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.
INTRODUCTION
Recently, the temperature replica exchange method1, 2 (tREM) [also called parallel tempering3 (PT)] has become a key workhorse for equilibrium sampling of a variety of complex systems across multidisciplinary fields.4, 5, 6, 7, 8 Simulating a set of replicas of the same system at a distribution of temperatures and swapping configurations among replicas offers an effective means of avoiding trapping in local minima and mitigating broken ergodicity at low temperatures.9
A great deal of effort10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 has been devoted to improving the efficiency of the standard tREM. One of the debated issues is its applicability to systems displaying strong phase transitions, around which metastable or unstable energy states intervene between two macroscopic phases.28 The standard tREM struggles to attain its maximum power in the vicinity of a first-order phase transition, in which canonical energy distributions are effectively disjointed by an energy gap corresponding to a latent heat. Since the acceptance probability of replica exchanges is determined by the energy overlap of neighboring replicas an energy gap between PT<Tc(E) and PT>Tc(E) around the critical temperature Tc, PT(E) being the canonical probability density function (PDF) at the temperature T, significantly impairs replica exchanges.
More fundamentally, the failure of the tREM in first-order phase transitions is intimately connected to an anomalous behavior of the microcanonical entropy, S(E), across the transition region. In many finite size systems, such as spins,29 nuclei fragmentations,30, 31 model proteins,32, 33, 34 and atomic clusters,8, 35, 36S(E) shows a convex dip, i.e., ∂2S∕∂E2>0,30 across the transition region, as sketched in Fig. 1a. Stemming from this convex “intruder” in S(E), the statistical temperature or microcanonical temperature,
(1) |
exhibits a negative slope region in Fig. 1b, the so called backbending or S-loop.32, 33, 35, 36 The existence of the backbending has been verified in recent experiments on nuclear fragmentation37 and cluster melting,38 and its physical origin has been attributed to avoiding a “static” phase coexistence due to the free energy cost forming interfaces.
The backbending in TS(E) manifests a bimodal structure in PT(E)∝e−βF(E,T), F(E,T)=E−TS(E) being the Helmhotz free energy density and β=[kBT]−1 (kB=1).39 With the stationary points, , of F(E,T) determined by an extremum condition, , with for T2<T<T1,30, 34PT(E) becomes double peaked at and with a minimum at . The stability condition, , the prime being a differentiation with respect to E, reveals that the intermediate energy states between and in Fig. 1b corresponding to the backbending region in TS(E) are intrinsically unstable for the canonical ensemble.
For a small system size, L, the canonical ensemble can sample both free energy minima, at and , across the free energy barrier at . However, as L increases, the backbending energy region becomes inaccessible due to a high free energy barrier, implying that tunneling transitions between the two macroscopic phases become unlikely, and PT(E) becomes localized around or , depending on whether T<Tc or not. Accordingly, the acceptance of replica exchanges for a pair of inverse temperatures, β and β′, close to βc=1∕Tc, becomes exponentially suppressed as
(2) |
where Δβ=β′−β and . Notice that ΔE=E′−E≈ΔE* for β>βc>β′ and ΔE≈−ΔE* for β<βc<β′ since PT(E) is centered at for β>βc and for β<βc. We conclude that the instability of the canonical ensemble to the negative slope region in TS(E) is the main cause of the poor acceptance of replica exchanges in the standard tREM, with increasing system size.
An obvious way to restore the full power of replica exchanges is to utilize a noncanonical ensemble, avoiding the instability to the negative slope region in TS(E), and allowing a unimodal energy distribution. The Gaussian ensemble approach40 and its parallel version41 accomplish this goal by multiplying the Boltzmann factor by a Gaussian in energy.
In this paper, we propose a general framework to systematically build up optimized noncanonical ensembles, transforming unstable or metastable energy states of canonical ensembles into stable states in the presence of the backbending in TS(E). Exploiting the one-to-one correspondence between the sampling weight and the effective temperatures,42, 43, 44, 45, 46 we develop an inverse mapping strategy, which determines a set of optimal generalized ensemble weights from the parametrized effective temperatures tailored to naturally bridge between ordered and disordered phases via a succession of unimodal energy distributions.
