Wang-Landau method: Difference between revisions

Revision as of 10:35, 9 July 2008

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 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 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:

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

;

where $E_{i}=E(X_{i})$ , and $\delta (x,y)$ is the Kronecker Delta.

If the probability distribution of energies is nearly unifom: $P(E)\simeq cte$ ; then

\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

@@ Line 1: / Line 1: @@
 The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau (Ref. 1) to compute the density of
 states, <math> \Omega (E) </math>, of [[Potts model|Potts models]];
@@ Line 6: / Line 5: @@
 == Sketches of the method ==
-The '''Wang-Landau method''' in its original version is a simulation technique designed to reach an uniform
+The Wang-Landau method, in its original version, is a [[Computer simulation techniques |simulation technique]] designed to achieve a uniform sampling of the energies of the system in a given range.
-sampling of the energies of the system in a given range.
 In a standard [[Metropolis Monte Carlo|Metropolis Monte Carlo]] in the [[canonical ensemble|canonical ensemble]]
-the probability of a given microstate, <math> X </math>  is given by:
+the probability of a given [[microstate]], <math> X </math>,  is given by:
-<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>;
+:<math> P(X) \propto \exp \left[ - E(X)/k_B T \right] </math>;
-whereas for the Wang-Landau procedure we can write:
+whereas for the Wang-Landau procedure one can write:
-<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;
+:<math> P(X) \propto \exp \left[ f(E(X)) \right] </math> ;
 where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes
-during the simulation in order get a prefixed distribution of energies (usually
+during the simulation in order produce a predefined distribution of energies (usually
 a uniform distribution); this is done by modifying the values of <math> f(E) </math>
 to reduce the probability of the energies that have been already ''visited'', i.e.
@@ Line 24: / Line 22: @@
 is uptdated as:
-<math> f^{new}(E_i) = f(E_i) - \Delta f </math> ;
+:<math> f^{new}(E_i) = f(E_i) - \Delta f </math> ;
-where it has been considered that the system has discrete values of the energy (as it
-happens in [[Potts model|Potts Models]]), and <math> \Delta f > 0  </math>
+where it has been considered that the system has discrete values of the energy (as happens in [[Potts model|Potts Models]]), and <math> \Delta f > 0  </math>.
 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
+approaches the one predefined. Notice that this simulation scheme does not produce
-an
+an equilibrium procedure, since it does not fulfil [[detailed balance]]. To overcome
-equilibrium procedure, since it does not fulfills detailed balance. To overcome
+this problem, the Wang-Landau procedure consists in the repetition of the scheme
-this problem the Wang-Landau procedure consists in the repetition of the scheme
+sketched above along several stages. In each subsequent stage the perturbation
-sketched above along several stages. In each subsequent stage the '''perturbation'''
+parameter <math> \Delta f </math> is reduced. So, for the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages can be considered as a good description of the actual equilibrium system, therefore:
-parameter <math> \Delta f </math> is reduced. So, at the last stages the function <math> f(E) </math> hardly changes and the simulation results of these last stages
-can be considered as a good description of the actual equilibrium system, therefore:
-<math> P(E) \propto e^{f(E)} \int d X_i \delta( E,  E_i ) = e^{f(E)} \Omega(E)
+:<math> P(E) \propto e^{f(E)} \int d X_i \delta( E,  E_i ) = e^{f(E)} \Omega(E)</math>;
-</math>;
 where <math> E_i = E(X_i) </math>, and <math> \delta(x,y) </math> is the
-[[Kronecker delta|Kronecker Delta]]
+[[Kronecker delta|Kronecker Delta]].
 If the probability distribution of energies is nearly unifom:
 <math> P(E) \simeq  cte </math>; then
-: <math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>
+:<math> \Omega(E) \propto \exp \left[ - f(E) \right] </math>
 == Extensions ==

Wang-Landau method: Difference between revisions

Revision as of 10:35, 9 July 2008

Sketches of the method

Extensions

References

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