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 updated 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

1. 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)
2. 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)
3. 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)
4. 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)
5. 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)
6. 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)
7. Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", J. Chem. Phys. 119, 12163 (2003)
8. 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)
9. R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E 75 046701 (2007)