Simulations on Potts spins with varying system size, L, demonstrate that optimally designed generalized ensembles combined with replica exchanges (gREM) yield a dramatic acceleration of tunneling transitions and enable a comprehensive sampling of the phase transition region. A detailed mechanism for an order of magnitude acceleration in tunneling is discussed, with a quantitative performance comparison to the Wang–Landau (WL) method. Finite size scaling analysis reveals that the mean tunneling time, τE, can be as favorable as ∝L2d characteristic of a perfect random walk in flat histogram methods.47, 48, 49
The paper is organized as follows: In Sec. 2, the theory of the gREM is presented, with detailed simulation protocols. In Sec. 3, the performance of the gREM is examined and quantitatively compared to that of WL sampling,48 for Potts spin systems under various simulation conditions. Conclusions and a brief summary are presented in Sec. 4.
GENERALIZED REPLICA EXCHANGE METHOD
Inverse mapping strategy
The key idea of the gREM is to construct a set of generalized ensemble weights, Wα(E,λα) (α=1,2,…,M), which, as the parameter λα varies, successively access the unstable energy region between and in Fig. 1b. Here α and M are the replica index and the number of replicas, respectively. An important relation in the design of optimal weights is the inverse mapping between the Wα and the effective temperature
(3) |
wα=−ln Wα being the generalized effective potential. Notice that the definition of the effective temperature in Eq. 3 is analogous to that of the statistical temperature in Eq. 1. Based on the one-to-one correspondence in Eq. 3, the effective temperature completely determines the sampling weight up to a constant through the inverse mapping,
(4) |
A necessary and sufficient condition on Tα(E;λα), such that unstable or metastable energy states of the canonical ensemble between and are transformed into stable ones with a unimodal PDF, is derived by identifying an extremum, , of a generalized free energy density, βFα(E)=wα(E)−S(E),
(5) |
and a stability condition
(6) |
where and , and the prime denotes differentiation with respect to E. The primary finding in both Eqs. 5, 6 is that the stationary points of Fα(E), , are determined as the crossing points between Tα(E;λα) and TS(E) in two-dimensional (E,T) space, and the parameter γα modulates the stability of via . Based on Eqs. 5, 6 we find that a unimodal distribution in the generalized PDF (GPDF), i.e., Pα(E)=e−βFα, can arise from forming the unique crossing point, , between TS(E) and Tα(E;λα), subject to .
More quantitatively, expanding the GPDF up to second order yields
(7) |
, illuminating that the “stable” in Fig. 2, with γα<γS, generates a Gaussian PDF centered at with the positive σγ even for a negative γS. As γα further decreases to −∞, the Gaussian in Eq. 7 approaches corresponding to the microcanonical ensemble case. On the other hand, Pα(E) becomes locally flat around , with γα=γS (“marginal” in Fig. 2). The crossing point, , becomes unstable for γα>γS as in the canonical ensemble.
Linear effective temperature: Tsallis weight
The simplest parametrization scheme for forming stable crossing points between Tα(E;λα) and TS(E) is to align linear effective temperatures in parallel with the constant slope, λ0 [see Fig. 3a], as
(8) |
the control parameter λα being the T-intercept at an arbitrarily chosen E0. To form the unique stable crossing point in each replica, γ0 must be less than the minimum slope , being the minimum slope of TS(E) for the sampled energy region. Since TS(E) is monotonically increasing except for the transition region, in most cases a proper γ0 is easily guessed from the approximate TS(E) by connecting a few points of , being an average energy of a short canonical run at T. For example, γ0 can be simply chosen as , T1 and TM being the lowest and highest temperature, and .
