Optimal replica exchange method combined with Tsallis weight sampling

Abstract

A unified framework integrating the generalized ensemble sampling associated with the Tsallis weight [C. Tsallis, J. Stat. Phys. 52, 479 (1988)] and the replica exchange method (REM) has been proposed to accelerate the convergence of the conventional temperature REM (t-REM). Using the effective temperature formulation of the Tsallis weight sampling, it is shown that the average acceptance probability for configurational swaps between neighboring replicas in the combination of Tsallis weight sampling and REM (Tsallis-REM) is directly proportional to an overlap integral of the energy distributions of neighboring replicas as in the t-REM. Based on this observation, we suggest a robust method to select optimal Tsallis parameters in the conventional parametrization scheme and present new parametrization schemes for the Tsallis-REM, which significantly improves the acceptance of configurational swaps by systematically modulating energy overlaps between neighboring replicas. The distinguished feature of our method is that all relevant parameters in the Tsallis-REM are automatically determined from the equilibrium phase simulation using the t-REM. The overall performance of our method is explicitly demonstrated for various simulation conditions for the Lennard-Jones 31 atom clusters, exhibiting a double-funneled energy landscape.

Figure 1

Statistical temperature T_S(U) and Tsallis effective temperatures ${\tilde{T}}_{\alpha }\left(U\right)$ in (a) parametrization scheme I with $\left(U_0<{\tilde{U}}_L,T_0=T_L\right)$ and (b) $\left(U_0>{\tilde{U}}_H,T_0=T_H\right)$. Here T_S(U) is assumed to be a linear function of U. The intermediate points of $\left({\tilde{U}}_{\alpha },t_{\alpha }\right)$ (1<α<M) are determined from the equilibrium phase in the conventional t-REM.

Figure 2

Statistical temperature T_S(U) and Tsallis effective temperatures ${\tilde{T}}_{\alpha }\left(U\right)$ in (a) parametrization scheme II with the constant slope ${\tilde{q}}_0$ and different T_α, and (b) scheme III with a set of multiple parameters $\left({\tilde{q}}_{\alpha },U_{\alpha },T_{\alpha }\right)$. For a linear T_S(U) both schemes II and III become equivalent when setting ${\tilde{q}}_0=\gamma {\tilde{q}}_{LH}$. However, in a curved T_S(U), scheme III is more effective than scheme II.

Figure 3

(a) Statistical temperature T_S(U) and Tsallis effective temperatures ${\tilde{T}}_{\alpha }^{\mathrm{I}}\left(U\right)$ with (U₀,T₀)=(−200,0.0098), and (b) Tsallis PDFs (line points) and the reweighted canonical PDFs (solid lines). From bottom to top α=10, 15, 20, 25, 30, and 35 at U=−135 in (a).

Figure 4

(a) Statistical temperature T_S(U) and Tsallis effective temperatures ${\tilde{T}}_{\alpha }^{\mathrm{I}}\left(U\right)$ with (U₀,T₀)=(−80,0.52), and (b) Tsallis PDFs (line points) and reweighted canonical PDFs (solid lines). From bottom to top α=10, 15, 20, 25, 30, and 35 at U=−135 in (a).

Figure 5

(a) p_acc(α) as a function of the replica index α and accumulated tunneling events in (b) replica space and (c) energy space with varying (U₀,T₀) in parametrization scheme I. N_τU has been counted for the bounded energy region [−133,−105].

Figure 6

(a) Statistical temperature T_S(U) and Tsallis effective temperatures ${\tilde{T}}_{\alpha }^{\mathrm{III}}\left(U\right)$ with γ=0.9, and (b) Tsallis PDFs (line points) and the reweighted canonical PDFs (solid lines). From bottom to top α=10, 15, 20, 25, 30, and 35 at U=−135 in (a).

Figure 7

(a) p_acc(α) as a function of the replica index α and accumulated tunneling events in (b) replica space and (c) energy space with varying ${\tilde{q}}_0$ in schemes II and γ in scheme III. N_τU has been determined for the bounded energy region [−133,−105].

Figure 8

(a) Statistical temperature T_S(U) and Tsallis effective temperatures ${\tilde{T}}_{\alpha }^{\mathrm{III}}\left(U\right)$ with γ=1.0 and M=10, and (b) Tsallis PDFs (line points) and the reweighted canonical PDFs (solid lines). From bottom to top α=2, 4, 6, 8, 9, and 10 at U=−135 in (a). In (a) ${\tilde{T}}_9\left(U\right)$ almost coincides with ${\tilde{T}}_{10}\left(U\right)$, resulting in the superimposed Tsallis PDFs in (b).

Figure 9

(a) p_acc(t_α) as a function of the temperature t_α and accumulated tunneling events in (b) replica space and (c) energy space with varying a number of replicas (M) in parametrization schemes II and III.

Figure 10

(a) Heat capacities determined by t-REM with M=35 for a total 10¹⁰ MC cycles and Tsallis-REM simulations with different parametrization schemes and M for a total 5×10⁹ MC cycles, and (b) magnified view around the Mackay and anti-Mackay transition regions.