The Wang and Landau algorithm, proposed by Fugao Wang and David P. Landau,[1] is a Monte Carlo method designed to estimate 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[2] and the folding of proteins.
[5] The Wang and Landau algorithm is used to obtain an estimate for the density of states of a system characterized by a cost function.
It uses a non-Markovian stochastic process which asymptotically converges to a multicanonical ensemble.
[1] (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.
Because Wang and Landau algorithm works in discrete spectra,[1] the spectrum
, such that Given this discrete spectrum, the algorithm is initialized by: The algorithm then performs a multicanonical ensemble simulation:[1] a Metropolis–Hastings random walk in the phase space of the system with a probability distribution given by
is incremented by one and the following update is performed: This is the crucial step of the algorithm, and it is what makes the Wang and Landau algorithm non-Markovian: the stochastic process now depends on the history of the process.
, that proposal is now more likely refused; in this sense, the algorithm forces the system to visit all of the spectrum equally.
However, this flatness depends on how well-approximated the calculated entropy is to the exact entropy, which naturally depends on the value of f.[7] To better and better approximate the exact entropy (and thus histogram's flatness), f is decreased after M proposal-acceptance steps: It was later shown that updating the f by constantly dividing by two can lead to saturation errors.
[7] A small modification to the Wang and Landau method to avoid this problem is to use the f factor proportional to
The analytical DOS is given by, by performing the last integral we obtain In general, the DOS for a multidimensional harmonic oscillator will be given by some power of E, the exponent will be a function of the dimension of the system.
Hence, we can use a simple harmonic oscillator potential to test the accuracy of Wang–Landau algorithm because we know already the analytic form of the density of states.
Molecular dynamics (MD) is usually preferable to Monte Carlo (MC), so it is desirable to have a MD algorithm incorporating the basic WL idea for flat energy sampling.
That algorithm is Statistical Temperature Molecular Dynamics (STMD), developed [8] by Jaegil Kim et al at Boston University.
An essential first step was made with the Statistical Temperature Monte Carlo (STMC) algorithm.
WLMC requires an extensive increase in the number of energy bins with system size, caused by working directly with the density of states.
upon integration of its inverse, allowing the use of larger energy bins than in WL.
STMC was compared with WL for the Ising model and the Lennard-Jones liquid.
Upon increasing energy bin size, STMC gets the same results over a considerable range, while the performance of WL deteriorates rapidly.
In sum, STMC needs fewer steps to obtain the same quality of results.
denotes the usual force derived from the potential energy.
and V. The forces are scaled as indicated, and the statistical temperature is updated every time step, using the same procedure as in STMC.
As the simulation converges to flat energy sampling, the running estimate
is called the kinetic temperature as it controls the velocities as usual, but does not enter the configurational sampling, which is unusual.
This simple change makes STMD entirely intensive and substantially improves performance for large systems.
Equilibrium information is impossible to obtain with a canonical simulation, as supercooling or superheating is necessary to cause the transition.