Once γ0 is fixed the dynamic range of λα is determined to cover the interesting temperature range between T1 and TM as λ1=T1 and , with . The first and Mth effective temperatures are chosen to cross and , respectively. Then, the intermediate values of λα (1<α<M) are determined by equally dividing the parameter space as
(9) |
and Δλ=(λM−λ1)∕(M−1).
Interestingly, the linear effective temperature of Eq. 8 produces a generic form of the Tsallis weight50, 51, 52
(10) |
Identifying γ0 and λα by (q−1) and , respectively, q being the nonextensivity parameter, recovers the original form of the Tsallis weight proposed in nonextensive statistical mechanics.50 Thus the gREM with the linear effective temperature of Eq. 8 is basically equivalent to the Tsallis-weight based REM, previously presented in the form of the generalized PT,14q-REM,15 and Tsallis-REM.53
However, it should be noted that the parametrized Tsallis weights in the gREM are targeted to transform unstable and metastable energy states of the canonical ensemble into stable ones, resulting in much narrower energy distributions than the canonical PDFs. On the other hand, other variants14, 15, 53, 54 of the Tsallis-weight based REM are mainly focused on producing more delocalized energy distributions to increase an acceptance of replica exchanges. This is why the gREM utilizes λα as a control parameter rather than q in the conventional implementation.
Another advantage of the gREM is that the inverse mapping enables a systematic selection of relevant parameters, which is particularly beneficial for the negative slope region of TS(E). The dynamic range of λα lies between λ1 and λM determined by two short canonical runs at T1 and TM, respectively. The inverse mapping is a general framework for the combination of any generalized ensemble sampling with replica exchanges. Depending on the profile of TS(E), characteristic of the phase transition, different types of effective temperatures can be designed to produce optimal weights. In the limit, γ0→0, the gREM recovers the tREM running on temperatures λα, denoting Wα(E;λα)∼e−(E−E0)∕λα.
Simulation protocols of the gREM
Detailed simulation protocols of the gREM are outlined as follows.
-
(i)
Perform short canonical runs at several temperatures between T1 and TM to determine the data set, . Select a proper γ0 to be less than and determine λα by employing Eq. 9 between λ1=T1 and , with .
-
(ii)Run the gREM simulation in each replica by making trial moves in configuration space with the acceptance,
Every fixed time step, attempt a replica exchange between neighboring replicas with the acceptance(11) (12) -
(iii)
Once a sufficiently long production run has been performed, calculate the entropy estimate by joining multiple generalized ensemble runs via the weighted histogram analysis method (WHAM).55
NUMERICAL SIMULATIONS AND RESULTS
To illustrate how effectively the gREM works around a typical first-order transition, we have chosen eight-state Potts spins as a benchmark, with the energy E=−∑⟨ij⟩δ(Si,Sj),56 in which the sum runs over the nearest-neighbor spins on an L×L square lattice and Kronecker δ takes the value one if Si=Sj and zero otherwise. Here Si=1,2,…, Q=8 are spin variables.
Characteristic features of the gREM
To determine the dynamic range of λα and the optimal value of γ0 we first performed short canonical Monte Carlo (MC) simulations for 2×104 MC sweeps (MCSs) at T1=0.7 and TM=0.8, which determine and. Here one MCS means L2 MC trial moves of all spins. Based on we performed various gREM simulations with varying γ0 for L=64, as summarized in Table 1. Replica exchanges were attempted every L MC moves and the ground state was used as the initial configuration in all replicas.
Table 1.
