Wang-Landau method - Revision history

Carl McBride: /* Outline of the method */ Slight tidy

2012-06-21T14:05:28Z

Outline of the method: Slight tidy

Carl McBride: Started a section for the EXEDOS adaptation

2012-06-21T14:04:30Z

Started a section for the EXEDOS adaptation

← Older revision		Revision as of 16:04, 21 June 2012
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	Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,		Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,
	and can be used to refine the numerical results.		and can be used to refine the numerical results.
			==EXEDOS==
			EXEDOS ('''ex'''panded '''e'''nsemble '''d'''ensity '''o'''f '''s'''tates) <ref>[http://dx.doi.org/10.1063/1.1508365 Evelina B. Kim, Roland Faller, Qiliang Yan, Nicholas L. Abbott, and Juan J. de Pablo "Potential of mean force between a spherical particle suspended in a nematic liquid crystal and a substrate", Journal of Chemical Physics '''117''' pp. 7781- (2002)]</ref>.
	==Applications==		==Applications==
	The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.		The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.

Carl McBride: /* Molecular dynamics */ Corrected an internal link

2011-01-12T13:47:37Z

Molecular dynamics: Corrected an internal link

← Older revision		Revision as of 15:47, 12 January 2011
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	where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].		where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].
	==Molecular dynamics==		==Molecular dynamics==
	The Wang-Landau method has been extended for use in [[molecular dynamics]] simulations, including the [[multicanonical method]] <ref>[http://dx.doi.org/10.1063/1.3517105 Hiromitsu Shimoyama, Haruki Nakamura, and Yasushige Yonezawa "Simple and effective application of the Wang–Landau method for multicanonical molecular dynamics simulation", Journal of Chemical Physics '''134''' 024109 (2011)]</ref>.		The Wang-Landau method has been extended for use in [[molecular dynamics]] simulations, including the [[Multicanonical ensemble \| multicanonical method]] <ref>[http://dx.doi.org/10.1063/1.3517105 Hiromitsu Shimoyama, Haruki Nakamura, and Yasushige Yonezawa "Simple and effective application of the Wang–Landau method for multicanonical molecular dynamics simulation", Journal of Chemical Physics '''134''' 024109 (2011)]</ref>.

	== Extensions ==		== Extensions ==
	The Wang-Landau method has inspired a number of simulation algorithms that		The Wang-Landau method has inspired a number of simulation algorithms that

Carl McBride: Started section on molecular dynamics

2011-01-12T11:50:31Z

Started section on molecular dynamics

← Older revision		Revision as of 13:50, 12 January 2011
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	where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].		where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].
			==Molecular dynamics==
			The Wang-Landau method has been extended for use in [[molecular dynamics]] simulations, including the [[multicanonical method]] <ref>[http://dx.doi.org/10.1063/1.3517105 Hiromitsu Shimoyama, Haruki Nakamura, and Yasushige Yonezawa "Simple and effective application of the Wang–Landau method for multicanonical molecular dynamics simulation", Journal of Chemical Physics '''134''' 024109 (2011)]</ref>.
	== Extensions ==		== Extensions ==
	The Wang-Landau method has inspired a number of simulation algorithms that		The Wang-Landau method has inspired a number of simulation algorithms that

Nice and Tidy: /* Extensions */ Added an internal link

2010-11-18T13:41:30Z

Extensions: Added an internal link

← Older revision		Revision as of 15:41, 18 November 2010
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	* [[Inverse Monte Carlo\|Inverse Monte Carlo]] methods <ref>[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (2004)]</ref> <ref name="wilding">[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003)]</ref>		* [[Inverse Monte Carlo\|Inverse Monte Carlo]] methods <ref>[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (2004)]</ref> <ref name="wilding">[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003)]</ref>
	* [[Computation of phase equilibria]] of fluids <ref name="Lomba1">[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E '''71''' 046132 (2005)]</ref> <ref name="Lomba2">[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]</ref> <ref name="Ganzenmuller">[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]</ref>		* [[Computation of phase equilibria]] of fluids <ref name="Lomba1">[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E '''71''' 046132 (2005)]</ref> <ref name="Lomba2">[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]</ref> <ref name="Ganzenmuller">[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]</ref>
	* Control of polydispersity by chemical potential ''tuning''<ref name="wilding"> </ref>		* Control of polydispersity by [[chemical potential]] ''tuning''<ref name="wilding"> </ref>
	=== Phase equilibria ===		=== Phase equilibria ===
	In the original version one computes the [[entropy\|entropy]] of the system as a function of		In the original version one computes the [[entropy\|entropy]] of the system as a function of
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	Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,		Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,
	and can be used to refine the numerical results.		and can be used to refine the numerical results.

