Generalized replica exchange method

Abstract

We present a powerful replica exchange method, particularly suited to first-order phase transitions associated with the backbending in the statistical temperature, by merging an optimally designed generalized ensemble sampling with replica exchanges. The key ingredients of our method are parametrized effective sampling weights, smoothly joining ordered and disordered phases with a succession of unimodal energy distributions by transforming unstable or metastable energy states of canonical ensembles into stable ones. The inverse mapping between the sampling weight and the effective temperature provides a systematic way to design the effective sampling weights and determine a dynamic range of relevant parameters. Illustrative simulations on Potts spins with varying system size and simulation conditions demonstrate a comprehensive sampling for phase-coexistent states with a dramatic acceleration of tunneling transitions. A significant improvement over the power-law slowing down of mean tunneling times with increasing system size is obtained, and the underlying mechanism for accelerated tunneling is discussed.

Figure 1

A schematic representation of (a) the convex dip i.e., ∂²S∕∂E²>0 in S(E) and (b) the backbending in T_S(E). Intermediate energy states between $E_u^1$ and $E_u^2$ are unstable in the canonical ensemble and become inaccessible as the system size increases.

Figure 2

Statistical temperature T_S(E) and effective temperatures $T_{\alpha }\left(E\right)=T_c+{\gamma }_{\alpha }\left(E-E_2^{*}\right)$ with varying γ₀, ${\gamma }_0=T_{\alpha }^{\prime }\left(E_2^{*}\right)$. Unstable energy states around $E_2^{*}$ in the canonical ensemble (γ₀=0) become stable in the generalized ensemble of Eq. 10 with ${\gamma }_0<{\gamma }_S^{\text{min}}=T_S^{\prime }\left(E_2^{*}\right)$. Marginal corresponds to ${\gamma }_0={\gamma }_S^{\text{min}}$.

Figure 3

(a) Most probable energy set $\left[E_{\alpha }^{*},T_{\alpha }^{*}\right]$ (squares) determined by the gREM₃ for 10⁷ MCS and T_S(E) (solid line) determined by the STMC simulation and effective temperatures T_α(E;λ_α), (b) resulting GPDFs P_α(E) and P_T(E) (solid line) for Potts spins with L=64, and (c) energy trajectories sampled by Eq. 10. In both (a) and (b), α=1, 5, 10, 15, 20, 25, and 30 from left to right. α=1, 5, 10, 13, 15, 17, 20, 25, and 30 from down to up in (c).

Figure 4

Simulated trajectories of α=1 and 30 in (a) energy and (b) replica space of the gREM₃, and in (c) energy and (d) replica space of the tREM for Potts spins with L=64.

Figure 5

(a) Most probable energy set $\left[E_{\alpha }^{*},T_{\alpha }^{*}\right]$ of the gREM₁ and the gREM₄ simulations, (b) acceptances probabilities p_acc(α), and (c) superimposed energy distributions P_T(E) of various gREM simulations in Table 1 with varying γ₀ for fixed L=64. Same colors are used for the same simulations in both (b) and (c).

Figure 6

(a) p_acc(α) as a function of $E_{\alpha }^{*}$, (b) resulting GPDFs, and (c) superimposed P_T(E) of the gREM^* simulations in Table 1 for L=64. In (b), α=1, 5, 10, 15, 20, 25, and 30 from left to right.

Figure 7

Accumulated tunneling transitions: N_E as a function of a total MCS in (a) the gREM and (b) the gREM^* simulations in Table 1 for L=64. In (a), N_E of the WL has been magnified by 400 times for visualization and the gREM^* simulations in (b) were performed with $T_{\alpha }^{*}\left(E;{\lambda }_{\alpha }^{*}\right)$ based on $\left[E_{\alpha }^{*},T_{\alpha }^{*}\right]$ of the gREM₄.

Figure 8

Time profiles of f_q=∑_iδ(S_i,q)∕L² in the gREM₃ in Table 1 for L=64. Notice that intermediate replicas of α=13, 15, and 17 exclusively sample mixed-phase configurations consist of a major spin state with f_q>0 and other minor spin states with f_q≈0.

Figure 9

(a) Log-log plots of mean tunneling times τ_E and L for the WL, and the gREM^I and the gREM^II simulations from top to bottom, and (b) profiles of T_S(E) estimated by $\left[E_{\alpha }^{*},T_{\alpha }^{*}\right]$ of the gREM^II simulations with varying L. In (a), lines are linear fits to the corresponding data points.

Figure 10

Heat capacities C_v(T) around T_c of the WL-MUCA and various gREM simulations for Potts spins at (a) L=64 and (b) L=128, and plots of β_c(L) with L⁻² for the gREM^II in Table 3, the WL, and the WL-MUCA simulations. In (b), the line is a linear fit to the data of the gREM^II.