Methods | L | M | γ0 | E0 | τE (MCS) | tS (MCS) (×108) |
---|---|---|---|---|---|---|
gREM1 | 64 | 30 | −0.000010 | −1.83 | 193.0 | 1.5 |
gREM2 | 64 | 30 | −0.000025 | −1.83 | 388.5 | 1.5 |
gREM3 | 64 | 30 | −0.000075 | −1.83 | 1511.5 | 1.5 |
gREM4 | 64 | 30 | −0.00015 | −1.83 | 3950.8 | 1.5 |
64 | 30 | −0.0000075 | −1.83 | 390.6 | 1.5 | |
64 | 30 | −0.000015 | −1.83 | 739.4 | 1.5 | |
64 | 30 | −0.000075 | −1.83 | 1553.7 | 1.5 | |
WL | 64 | 2.34×105 | 1.53 |
Setting in Eq. 8, the dynamic range of λα, between λ1=T1 and , depends on γ0. For γ0=−0.000 075 corresponding to the gREM3 in Table 1, λ1≈0.7 and λM≈1.1. Resulting effective temperatures (solid lines) in Fig. 3a are chosen to fully cover the phase transition region, including the backbending region between and in TS(E). For comparison, we also plot the exact TS(E), which was determined by the statistical temperature Monte Carlo (STMC) algorithm49 for 2×108 MCS. All relevant parameters in the gREM3 have been chosen based on short canonical runs at T1 and TM and full knowledge of TS(E) is not necessary.
Since Tα(E;λα) was designed to form a unique, stable crossing point, , with TS(E), the resulting GPDFs in Fig. 3b are rapidly localized around with a Gaussian shape, and naturally bridge between ordered and disordered phases with uimodal energy distributions across the transition region. Since Pα(E) is sharply peaked at , , the set of most probable energies, , asymptotically converge toward a locus of TS(E). Indeed, the profile of shows a perfect coincidence with TS(E) determined by STMC, and exactly correspond to crossing points between TS(E) and Tα(E;λα) in Fig. 3a. For convenience, the most probable energy was approximated by the average energy summed over the αth replica.
The superimposed energy distribution, PT(E)=(1∕M)∑αPα(E), shows an almost uniform sampling across the backbending region with a characteristic structure, resulting from narrowly peaked Pα(E). Each energy trajectory sampled by Wα(E;λα) in Fig. 3c is constantly fluctuating around , explicitly illustrating that the gREM achieves a comprehensive sampling for phase-coexistent states by transforming unstable energy states of the canonical ensembles into stable ones.
Actual replica trajectories initiated from α=1 and 30 in Figs. 4a, 4b start to sample the whole dynamic energy range after 3×104 MCS and exhibit very frequent tunneling transitions in both energy and replica spaces via the localized GPDFs across the transition region. For comparison we also performed a tREM simulation with M=30, employing an equidistant temperature allocation for the same temperature range between T1 and TM. Two effectively disjointed sampling domains with no replica exchanges around α=14 are apparent in Figs. 4c, 4d due to a vanishing energy overlap between far-separated energy distributions of neighboring replicas at Tc.
γ0 dependence of the gREM
Varying γ0 directly affects the gREM simulation by changing the width of GPDFs and the distribution of . These effects are entangled in Eq. 8 since decreasing γ0 produces a more dense population of for the transition region, as demonstrated in Fig. 5a, but a narrower GPDF in each replica due to the increased . The collapse of of both gREM1 and gREM4 into TS(E) determined by STMC reveals that the most probable energy set indeed traces a locus of TS(E).
Due to the opposing effects of γ0 for energy overlaps between neighboring replicas, the acceptance probability, pacc(α), for replica exchanges between α and (α+1) shows a nonmonotonic γ0-dependence in Fig. 5b. The tREM shows no replica exchanges around Tc≈0.745 46, with a minimum in pacc(α) at α=14 (T14=0.7448 and T15=0.7482). On the other hand, a systematic enhancement of pacc(α) shows up around α=14 in the gREM as the effective temperatures form stable crossing points in the transition region, upon decreasing γ0 to −0.000 025. A further decrease to γ0=−0.000 15 yields an almost uniform acceptance, but an overall reduction in pacc(α) due to significantly narrowed GPDFs.
The γ0-dependence of the gREM is also clearly captured in the superimposed energy distributions, PT(E), in Fig. 5c, illustrating that phase-coexistent energy states between −1.5L2 and −1.1L2 are only accessible for . The existence of the threshold value reveals that the success of the gREM relies on the transformation of unstable energy states of canonical ensembles into stable ones with optimized noncanonical ensembles. As steeper effective temperatures produce a denser population of in the transition region, PT(E) becomes more flattened for the entire energy range, with characteristic oscillatory structures, stemming from sharply peaked Pα(E) as in the gREM4.