	==Applications==		==Applications==
	The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.		The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.

Nice and Tidy at 10:34, 18 November 2010

2010-11-18T10:34:05Z

← Older revision		Revision as of 12:34, 18 November 2010
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	<math> E </math>.		<math> E </math>.

	== ~~Sketches~~ of the method ==		== Outline of the method ==
	The Wang-Landau method, in its original version, is a [[Computer simulation techniques \|simulation technique]] designed to achieve a uniform sampling of the energies of the system in a given range.		The Wang-Landau method, in its original version, is a [[Computer simulation techniques \|simulation technique]] designed to achieve a uniform sampling of the energies of the system in a given range.
	In a standard [[Metropolis Monte Carlo\|Metropolis Monte Carlo]] in the [[canonical ensemble\|canonical ensemble]]		In a standard [[Metropolis Monte Carlo\|Metropolis Monte Carlo]] in the [[canonical ensemble\|canonical ensemble]]
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	The Wang-Landau method has inspired a number of simulation algorithms that		The Wang-Landau method has inspired a number of simulation algorithms that
	use the same strategy in different contexts. For example:		use the same strategy in different contexts. For example:
	* [[Inverse Monte Carlo\|Inverse Monte Carlo]] methods <ref>[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 ~~(5 pages)~~ (2004)]</ref> <ref name="wilding">[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003)]</ref>		* [[Inverse Monte Carlo\|Inverse Monte Carlo]] methods <ref>[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (2004)]</ref> <ref name="wilding">[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003)]</ref>
	* [[Computation of phase equilibria]] of fluids <ref name="Lomba1">[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E '''71''' 046132 (2005)]</ref> <ref name="Lomba2">[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]</ref> <ref name="Ganzenmuller">[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]</ref>		* [[Computation of phase equilibria]] of fluids <ref name="Lomba1">[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E '''71''' 046132 (2005)]</ref> <ref name="Lomba2">[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]</ref> <ref name="Ganzenmuller">[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]</ref>
	* Control of polydispersity by chemical potential ''tuning''<ref name="wilding"> </ref>		* Control of polydispersity by chemical potential ''tuning''<ref name="wilding"> </ref>

Nice and Tidy: Changed references to Cite format

2010-11-18T10:31:28Z

Changed references to Cite format

← Older revision		Revision as of 12:31, 18 November 2010
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	The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau (~~Ref~~. ~~1-2~~) to compute the density of		The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau <ref>[http://dx.doi.org/10.1103/PhysRevLett.86.2050 Fugao Wang and D. P. Landau "Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States", Physical Review Letters '''86''' pp. 2050-2053 (2001)]</ref>
	states, <math> \Omega (E) </math>, of [[Potts model\|Potts models]];		<ref>[http://dx.doi.org/10.1103/PhysRevE.64.056101 Fugao Wang and D. P. Landau "Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram", Physical Review E '''64''' 056101 (2001)]</ref>
			to compute the density of states, <math> \Omega (E) </math>, of [[Potts model\|Potts models]];
	where <math> \Omega(E) </math> is the number of [[microstate \|microstates]] of the system having energy		where <math> \Omega(E) </math> is the number of [[microstate \|microstates]] of the system having energy
	<math> E </math>.		<math> E </math>.
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	:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;		:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;

	where :<math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the		where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the
	[[Kronecker delta\|Kronecker Delta]], and <math> g(E) </math> is the fraction of		[[Kronecker delta\|Kronecker Delta]], and <math> g(E) </math> is the fraction of
	microstates with energy <math> E </math> obtained in the sampling.		microstates with energy <math> E </math> obtained in the sampling.
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	where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].		where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].




	== Extensions ==		== Extensions ==
	The Wang-Landau method has inspired a number of simulation algorithms that		The Wang-Landau method has inspired a number of simulation algorithms that
	use the same strategy in different contexts. For example:		use the same strategy in different contexts. For example:
	* [[Inverse Monte Carlo\|Inverse Monte Carlo]] methods (~~Refs 4-6~~)		* [[Inverse Monte Carlo\|Inverse Monte Carlo]] methods <ref>[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (2003)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (5 pages) (2004)]</ref> <ref name="wilding">[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003)]</ref>
	* [[Computation of phase equilibria]] of fluids (~~Refs 7~~-9)		* [[Computation of phase equilibria]] of fluids <ref name="Lomba1">[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E '''71''' 046132 (2005)]</ref> <ref name="Lomba2">[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]</ref> <ref name="Ganzenmuller">[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]</ref>
	* Control of polydispersity by chemical potential ''tuning'' ~~(Ref 6)~~		* Control of polydispersity by chemical potential ''tuning''<ref name="wilding"> </ref>

