Dynamical traps in Wang-Landau sampling of continuous systems: Mechanism and solution

We study the mechanism behind dynamical trappings experienced during Wang-Landau sampling of continuous systems reported by several authors. Trapping is caused by the random walker coming close to a local energy extremum, although the mechanism is different from that of the critical slowing-down encountered in conventional molecular dynamics or Monte Carlo simulations. When trapped, the random walker misses the entire or even several stages of Wang-Landau modification factor reduction, leading to inadequate sampling of the configuration space and a rough density of states, even though the modification factor has been reduced to very small values. Trapping is dependent on specific systems, the choice of energy bins, and the Monte Carlo step size, making it highly unpredictable. A general, simple, and effective solution is proposed where the configurations of multiple parallel Wang-Landau trajectories are interswapped to prevent trapping. We also explain why swapping frees the random walker from such traps. The efficacy of the proposed algorithm is demonstrated.

Figure 1

Spikes in the DOS for (a) the frustrated $XY$ model ($L=16$) and (b) the 8-mer poly-alanine. For both, we performed 10 stages of WLS, where each stage is simulated for $5\times {10}^6$ updates per spin/angle, and the modification factor $lnf$ is halved at at each stage. Shown are the DOS values from a single trajectory at the end of the tenth stage.

Figure 2

Schematic illustrating how the random walker gets trapped inside a spiked bin.

Figure 3

Comparison of behaviors between a trapped and normal segments of the trajectory for the $XY$ model ($L=16$). (a) Autocorrelation function and infinity norm of force along the segments. (b) Scatterplot of proposed energy change $\mathrm{\Delta }E$ versus proposed angle change $\mathrm{\Delta }\theta$. (c) Energy distribution of accepted moves.

Figure 4

(a) Specific heat capacity per spin of the $XY$ model for a fixed set of $J_{ij}$ ($L=16$). The $\langle c_v\rangle$ for $T=250$ and $T=\infty$ is calculated by averaging over 32 trajectories. For replica-exchange Monte Carlo (open circles), $\langle c_v\rangle$ is computed by averaging over 25 independent runs. Inset: Closeup view near zero temperature of $T=250$ (filled circles) and replica-exchange (open circles). Error bars indicate the standard deviation. The bars for replica exchange are smaller than the circles. (b) For $T=\infty$. Converged: $\langle c_v\rangle$, averaged over 14 converged trajectories. Unconverged: $\langle c_v\rangle$, averaged over 18 unconverged trajectories. Error bars indicate the standard deviation.

Figure 5

Distribution of spike locations with respect to energy for the frustrated $XY$ model ($L=16$). Inset: Energy probability distribution $P\left(E\right)=g\left(E\right)e^{-E/T}$ at temperature $T=0.01$ (corresponding to $E\approx -346$, where spiking is dominant) and at $T=0.4$ (phase transition temperature).