Wang-Landau method

The Wang-Landau method was proposed by F. Wang and D. P. Landau (Ref. 1) to compute the density of states, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega (E) } , of Potts models; where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(E) } is the number of microstates of the system having energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X } is given by:

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

whereas for the Wang-Landau procedure we can write:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(X) \propto \exp \left[ f(E(X)) \right] }  ;

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(E) } is a function of the energy. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(E) } to reduce the probability of the energies that have been already visited, i.e. If the current configuration has energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_i } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(E_i) } is uptdated as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta f } is reduced. So, at the last stages the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

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

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_i = E(X_i) } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(x,y) } is the Kronecker Delta

If the probability distribution of energies is nearly unifom: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(E) \simeq cte } ; then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(E) \propto \exp \left[ - f(E) \right] }

Extensions

The Wang-Landau method has inspired a number of simulation algorithms that use the same strategy in different contexts.

References

1. 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)
2. 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)
3. 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)
4. 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)
5. R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E 75 046701 (2007)