Wang-Landau method

The Wang-Landau method was proposed by F. Wang and D. P. Landau (Ref. 1) to compute the density of states, $\Omega (E)$ , of Potts models; where $\Omega (E)$ is the number of microstates of the system having energy $E$ .

Sketches of the method

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 in the canonical ensemble the probability of a given microstate, $X$ is given by:

$P(X)\propto \exp \left[-E(X)/k_{B}T\right]$ ;

whereas for the Wang-Landau procedure we can write:

$P(X)\propto \exp \left[f(E(X))\right]$ ;

where $f(E)$ is a function of the energy. $f(E)$ changes during the simulation in order get a prefixed distribution of energies (usually a uniform distribution); this is done by modifying the values of $f(E)$ to reduce the probability of the energies that have been already visited, i.e. If the current configuration has energy $E_{i}$ , $f(E_{i})$ is uptdated as:

$f^{new}(E_{i})=f(E_{i})-\Delta f$ ;

where it has been considered that the system has discrete values of the energy (as it happens in Potts Models), and $\Delta f>0$

Such a simple scheme is continued until the shape of the energy distribution approaches the prefixed one. Notice that this simulation scheme does not produces an equilibrium procedure, since it does not fulfills 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 $\Delta f$ is reduced. So, at the last stages the function $f(E)$ hardly changes and the simulation results of these last stages can be considered as a good description of the actual equilibrium system, therefore: