Statistical mechanics, structure of matter Definition

Wang–Landau algorithm

Builds on Monte Carlo method in statistical physics Boltzmann distribution Ising model

Definition

Let $S$ be a finite set of configurations, and let $H \colon S \to \mathcal{E}$ be an energy function onto a finite set $\mathcal{E}$ . The density of states is $g(E) = \abs{\set{\sigma \in S : H(\sigma) = E}}$ . Fix a symmetric proposal rule on $S$ , a flatness threshold $\theta \in (0, 1)$ and a final tolerance $\epsilon > 0$ . The Wang–Landau algorithm keeps an estimate $\ln \tilde g(E)$ and a histogram $h(E)$ for $E \in \mathcal{E}$ , and a modification factor $\ln f$ .

Start with $\ln \tilde g(E) = 0$ and $h(E) = 0$ for all $E$ , $\ln f = 1$ , and any configuration $\sigma$ .
Propose $\sigma'$ from $\sigma$ and accept it with probability $\min\big(1, \tilde g(H(\sigma)) / \tilde g(H(\sigma'))\big)$ .
After each proposal, let $E$ be the energy of the current configuration, and increase $\ln \tilde g(E)$ by $\ln f$ and $h(E)$ by $1$ .
When $h$ is flat, that is, $h(E) \ge \theta \cdot \frac{1}{\abs{\mathcal{E}}} \sum_{E'} h(E')$ for all $E$ , reset $h$ to $0$ and replace $\ln f$ by $\frac12 \ln f$ .
Stop when $\ln f < \epsilon$ ; the output is $\tilde g$ , an estimate of $g$ up to a constant factor.

Remarks

With a fixed $\tilde g$ , the walk of clause 2 is at energy $E$ with long-run probability proportional to $g(E) / \tilde g(E)$ . So if $\tilde g = g$ , it visits each energy level equally often. The updates of clause 3 push $\tilde g$ towards $g$ , and the shrinking factor $f$ makes the corrections finer. The constant is fixed by a known value, such as $\sum_E g(E) = \abs{S}$ or the number of ground states. Example: run the method for the Ising model on a finite graph. Then the partition function at every inverse temperature $\beta$ is estimated by $\sum_E \tilde g(E) e^{-\beta E}$ , and this gives the Boltzmann distribution of the energy. Non-example: with $\ln f$ kept fixed, the estimate never converges, because each visit changes $\tilde g$ by a fixed amount and the statistical error stays of order $\ln f$ . The method is not a Markov chain with a fixed stationary distribution, since the weights change during the run. With the halving schedule the error can saturate. A schedule in which $\ln f$ decreases like $1/t$ in the late stage is used to avoid this.

Reading path

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Boltzmann distribution concept
Ising model example
Monte Carlo method in statistical physics concept
Wang–Landau algorithm definition

Tag	`AG9ZVFQ1WW`
URI	`m2://AG9ZVFQ1WW`
Kind	definition
Writing	W0 stub
MSC	82M31, 65C05
Defines	Wang–Landau algorithm
Uses	Monte Carlo method in statistical physics, Boltzmann distribution, Ising model
Changed	3 November 2026

Wang–Landau algorithm

Definition

Remarks

Reading path

Further reading

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