Grand canonical Monte Carlo - Revision history

Lgmac: Corrected typo (L subindex inserted) ;)

2019-12-11T02:20:40Z

Corrected typo (L subindex inserted) ;)

← Older revision		Revision as of 04:20, 11 December 2019
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	:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P		:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P
	f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math>		f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math>

	where the <math>U</math> subindex stands for ''unlabelled'', the coordinates in parenthesis indicate that the positions are not attributed to any particular choice of labelling and the sum runs over all possible particle label permutations.		where the <math>U</math> subindex stands for ''unlabelled'', the coordinates in parenthesis indicate that the positions are not attributed to any particular choice of labelling and the sum runs over all possible particle label permutations.

Carl McBride: Added a See also section

2014-04-01T14:19:51Z

Added a See also section

← Older revision		Revision as of 16:19, 1 April 2014
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	The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.		The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.
			==See also==
			*[[Mass-stat]]
	== References ==		== References ==
	<references/>		<references/>

134.109.16.22: /* Theoretical basis */ fixed typo and link

2012-11-22T15:29:04Z

Theoretical basis: fixed typo and link

← Older revision		Revision as of 17:29, 22 November 2012
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	:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math>		:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math>

	where <math>N</math> is the total number of particles, <math>\mu</math> is the [[chemical potential]], <math>\beta := 1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].		where <math>N</math> is the total number of particles, <math>\mu</math> is the [[chemical potential]], <math>\beta := 1/k_B T</math> and <math>\Lambda</math> is the [[De Broglie thermal wavelength\|de Broglie thermal wavelength]].

	The sub-index ''L'' makes emphasis on a particular definition of [[microstate]] in which particles are assumed to be distinguishable. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles has no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:		The sub-index ''L'' makes emphasis on a particular definition of [[microstate]] in which particles are assumed to be distinguishable. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles has no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:

134.109.16.22: /* Theoretical basis */ typos corrected

2012-11-22T15:04:37Z

Theoretical basis: typos corrected

← Older revision		Revision as of 17:04, 22 November 2012
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	f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math>		f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math>

	where the <math>U</math> subindex stands for ''unlabelled'', the coordinates in ~~parentheis~~ indicate that the positions are not ~~atributted~~ to any particular choice of labelling and the sum runs over all ~~posible~~ particle label permutations.		where the <math>U</math> subindex stands for ''unlabelled'', the coordinates in parenthesis indicate that the positions are not attributed to any particular choice of labelling and the sum runs over all possible particle label permutations.

	Upon trial insertion of an extra particle, one obtains:		Upon trial insertion of an extra particle, one obtains:

141.108.1.152: /* Theoretical basis */

2012-06-25T15:25:04Z

Theoretical basis

← Older revision		Revision as of 17:25, 25 June 2012
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	probability), while <math>f(o)</math> is the probability density of state <math>o</math>.		probability), while <math>f(o)</math> is the probability density of state <math>o</math>.

	As noted by Norman and Filinov <ref>G. E. Norman and V. S. Filinov "Investigations of phase transitions by a Monte-Carlo method", High Temperature '''7''' pp. 216-222 (1969)</ref>, evaluation of the proper acceptance rules requires very ~~carefull~~ interpretation of the (~~clasical~~) grand canonical probability density:		As noted by Norman and Filinov <ref>G. E. Norman and V. S. Filinov "Investigations of phase transitions by a Monte-Carlo method", High Temperature '''7''' pp. 216-222 (1969)</ref>, evaluation of the proper acceptance rules requires very careful interpretation of the (classical) grand canonical probability density:

	:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math>		:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math>

Carl McBride: Changed reference to Cite format.

2010-04-07T13:50:00Z

Changed reference to Cite format.

← Older revision		Revision as of 15:50, 7 April 2010
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	randomly. The trial move is then accepted or rejected according to the		randomly. The trial move is then accepted or rejected according to the
	usual Monte Carlo lottery.		usual Monte Carlo lottery.
	As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability		As usual, a trial move from an original state (<math>o</math>) to a new state (<math>n</math>) is accepted with probability

	:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math>		:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math>
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	probability), while <math>f(o)</math> is the probability density of state <math>o</math>.		probability), while <math>f(o)</math> is the probability density of state <math>o</math>.

