Wang-Landau method

The Wang-Landau method was proposed by F. Wang and D. P. Landau (Ref. 1-2) 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 achieve a 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 one 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 produce a predefined 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 happens in Potts Models), and $\Delta f>0$ .

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 $\Delta f$ is reduced. So, for 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:

g(E)\propto e^{f(E)}\int dX_{i}\delta (E,E_{i})=e^{f(E)}\Omega (E)

;

where $E_{i}=E(X_{i})$ , $\delta (x,y)$ is the Kronecker Delta, and $g(E)$ is the fraction of microstates with energy $E$ obtained in the sampling.

If the probability distribution of energies, $g(E)$ , is nearly flat (if a uniform distribution of energies is the target), i.e.

g(E_{i})\simeq 1/n_{E};

; for each value

E_{i}

in the selected range,

with $n_{E}$ being the total number of discrete values of the energy in the range, then the density of states will be given by:

\Omega (E)\propto \exp \left[-f(E)\right]

Microcanonical thermodynamics

Once one knows $\Omega (E)$ with accuracy, one can derive the thermodynamics of the system, since the entropy in the microcanonical ensemble is given by:

S\left(E\right)=k_{B}\log \Omega (E)

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 methods
Computation of phase equilibria of fluids (Refs 4-6)
Control of polydispersity by chemical potential tuning (Ref 7)

References

Wang-Landau method

Contents

Sketches of the method

Microcanonical thermodynamics

Extensions

References

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Wang-Landau method

Sketches of the method

Microcanonical thermodynamics

Extensions

References

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