Densities of States: The Wang-Landau Monte Carlo Method

One of the most interesting recent developments in molecular simulation are the so-called ``density of states'' methods, first implemented by Wang and Landau [22,23]. In principle, these techniques provide a route to calculating the density of states of a system of interest, $\Omega (E)$ (Eq. 19). ⁶ Determining $\Omega$ gives a full understanding of the thermodynamics of a system, because the entropy is calculable directly as

$$S = k_BT\ln\Omega$$ (342)

So, it is not undesirable at all to be able to compute $\Omega$.

Now, we know that $\Omega$ is buried in the probability distribution of energy that arises from standard NVT Monte Carlo simulation:

$$P(E) = \Omega(E)\exp\left(-\beta E\right)\left/\sum_{i}\Omega(E_i)\exp\left(-\beta E_i\right)\right.$$ (343)

If we run a simple NVT MC simulation and populate a histogram of total energy, $H(E)$, we can in principle compute $\Omega (E)$ by first normalizing $H(E)$ $\rightarrow P(E)$ and multiplying each entry $i$ by the factor $\exp\left(+\beta E_i\right)$. The problem in practice is that states for which the Boltzmann factor ( $\exp\left(-\beta E_i\right)$) is low are rarely if ever visited in reasonable time, and the statistical strength of $H(E)$ in such regions of energy is therefore poor. Indeed, conventional MC is designed to not cover all of energy space, but to perform importance sampling of configurational space. Furthermore, if the Markov process is trapped in a local minimum on the potential energy hypersurface, barrier states with vanishingly small Boltzmann factors effectively prevent the escape of the process, preventing an adequate sampling of the potential energy hypersurface. For these reasons, it is desirable to perform a random walk in energy rather than configurational space.

In this section, we'll review the two papers by Wang and Landau which introduced their technique by demonstrating how to compute $\Omega$ for Ising and Potts systems by conducting random walks in energy space. These are lattice systems; more recent work has been focused on developing efficient continuous-space versions of the technique (E.g., [24,25]).