	As noted by Filinov, evaluation of the proper acceptance rules requires very carefull interpretation of the (clasical) grand canonical probability density:		As noted by Norman and Filinov <ref>G. E. Norman and V. S. Filinov "Investigations of phase transitions by a Monte-Carlo method", High Temperature '''7''' pp. 216-222 (1969)</ref>, evaluation of the proper acceptance rules requires very carefull interpretation of the (clasical) grand canonical probability density:

	:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math>		:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math>
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	The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.		The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.

	== References ==		== References ==
	~~# G. E. Norman and V. S. Filinov "Investigations of phase transitions by a Monte-Carlo method", High Temperature~~ '''7''' ~~pp. 216-222 (1969)~~		<references/>
	#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]		'''Related reading'''
	#[http://dx.doi.org/10.1080/00268977500100221 D. J. Adams "Grand canonical ensemble Monte Carlo for a Lennard-Jones fluid", Molecular Physics '''29''' pp. 307-311 (1975)]		*[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]
	#[http://dx.doi.org/10.1063/1.2839302 Attila Malasics, Dirk Gillespie, and Dezső Boda "Simulating prescribed particle densities in the grand canonical ensemble using iterative algorithms", Journal of Chemical Physics '''128''' 124102 (2008)]		*[http://dx.doi.org/10.1080/00268977500100221 D. J. Adams "Grand canonical ensemble Monte Carlo for a Lennard-Jones fluid", Molecular Physics '''29''' pp. 307-311 (1975)]
			*[http://dx.doi.org/10.1063/1.2839302 Attila Malasics, Dirk Gillespie, and Dezső Boda "Simulating prescribed particle densities in the grand canonical ensemble using iterative algorithms", Journal of Chemical Physics '''128''' 124102 (2008)]
	[[Category: Monte Carlo]]		[[Category: Monte Carlo]]

131.188.166.124 at 12:46, 7 April 2010

2010-04-07T12:46:06Z

← Older revision		Revision as of 14:46, 7 April 2010
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	:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =		:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =
	\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} \qquad\qquad\text{(4)}</math>		\frac{\alpha( N+1 \rightarrow N )}{\alpha( N \rightarrow N+1)} = \frac{V}{N+1} \qquad\qquad\text{(4)}</math>

	Substitution of Eq.(3) and Eq.(4) into Eq.(1) yields the		Substitution of Eq.(3) and Eq.(4) into Eq.(1) yields the

Carl McBride: Added another acronym

2008-10-22T13:55:49Z

Added another acronym

← Older revision		Revision as of 15:55, 22 October 2008
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	'''Grand-canonical ensemble Monte Carlo''' (GCEMC) is a very versatile and powerful [[Monte Carlo]] technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface \|interfacial]] phenomena, in the last decade		'''Grand-canonical ensemble Monte Carlo''' (GCEMC or GCMC) is a very versatile and powerful [[Monte Carlo]] technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface \|interfacial]] phenomena, in the last decade
	[[grand canonical ensemble \| grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo \|configurational bias]] grand canonical technique has very much improved the situation.		[[grand canonical ensemble \| grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo \|configurational bias]] grand canonical technique has very much improved the situation.

Carl McBride: Monte Carlo in the grand-canonical ensemble moved to Grand canonical Monte Carlo: Moved to a more 'standard' name

2008-10-22T13:53:23Z

Monte Carlo in the grand-canonical ensemble moved to Grand canonical Monte Carlo: Moved to a more 'standard' name

← Older revision	Revision as of 15:53, 22 October 2008
(No difference)

Carl McBride: /* Theoretical basis */ Added an internal link

2008-07-07T11:17:59Z

Theoretical basis: Added an internal link

← Older revision		Revision as of 13:17, 7 July 2008
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	where <math>N</math> is the total number of particles, <math>\mu</math> is the [[chemical potential]], <math>\beta := 1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].		where <math>N</math> is the total number of particles, <math>\mu</math> is the [[chemical potential]], <math>\beta := 1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].

	The sub-index ''L'' makes emphasis on a particular definition of microstate in which particles are assumed to be distinguishable. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles has no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:		The sub-index ''L'' makes emphasis on a particular definition of [[microstate]] in which particles are assumed to be distinguishable. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles has no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:

	:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P		:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P