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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
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| states, <math> \Omega (E) </math>, of [[Potts model|Potts models]];
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| where <math> \Omega(E) </math> is the number of [[microstate |microstates]] of the system having energy
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| <math> E </math>.
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| == Sketches of the method ==
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| 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.
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| In a standard [[Metropolis Monte Carlo|Metropolis Monte Carlo]] in the [[canonical ensemble|canonical ensemble]]
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| 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>;
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| whereas for the Wang-Landau procedure one can write:
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| :<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
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| during the simulation in order produce a predefined distribution of energies (usually
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| a uniform distribution); this is done by modifying the values of <math> f(E) </math>
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| to reduce the probability of the energies that have been already ''visited'', i.e.
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| If the current configuration has energy <math> E_i </math>, <math> f(E_i) </math>
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| is updated as:
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| :<math> f^{new}(E_i) = f(E_i) - \Delta f </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
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| approaches the one predefined. Notice that this simulation scheme does not produce
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| an equilibrium procedure, since it does not fulfil [[detailed balance]]. To overcome
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| this problem, the Wang-Landau procedure consists in the repetition of the scheme
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| sketched above along several stages. In each subsequent stage the perturbation
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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:
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|
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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>;
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|
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| where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the
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| [[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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|
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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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|
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| :<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>
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|
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| === Microcanonical thermodynamics ===
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| Once one knows <math> \Omega(E) </math> with accuracy, one can derive the thermodynamics
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| of the system, since the [[entropy|entropy]] in the [[microcanonical ensemble|microcanonical ensemble]] is given by:
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| :<math> S \left( E \right) = k_{B} \log \Omega(E) </math>
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|
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| where <math> k_{B} </math> is the [[Boltzmann constant | Boltzmann constant]].
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| == 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 |
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| * Control of polydispersity by chemical potential ''tuning'' (Ref 6) | | * Control of polydispersity by chemical potential ''tuning'' (Ref 6) |
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| ==References== | | === Computation of phase equilibria === |
| #[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", Phys. Rev. Lett. '''86''', 2050 - 2053 (2001) ]
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| #[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)]
| | The Wang-Landau procedure can be adapted to compute different thermodynamical potentials. |
| #[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)]
| | In the original paper the [[entropy|entropy]] is computed as a function of the [[internal energy|internal energy]]. |
| #[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)]
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| #[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)]
| | For instance, in Refs 7-9 it is shown how |
| #[http://dx.doi.org/10.1063/1.1626635 Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", J. Chem. Phys. '''119''', 12163 (2003) ]
| | 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]] |
| #[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", Phys. Rev. E '''71''', 046132 (2005) ]
| | and volume. |
| #[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" J. Chem. Phys. '''126''', 244510 (2007) ]
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| #[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)]
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| #[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)]
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| #[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]] | |
| [[category: computer simulation techniques]]
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