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The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau | The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau (Ref. 1-2) to compute the density of | ||
states, <math> \Omega (E) </math>, of [[Potts model|Potts models]]; | |||
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>. | ||
== | == 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 [[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]] | ||
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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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 | is uptdated 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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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 one predefined. Notice that this simulation scheme does not produce | ||
an equilibrium procedure, since it does not | an equilibrium procedure, since it does not fulfil [[detailed balance]]. To overcome | ||
this problem, the Wang-Landau procedure consists in the repetition of the scheme | this problem, the Wang-Landau procedure consists in the repetition of the scheme | ||
sketched above along several stages. In each subsequent stage the perturbation | 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: | 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> | :<math> P(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>, | where <math> E_i = E(X_i) </math>, and <math> \delta(x,y) </math> is the | ||
[[Kronecker delta|Kronecker Delta]] | [[Kronecker delta|Kronecker Delta]]. | ||
If the probability distribution of energies | If the probability distribution of energies is nearly unifom: | ||
<math> P(E) \simeq cte </math>; then | |||
:<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math> | :<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math> | ||
== | == Extensions == | ||
The Wang-Landau method has inspired a number of simulation algorithms that | |||
use the same strategy in different contexts. | |||
* [[Inverse Monte Carlo|Inverse Monte Carlo]] methods | |||
* Computation of the phase equibria of fluids | |||
* Control of polydispersity by chemical potential ''tunning'' | |||
==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", Phys. Rev. Lett. 86, 2050 - 2053 (2001) ] | |||
''' | #[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)] | ||
#[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/10.1103/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]] |