Flat-histogram method comparison on the two-dimensional Ising model

We compare the convergence of several flat-histogram methods applied to the two-dimensional Ising model, including the recently introduced stochastic approximation with a dynamic update factor (SAD) method. We compare this method to the Wang-Landau (WL) method, the $1/t$ variant of the WL method, and standard stochastic approximation Monte Carlo (SAMC). In addition, we consider a procedure WL followed by a “production run” with fixed weights that refines the estimation of the entropy. We find that WL followed by a production run does converge to the true density of states, in contrast to pure WL. Three of the methods converge robustly: SAD, $1/t$-WL, and WL followed by a production run. Of these, SAD does not require a priori knowledge of the energy range. This work also shows that WL followed by a production run performs superior to other forms of WL while ensuring both ergodicity and detailed balance.

Figure 1

The update factor ${\gamma }_t$ versus the iteration number for the $N=32\times 32$ system.

Figure 2

The specific heat capacity versus the reciprocal temperature $\beta$ for the $N=32\times 32$ system for each histogram method at ${10}^9$ moves.

Figure 3

The maximum error in the specific heat capacity for each method for the $N=32\times 32$ and $T_{min}=1$ as a function of number of iterations run. The maximum error is averaged over eight independent simulations, and the best and worst simulations for each method are shown as a semitransparent shaded area.

Figure 4

The update factor ${\gamma }_t$ versus the iteration number for the $N=128\times 128$ system.

Figure 5

The maximum error in the specific heat capacity for each method for the $N=128\times 128$ and $T_{min}=1$ as a function of number of iterations run. The maximum error is averaged over eight independent simulations, and the best and worst simulations for each method are shown as a semitransparent shaded area.