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 fulfil 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:

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

where $k_{B}$ is the Boltzmann constant.

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 (Refs 4-6)
• Computation of phase equilibria of fluids (Refs 7-9)
• Control of polydispersity by chemical potential tuning (Ref 6)

Phase equilibria

In the original version one computes the entropy of the system as a function of the internal energy, $E$, for fixed conditions of volume, and number of particles. In Refs (7-9) it is shown how the procedure can be applied to compute other thermodynamic potentials that can be used later to locate phase transitions. For instance one can compute the Helmholtz energy function , $A\left(N|V,T\right)$ as a function of the number of particle $N$ for fixed conditions of volume, $V$, and temperature, $T$.

It can be convenient (Refs 7-8) to supplement the Wang-Landau algorithm, which does not fulfil detailed balance, with an equilibrium simulation. In this equilibrium simulation one can use the final result for $f\left(E\right)$ (or $f\left(N\right)$ to weight the probability of the different configurations.

References

1. 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)
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. 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 (6 pages) (2003)
5. N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Physical Review E 70 021203 (5 pages) (2004)
6. Nigel B. Wilding "A nonequilibrium Monte Carlo approach to potential refinement in inverse problems", Journal of Chemical Physics 119, 12163 (2003)
7. E. Lomba, C. Martín, and N. G. Almarza, "Simulation study of the phase behavior of a planar Maier-Saupe nematogenic liquid", Physical Review E E 71 046132 (2005)
8. 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)
9. 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)
10. 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)
11. R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E 75 046701 (2007)