Wang and Landau algorithm

The Wang and Landau algorithm, proposed by Fugao Wang and David P. Landau, is a Monte Carlo method designed to calculate the density of states of a system. The method performs a non-Markovian random walk to build the density of states by quickly visiting all the available energy spectrum. The Wang and Landau algorithm is an important method to obtain the density of states required to perform a multicanonical simulation.

The Wang–Landau algorithm can be applied to any system which is characterized by a cost (or energy) function. For instance, it has been applied to the solution of numerical integrals and the folding of proteins. The Wang-Landau sampling is related to the metadynamics algorithm.

The Wang and Landau algorithm is used to obtain the density of states of a system characterized by a cost function. It uses a non-Markovian which asymptotically converges to a multicanonical ensemble. (I.e. to a Metropolis-Hastings algorithm with sampling distribution inverse to the density of states.) The major consequence is that this sampling distribution leads to a simulation where the energy barriers are invisible. This means that the algorithm visits all the accessible states (favorable and less favorable) much faster than a Metropolis algorithm.

Consider a system defined on a phase space ${\displaystyle \Omega }$, and a cost function, E, (e.g. the energy), bounded on a spectrum ${\displaystyle E\in \Gamma =[E_{\min },E_{\max }]}$, which has an associated density of states ${\displaystyle \rho (E)\equiv \exp(S(E))}$, which is to be computed. Because Wang and Landau algorithm works in discrete spectra, the spectrum ${\displaystyle \Gamma }$ is divided in N discrete values with a difference between them of ${\displaystyle \Delta }$, such that

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