	=== Phase equilibria ===		=== Phase equilibria ===

	In the original version one computes the [[entropy\|entropy]] of the system as a function of		In the original version one computes the [[entropy\|entropy]] of the system as a function of
	the [[internal energy\|internal energy]], <math> E </math>, for fixed conditions of volume,		the [[internal energy\|internal energy]], <math> E </math>, for fixed conditions of volume,
	and number of particles.		and number of particles.
	In Refs ~~(7-9)~~ it is shown how the procedure can be applied to compute other thermodynamic		In Refs. <ref name="Lomba1"> </ref><ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref> it was shown how the procedure can be applied to compute other thermodynamic
	potentials that can be used ~~later~~ to locate [[phase transitions]]. For instance one		potentials that can be subsequently used to locate [[phase transitions]]. For instance, one
	can compute the [[Helmholtz energy function \| Helmholtz energy function ]],		can compute the [[Helmholtz energy function \| Helmholtz energy function ]],
	<math> A \left( N \| V, T \right) </math> as a function of the number of particle <math> N </math>		<math> A \left( N \| V, T \right) </math> as a function of the number of particle <math> N </math>
	for fixed conditions of volume, <math> V </math>, and [[temperature\|temperature]], <math> T </math>.		for fixed conditions of volume, <math> V </math>, and [[temperature\|temperature]], <math> T </math>.

	=== Refinement of the results ===		=== Refinement of the results ===
	It can be convenient ~~(Refs 7-8)~~ to supplement the Wang-Landau algorithm, which does not fulfil detailed balance,		It can be convenient to supplement the Wang-Landau algorithm, which does not fulfil [[detailed balance]],
	with an equilibrium simulation. In this equilibrium simulation one can use		with an equilibrium simulation <ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref>. In this equilibrium simulation one can use
	the final result for <math> f\left( E \right) </math> (or <math> f\left( N \right) </math>) extracted from		the final result for <math> f\left( E \right) </math> (or <math> f\left( N \right) </math>) extracted from
	the Wang-Landau technique as a fixed function to weight		the Wang-Landau technique as a fixed function to weight
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	and can be used to refine the numerical results.		and can be used to refine the numerical results.
	==Applications==		==Applications==
	The WL algorithm has been applied successfully to several problems in physics, biology, chemistry.		The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.
	==See also==		==See also==
	*[[Statistical-temperature simulation algorithm]]		*[[Statistical-temperature simulation algorithm]]
	==References==		==References==
	~~#[http://dx.doi.org~~/~~10.1103/PhysRevLett.86.2050 Fugao Wang and D. P. Landau "Efficient, Multiple-Range Random Walk Algorithm to Calculate the Density of States", Physical Review Letters '''86''' pp. 2050-2053 (2001)]~~		<references/>
	#[http://dx.doi.org/10.1103/PhysRevE.64.056101 Fugao Wang and D. P. Landau "Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram", Physical Review E '''64''' ~~056101 (2001)]~~		'''Related reading'''
	#[http://dx.doi.org/10.1119/1.1707017 D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling", American Journal of Physics '''72''' pp. 1294-1302 (2004)]		*[http://dx.doi.org/10.1119/1.1707017 D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling", American Journal of Physics '''72''' pp. 1294-1302 (2004)]
	#[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Physical Review E '''68''' 011202 (6 pages) (2003)]		*[http://dx.doi.org/10.1063/1.2803061 R. E. Belardinelli and V. D. Pereyra "Wang-Landau algorithm: A theoretical analysis of the saturation of the error", Journal of Chemical Physics '''127''' 184105 (2007)]
	#[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E '''70''' 021203 (5 pages) (2004)]		*[http://dx.doi.org/10.1103/PhysRevE.75.046701 R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E '''75''' 046701 (2007)]
	~~#[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics '''119''', 12163 (2003) ]~~
	#[http://dx.doi.org/10.1103/PhysRevE.71.046132 E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E E '''71''' 046132 (2005) ]
	#[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martín, and C. McBride, "Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007) ]
	#[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]
	#[http://dx.doi.org/10.1063/1.2803061 R. E. Belardinelli and V. D. Pereyra "Wang-Landau algorithm: A theoretical analysis of the saturation of the error", Journal of Chemical Physics '''127''' 184105 (2007)]
	#[http://dx.doi.org/10.1103/PhysRevE.75.046701 R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E '''75''' 046701 (2007)]
	[[category: Monte Carlo]]		[[category: Monte Carlo]]
	[[category: computer simulation techniques]]		[[category: computer simulation techniques]]

