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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>
| | == Extensions == |
| <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 has inspired a number of simulation algorithms that |
| to compute the density of states, <math> \Omega (E) </math>, of [[Potts model|Potts models]];
| | use the same strategy in different contexts. For example: |
| where <math> \Omega(E) </math> is the number of [[microstate |microstates]] of the system having energy
| | * [[Inverse Monte Carlo|Inverse Monte Carlo]] methods (Refs 4-6) |
| <math> E </math>.
| | * [[Computation of phase equilibria]] of fluids (Refs 7-9) |
| | | * Control of polydispersity by chemical potential ''tuning'' (Ref 6) |
| == 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. | |
| 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:
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|
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|
| :<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>
| | === Computation of phase equilibria === |
|
| |
|
| whereas for the Wang-Landau procedure one can write:
| | The Wang-Landau procedure can be adapted to compute different thermodynamical potentials. |
| | In the original paper the [[entropy|entropy]] is computed as a function of the [[internal energy|internal energy]]. |
|
| |
|
| :<math> P(X) \propto \exp \left[ f(E(X)) \right] </math>
| | For instance, in Refs 7-9 it is shown how |
| | | to sample the [[Helmholtz energy function|Helmholtz energy function]] as a function of the number of particles, <math> N </math>, for fixed conditions of [[temperature|temperature]] |
| where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes
| | and volume. |
| 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>
| |
| 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>
| |
| is updated as:
| |
| | |
| :<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>.
| |
| | |
| 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
| |
| an equilibrium procedure, since it does not fulfill [[detailed balance]]. To overcome
| |
| this problem, the Wang-Landau procedure consists in the repetition of the scheme
| |
| sketched above along several stages. In each subsequent stage the perturbation
| |
| 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>
| |
| | |
| 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
| |
| microstates with energy <math> E </math> obtained in the sampling.
| |
| | |
| If the probability distribution of energies, <math> g(E) </math>, is nearly flat (if a uniform distribution of energies is the target), i.e.
| |
| : <math> g(E_i) \simeq 1/n_{E} ; </math>; for each value <math> E_i </math> in the selected range,
| |
| with <math> n_{E} </math> being the total number of discrete values of the energy in the range, then the density of
| |
| states will be given by:
| |
| | |
| :<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>
| |
| | |
| === Microcanonical thermodynamics ===
| |
| | |
| Once one knows <math> \Omega(E) </math> with accuracy, one can derive the thermodynamics
| |
| of the system, since the [[entropy|entropy]] in the [[microcanonical ensemble|microcanonical ensemble]] is given by:
| |
| | |
| :<math> S \left( E \right) = k_{B} \log \Omega(E) </math>
| |
| | |
| 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 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 ==
| |
| The Wang-Landau method has inspired a number of simulation algorithms that
| |
| 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 (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>
| |
| * Control of polydispersity by [[chemical potential]] ''tuning''<ref name="wilding"> </ref>
| |
| === Phase equilibria ===
| |
| 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,
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| and number of particles.
| |
| 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
| |
| 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>
| |
| for fixed conditions of volume, <math> V </math>, and [[temperature|temperature]], <math> T </math>. | |
| === Refinement of the results ===
| |
| It can be convenient to supplement the Wang-Landau algorithm, which does not fulfil [[detailed balance]],
| |
| 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 Wang-Landau technique as a fixed function to weight
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| the probability of the different configurations.
| |
| 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. | |
| ==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==
| |
| The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}.
| |
| ==See also==
| |
| *[[Statistical-temperature simulation algorithm]]
| |
| ==References==
| |
| <references/>
| |
| '''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/10.1063/1.2803061 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)]
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| [[category: Monte Carlo]]
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| [[category: computer simulation techniques]]
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