Editing Wang-Landau method
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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 one 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 | where <math> f(E) </math> is a function of the energy. <math> f(E) </math> changes | ||
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is updated as: | is updated 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 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>. | ||
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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, 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: | ||
:<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math> | :<math> g(E) \propto e^{f(E)} \int d X_i \delta( E, E_i ) = e^{f(E)} \Omega(E)</math>; | ||
where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the | where <math> E_i = E(X_i) </math>, <math> \delta(x,y) </math> is the | ||
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where <math> k_{B} </math> is the [[Boltzmann constant | Boltzmann constant]]. | where <math> k_{B} </math> is the [[Boltzmann constant | Boltzmann constant]]. | ||
== Extensions == | == Extensions == | ||
The Wang-Landau method has inspired a number of simulation algorithms that | The Wang-Landau method has inspired a number of simulation algorithms that | ||
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Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency, | Such a strategy simplifies the estimation of error bars, provides a good test of the results consistency, | ||
and can be used to refine the numerical results. | and can be used to refine the numerical results. | ||
==Applications== | ==Applications== | ||
The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}. | The Wang-Landau algorithm has been applied successfully to several problems in physics{{reference needed}}, biology{{reference needed}}, and chemistry{{reference needed}}. |