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| 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> | | {{Stub-general}} |
| <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>
| | The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau (Ref. 1) to compute the density of |
| to compute the density of states, <math> \Omega (E) </math>, of [[Potts model|Potts models]]; | | 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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| == Outline of the method == | | == Sketches 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 simulation technique designed to reach an 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]] |
| the probability of a given [[microstate]], <math> X </math>, is given by: | | the probability of a given microstate, <math> X </math> is given by: |
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| :<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>; |
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| whereas for the Wang-Landau procedure one can write: | | whereas for the Wang-Landau procedure we can write: |
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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> ; |
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| 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 get a prefixed 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> |
| is updated as: | | is uptdated as: |
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| :<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 it |
| | happens in [[Potts model|Potts Models]]), and <math> \Delta f > 0 </math> |
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| 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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| Such a simple scheme is continued until the shape of the energy distribution | | Such a simple scheme is continued until the shape of the energy distribution |
| approaches the one predefined. Notice that this simulation scheme does not produce | | approaches the prefixed one. Notice that this simulation scheme does not produces |
| an equilibrium procedure, since it does not fulfill [[detailed balance]]. To overcome | | an |
| this problem, the Wang-Landau procedure consists in the repetition of the scheme | | equilibrium procedure, since it does not fulfills detailed balance. To overcome |
| sketched above along several stages. In each subsequent stage the perturbation | | this problem the Wang-Landau procedure consists in the repetition of the scheme |
| 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: | | sketched above along several stages. In each subsequent stage the '''perturbation''' |
| | | parameter <math> \Delta f </math> is reduced. So, at the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages |
| :<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>
| | can be considered as a good description of the actual equilibrium system, therefore: |
| | |
| 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
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| microstates with energy <math> E </math> obtained in the sampling.
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| | |
| If the probability distribution of energies, <math> g(E) </math>, is nearly flat (if a uniform distribution of energies is the target), i.e.
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| : <math> g(E_i) \simeq 1/n_{E} ; </math>; for each value <math> E_i </math> in the selected range,
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| with <math> n_{E} </math> being the total number of discrete values of the energy in the range, then the density of
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| states will be given by:
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| | |
| :<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>
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| === Microcanonical thermodynamics === | | <math> P(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E) |
| | </math>; |
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| Once one knows <math> \Omega(E) </math> with accuracy, one can derive the thermodynamics
| | where <math> E_i = E(X_i) </math>, and <math> \delta(x,y) </math> is the |
| of the system, since the [[entropy|entropy]] in the [[microcanonical ensemble|microcanonical ensemble]] is given by:
| | [[Kronecker delta|Kronecker Delta]] |
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| :<math> S \left( E \right) = k_{B} \log \Omega(E) </math> | | If the probability distribution of energies is nearly unifom: |
| | <math> P(E) \simeq cte </math>; then |
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| where <math> k_{B} </math> is the [[Boltzmann constant | Boltzmann constant]].
| | : <math> \Omega(E) \propto \exp \left[ - f(E) \right] </math> |
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| ==Molecular dynamics==
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| 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>.
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| == Extensions ==
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| The Wang-Landau method has inspired a number of simulation algorithms that
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| use the same strategy in different contexts. For example:
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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>
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| * [[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>
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| * Control of polydispersity by [[chemical potential]] ''tuning''<ref name="wilding"> </ref>
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| === Phase equilibria ===
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| In the original version one computes the [[entropy|entropy]] of the system as a function of
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| the [[internal energy|internal energy]], <math> E </math>, for fixed conditions of volume,
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| and number of particles.
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| 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
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| potentials that can be subsequently used to locate [[phase transitions]]. For instance, one
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| can compute the [[Helmholtz energy function | Helmholtz energy function ]],
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| <math> A \left( N | V, T \right) </math> as a function of the number of particle <math> N </math>
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| for fixed conditions of volume, <math> V </math>, and [[temperature|temperature]], <math> T </math>.
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| === Refinement of the results ===
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| It can be convenient to supplement the Wang-Landau algorithm, which does not fulfil [[detailed balance]],
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| with an equilibrium simulation <ref name="Lomba2"> </ref><ref name="Ganzenmuller"> </ref>. In this equilibrium simulation one can use
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| the final result for <math> f\left( E \right) </math> (or <math> f\left( N \right) </math>) extracted from
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| the Wang-Landau technique as a fixed function to weight
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| the probability of the different configurations.
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| Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency,
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| and can be used to refine the numerical results.
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| ==EXEDOS==
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| 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>.
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| ==Applications==
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| The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.
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| ==See also==
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| *[[Statistical-temperature simulation algorithm]]
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| ==References== | | ==References== |
| <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/10.1063/1.2794042 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/10.1063/1.2803061 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)]
| | #[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]] |