Forming more stable crossing points, , in the transition region is crucial for sampling phase-coexistent states and smoothly joining ordered and disordered phases in the gREM. However, narrowed GPDFs with decreasing γ0 is not desirable for pacc(α), as shown in Fig. 5a. To resolve this problem we performed additional simulations, denoted by the gREM* in Table 1, employing new effective temperatures as
(13) |
with . In Eq. 13, newly assigned control parameters make cross the most probable energy set of the gREM4, maintaining a dense population of for the transition region irrespective of γ0.
As seen in Fig. 6a, a significant enhancement of pacc(α), with an overall uniform acceptance, is obtained for , in comparison with gREM3, even with the same γ0=−0.000 075. The increase in pacc(α) for the transition region is more dramatic with a lower γ0, as in and , corresponding to γ0=−0.000 007 5 and −0.000 015, respectively. The systematic increase in pacc(α) is mainly due to more delocalized GPDFs in the gREM* simulations with increasing γ0, while maintaining centers of Gaussians at nearly the same , as illustrated in Fig. 6b. More delocalized energy distributions maximize energy overlaps among replicas, leading to a significant acceleration of replica exchanges. As a result, the uniform energy sampling in PT(E) remains unchanged in Fig. 6c regardless of γ0 and oscillatory structures are smoothed out. This implies that an optimal performance of the gREM is achieved by first determining densely populated with a steeper γ0 and then switching to a production run with Eq. 13 at .
Accelerated tunneling transitions
To systematically quantify the performance of the gREM simulations we compute the number of tunneling transitions in energy, denoted by NE, measuring how often all replicas make transitions from one phase to the other.25, 47, 53, 57, 58 We also calculate the mean tunneling time, τE, by determining the inverse of a linear slope of NE in production phases of the gREM simulations. Tunneling transitions were counted between the two boundary energies of −1.80L2 and −0.85L2.
Except for an equilibration stage, NE in Fig. 7a is linearly increasing as a function of the total simulation time. This implies that barrier crossing rates, which are proportional to the slope of NE, are almost constant throughout the simulations. The most frequent tunneling transitions were observed in gREM1 with γ0=−0.000 01, and NE systematically decreases as γ0 decreases from , even with a more comprehensive sampling and a better pacc(α) for the phase transition region in Figs. 5a, 5b.
The attenuation of NE with decreasing γ0 is mainly due to the increased number of intermediate replicas in the transition region, causing tunneling to require more consecutive replica exchanges. On the other hand, tunneling in the gREM with γ0 close to is rather rapid since a couple of replica exchanges among sparsely distributed intermediate replicas leads to a tunneling even with a relatively poor pacc(α). However, the accelerated NE with does not necessarily lead to an enhanced convergence because the transition region is poorly sampled due to a lack of replicas, as seen in PT(E) of Fig. 5c.
The effect of γ0 on NE is more apparent in the gREM* simulations in Fig. 7b, in which the location of intermediate replicas around the transitions region is almost frozen. As expected, increasing γ0, yielding more delocalized GPDFs centered at the same of the gREM4, results in a monotonic acceleration of NE, while retaining a faithful sampling in the transition region even with γ0 close to . Approximately a tenfold speedup of NE is observed in compared to gREM4.
For a comparison we also performed the entropic version of the WL sampling48 for the energy range between −1.83L2 and −0.83L2. The entropy estimate has been refined up to fd=f−1=10−9, employing the dynamic update scheme . Following the original convention, the modification factor was reduced to f once , starting from 1.01, with being the average energy histogram. Tunneling transitions in Fig. 7a, which has been magnified 400 times for visualization, show a steep rise at the initial stage of simulation, in which the nonvanishing modification factor ln f constantly biases the system to move away from visited energy regions. The reduction in f is very rapid initially, and fd reaches 10−6 after 3.3×107 MCS, but significantly slows down as fd becomes smaller, resulting in a total of 1.5×108 MCS up to fd=10−9.
