Efficient simulation protocol for determining the density of states: Combination of replica-exchange Wang-Landau method and multicanonical replica-exchange method

By combining two generalized-ensemble algorithms, the replica-exchange Wang-Landau (REWL) method and the multicanonical replica-exchange method (MUCAREM), we propose an effective simulation protocol to determine the density of states with high accuracy. The new protocol is referred to as REWL-MUCAREM, and REWL is first performed and then MUCAREM is performed. In order to verify the effectiveness of our protocol, we performed simulations of a square-lattice Ising model using the three methods, namely REWL, MUCAREM, and REWL-MUCAREM. The results showed that the density of states obtained by REWL-MUCAREM is more accurate than that is estimated by the two methods separately.

Figure 1

The specific heat. Panel (1) in (a) gives the exact solutions which were calculated by the exact DOS [40, 41]. Panels (2), (3), and (4) were obtained by simulations with $L=8,16,32,64,$ and 128 by MUCAREM, REWL, and REWL-MUCAREM, respectively. (b) shows the differences between the simulation results and exact one, $\mathrm{\Delta }C\left(T\right)=C_{\mathrm{sim}}\left(T\right)-C_{\mathrm{EXACT}}\left(T\right)$.

Figure 2

Mean local flatness $G\left(E\right)$ in Eq. (14) (red curves). The total number of spins is $N=32\times 32$. $G\left(E\right)$ were obtained from the simulations by (a) MUCAREM, (b) REWL, and (c) REWL-MUCAREM. The error bars (vertical yellow bars) were obtained by the standard error estimation. The best estimated DOS will give $G\left(E\right)=1$.

Figure 3

Global flatness $F$ defined in Eq. (15) as a function of the total number of spins, $N$. If $g_{\mathrm{sim}}^{\left\{i\right\}}\left(E\right)$ is equal to $g_{\mathrm{EXACT}}\left(E\right)$ in the entire energy range, then $F$ takes a value 1.

Figure 4

Global flatness $F$ defined in Eq. (15) as a function of the total number of spins, $N$. The error bars of MUCAREM3 except for $N=64\times 64$ were sufficiently small compared to the symbols.

Figure 5

(a) The number of degrees of orientation per water molecule $W_1$ as a function of the inverse of the number of total water molecules. The purple data points correspond to $N=1600,2880,4704$. The green fit curve was estimated by Eq. (B2). (b) The temperature dependence of average helix length of deca-alanine. Deca-alanine takes a coil structure at high temperatures (see, e.g., the right structure) and takes an $\alpha$-helix structure at low temperatures (see, e.g., the left structure).