982)]
	#[http://dx.doi.org/10.1080/002689798169104 David S. ~~corti~~ "Alternative derivation of Widom's test particle insertion method using the small system grand canonical ensemble", Molecular Physics '''93''' pp. 417-420 (1998)]		#[http://dx.doi.org/10.1080/002689798169104 David S. Corti "Alternative derivation of Widom's test particle insertion method using the small system grand canonical ensemble", Molecular Physics '''93''' pp. 417-420 (1998)]
	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Carl McBride: New page: The excess chemical potential is given by : $\mu^{ex} = -k_BT \ln \langle e^{-\Phi/k_bT}\rangle_N$ where $k_B$ is the Boltzmann constant, ''T'' is the [[temp...

2007-09-22T16:57:42Z

New page: The excess chemical potential is given by :<math>\mu^{ex} = -k_BT \ln \langle e^{-\Phi/k_bT}\rangle_N</math> where <math>k_B</math> is the Boltzmann constant, ''T'' is the [[temp...

New page

The excess [[chemical potential]] is given by

:<math>\mu^{ex} = -k_BT \ln \langle e^{-\Phi/k_bT}\rangle_N</math>

where <math>k_B</math> is the [[Boltzmann constant]], ''T'' is the [[temperature]], ''N'' is the number of particles within the system and <math>\Phi</math> is the [[intermolecular pair potential]].
==References==
#[http://dx.doi.org/10.1063/1.1734110 B. Widom "Some Topics in the Theory of Fluids", Journal of Chemical Physics '''39''' pp. 2808-2812 (1963)]
#[http://dx.doi.org/10.1021/j100395a005 B. Widom "Potential-distribution theory and the statistical mechanics of fluids", Journal of Physical Chemistry '''86''' pp. 869 - 872 (1982)]
#[http://dx.doi.org/10.1080/002689798169104 David S. corti "Alternative derivation of Widom's test particle insertion method using the small system grand canonical ensemble", Molecular Physics '''93''' pp. 417-420 (1998)]
[[category: computer simulation techniques]]