The most remarkable point in Fig. 7a is the dramatic acceleration of NE in the gREM simulations over the WL. Examining the τE, in Table 1, reveals a speedup of tunneling transitions in the poorest case of our algorithm (gREM4) of almost 60 times, increasing up to about 150 times for gREM3. The acceleration of NE is more significant for the gREM* simulations, in which replicas are more densely populated for the backbending region regardless of γ0 with Eq. 13. The ratio of tunneling times, , is about 600 in the . Here we only considered the simulations having a comprehensive sampling for the transition region, as in WL sampling.
What is the underlying mechanism for an order of magnitude acceleration of NE in the gREM over the WL? The bottleneck of barrier crossings in Potts spins is the formation of phase-coexistent states with interfacial regions, which becomes exponentially suppressed in the canonical ensemble. Flat histogram methods such as the multicanonical (MUCA) sampling,47 WL,48 and STMC (Ref. 49) alleviate this exponential slowing down by enhancing mixed-phase configurations, employing a sampling weight inversely proportional to the density of states. However, in a single replica simulation of the flat histogram type, forming mixed-phase configurations starting from ordered or disordered phases is intrinsically sequential and cumulative, and τE is mainly limited by the diffusion rate in energy space.
On the other hand, some intermediate replicas located at the transition region in the gREM always retain phase-coexistent states and naturally bridge the ordered and disordered phases through replica exchanges, as illustrated in Figs. 8a, 8b, 8c, 8d, 8e, 8f, in which fractions of spin states, fq=∑iδ(Si,q)∕L2, in the gREM3, were plotted as a function of MCS for different α. In the ordered phase (α=1), most spins occupy a single spin state, while all spin states are equally probable in the disordered phase (α=30). Intermediate replicas of α=13, 15, and 17 exclusively sample mixed-phase configurations characterized by a majority state with fq>0.5 and other states with fq≈0; these mixed-phase states are preserved throughout the simulation. We conclude that the main limiting factor for tunneling in the gREM is the efficacy of replica exchanges among intermediate replicas sampling phase-coexistent states.
Scaling behavior of τE and TS(E)
In a completely unbiased random walk in energy, corresponding to an ideal limit of flat histogram methods, τE scales as L2d, with d being the dimension of the system (d=2 in this paper), since the dynamic energy range expands as Ld. In an actual simulation, τE scales like L2d+z due to a deviation from a perfect random walk.47, 57 The value of z depends on the system and is known to be 0.65 for Potts spins and 0.73 for Ising spins, from studies using the MUCA (Ref. 47) and the flat histogram sampling exploiting the exact S(E),57 respectively.
As summarized in Table 2, we performed several WL simulations, varying L from 16 to 128. The dynamic energy range was restricted between and determined by the gREM simulation at the same L. A similar scaling behavior appears in the WL simulation for Potts spins, as seen in Fig. 9a, in which the log-log plot of τE and L shows a clear linear relationship. A slightly larger exponent of z=0.77, in comparison with that of the MUCA,47 might be due to the difference in energy boundaries for tunneling. In our study, transitions were counted between and , but tunneling transitions in the MUCA were counted between two free energy maxima at Tc, which have a much smaller separation.
Table 2.
L | 16 | 32 | 50 | 64 | 80 | 100 | 128 |
---|---|---|---|---|---|---|---|
τE (×105) (MCS) | 0.055 | 0.35 | 1.31 | 2.34 | 4.52 | 8.44 | 18.12 |
tS (×107) (MCS) | 0.98 | 3.16 | 7.47 | 1.51 | 1.69 | 24.5 | 32.2 |
To examine scaling properties of the gREM we performed two different sets of simulations with varying L, as summarized in Table 3. Simulations in set I denoted by gREMI are associated with the effective temperature of Eq. 8 with M=30 at the fixed . Set II denoted by the gREMII was performed with the scaled and M=50. Since the backbending region in TS(E) becomes flat with increasing L [see Fig. 9b], in the gREMII is systematically lowered by the scaling factor (64∕L)2, producing more delocalized GPDFs and enhancing pacc(α) for the transition region. Since the simulation with L=128 in the gREMI did not show statistically meaningful transitions, it was excluded in the scaling analysis.