Pojeda at 20:13, 17 November 2010

2010-11-17T20:13:05Z

← Older revision		Revision as of 22:13, 17 November 2010
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	:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;		:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;

	where :<math> f(E) </math> is a function of the energy. :<math> f(E) </math> changes		where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes
	during the simulation in order produce a predefined distribution of energies (usually		during the simulation in order produce a predefined distribution of energies (usually
	a uniform distribution); this is done by modifying the values of :<math> f(E) </math>		a uniform distribution); this is done by modifying the values of <math> f(E) </math>
	to reduce the probability of the energies that have been already ''visited'', i.e.		to reduce the probability of the energies that have been already ''visited'', i.e.
	If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>		If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>
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	where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].		where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].




	== Extensions ==		== Extensions ==

Pojeda at 19:58, 17 November 2010

2010-11-17T19:58:30Z

← Older revision		Revision as of 21:58, 17 November 2010
Line 15:		Line 15:
	:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;		:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;

	where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes		where :<math> f(E) </math> is a function of the energy. :<math> f(E) </math> changes
	during the simulation in order produce a predefined distribution of energies (usually		during the simulation in order produce a predefined distribution of energies (usually
	a uniform distribution); this is done by modifying the values of <math> f(E) </math>		a uniform distribution); this is done by modifying the values of :<math> f(E) </math>
	to reduce the probability of the energies that have been already ''visited'', i.e.		to reduce the probability of the energies that have been already ''visited'', i.e.
	If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>		If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>
Line 35:		Line 35:
	:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;		:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;

	where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the		where :<math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the
	[[Kronecker delta\|Kronecker Delta]], and <math> g(E) </math> is the fraction of		[[Kronecker delta\|Kronecker Delta]], and <math> g(E) </math> is the fraction of
	microstates with energy <math> E </math> obtained in the sampling.		microstates with energy <math> E </math> obtained in the sampling.

Pojeda at 14:40, 17 November 2010

2010-11-17T14:40:58Z

← Older revision		Revision as of 16:40, 17 November 2010
Line 81:		Line 81:
	Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,		Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,
	and can be used to refine the numerical results.		and can be used to refine the numerical results.
			==Applications==
			The WL algorithm has been applied successfully to several problems in physics, biology, chemistry.
	==See also==		==See also==
	*[[Statistical-temperature simulation algorithm]]		*[[Statistical-temperature simulation algorithm]]

← Older revision		Revision as of 16:05, 21 June 2012
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	the probability of a given [[microstate]], <math> X </math>, is given by:		the probability of a given [[microstate]], <math> X </math>, is given by:

	:<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>;		:<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>

	whereas for the Wang-Landau procedure one can write:		whereas for the Wang-Landau procedure one can write:

	:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;		:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math>

	where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes		where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes
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	is updated as:		is updated as:

	:<math> f^{new}(E_i) = f(E_i) - \Delta f </math> ;		:<math> f^{new}(E_i) = f(E_i) - \Delta f </math>

	where it has been considered that the system has discrete values of the energy (as happens in [[Potts model\|Potts Models]]), and <math> \Delta f > 0 </math>.		where it has been considered that the system has discrete values of the energy (as happens in [[Potts model\|Potts Models]]), and <math> \Delta f > 0 </math>.
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	parameter <math> \Delta f </math> is reduced. So, for the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages can be considered as a good description of the actual equilibrium system, therefore:		parameter <math> \Delta f </math> is reduced. So, for the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages can be considered as a good description of the actual equilibrium system, therefore:

	:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>;		:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>

	where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the		where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the
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	where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].		where <math> k_{B} </math> is the [[Boltzmann constant \| Boltzmann constant]].

	==Molecular dynamics==		==Molecular dynamics==
	The Wang-Landau method has been extended for use in [[molecular dynamics]] simulations, including the [[Multicanonical ensemble \| multicanonical method]] <ref>[http://dx.doi.org/10.1063/1.3517105 Hiromitsu Shimoyama, Haruki Nakamura, and Yasushige Yonezawa "Simple and effective application of the Wang–Landau method for multicanonical molecular dynamics simulation", Journal of Chemical Physics '''134''' 024109 (2011)]</ref>.		The Wang-Landau method has been extended for use in [[molecular dynamics]] simulations, including the [[Multicanonical ensemble \| multicanonical method]] <ref>[http://dx.doi.org/10.1063/1.3517105 Hiromitsu Shimoyama, Haruki Nakamura, and Yasushige Yonezawa "Simple and effective application of the Wang–Landau method for multicanonical molecular dynamics simulation", Journal of Chemical Physics '''134''' 024109 (2011)]</ref>.