Editing Wang-Landau method

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>
<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 
<math> E </math>.

== 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:

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

whereas for the Wang-Landau procedure one can write:

:<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
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, 
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 
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
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.

==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)]
[[category: Monte Carlo]]
[[category: computer simulation techniques]]