Table 3.
Methods | L | (MCS) | (MCS) | tS (MCS) | |
---|---|---|---|---|---|
gREM1 | 16 | −0.0012 | 48.1 | 43.7 | 1.0×107 |
gREM2 | 32 | −0.0003 | 274.7 | 224.2 | 7.0×107 |
gREM3 | 50 | −0.00012 | 851.0 | 487.1 | 1.5×108 |
gREM4 | 64 | −0.000075 | 1511.7 | 755.3 | 1.5×108 |
gREM5 | 80 | −0.000048 | 2710.2 | 1371.1 | 1.5×108 |
gREM6 | 100 | −0.000031 | 4739.1 | 1920.2 | 1.5×108 |
gREM7 | 128 | −0.0000187 | 2998.3 | 1.5×108 | |
128 | −0.0000038 | 139.5 | 1.5×108 |
The comparison of τE in Table 3 reveals that the speedup in NE with increasing L is even more remarkable in both gREMI and gREMII. The ratio increases from about 102 order at L=32 to 104 order at L=128. An apparent linear behavior in the log-log plot of L and τE is seen in the gREMI of Fig. 9a. The scaling behavior follows , implying that the acceleration of NE in the gREMI is about 102 order up to L=100. When more crossing points are created in the phase transition region with M=50, the growth of τE as a function of L is weaker, with a lower slope in the log-log plot in the gREMII, yielding . The scaling behavior of the gREM approaches that of a random walk in energy as more replicas are built up in the phase transition region.57
The profiles of TS(E) represented by the most probable energy set show a clear backbending for L=16 and 32 in Fig. 9b, but the extent of the backbending gradually declines with increasing L and becomes almost flat at L=128. In contrast to the backbending in the microcanonical caloric curve, TS(E), the inverse of the internal energy, T=U−1(E), which corresponds to a caloric curve in the canonical ensemble, monotonically increases across the phase transition region, implying that statistical ensembles are not equivalent in finite size systems.29, 30
Thermodynamics
After a sufficient long production run the gREM determines the entropy estimate, , by joining multiple replica simulations via the WHAM.55 Once is determined, all canonical thermodynamic properties, such as the internal energy, U(T), and the heat capacity, Cv(T), are completely determined. As demonstrated in Fig. 10a, heat capacities (lines) determined by various gREM simulations in Table 1 for L=64 are collapsed into a single curve with good agreement with the WL-MUCA simulation (squares), irrespective of simulation conditions. Also, heat capacities at L=128 in Fig. 10b determined by both gREM7 and in Table 2 are almost indistinguishable from that of the WL-MUCA even with a significantly shorter simulation time.
Here WL-MUCA denotes the MUCA sampling with the frozen refined by the WL up to fd=10−9. We found that thermodynamic properties determined by a bare gives a systematic discrepancy from those of the gREM simulations, as demonstrated in the plot of βc with L−2 in Fig. 10b. Indeed, it is found that energy distributions PMU(E) obtained by the MUCA sampling with the frozen for 108 MCS show a considerable deviation from a uniform sampling, implying that the WL still contains an error stemming from the refinement process.59 We correct this error by applying the reweighting process for the entropy estimate as . As shown in Figs. 10a, 10b the reweighted produce thermodynamics consistent with that of the gREM.
At finite L the singularities and discontinuities in first-order phase transitions of infinite systems are smeared out,60 and backbending emerges in TS(E). With increasing L, TS(E) becomes flat, rounded transitions transform into sharp ones, and the size dependent critical temperature, βc(L)=1∕Tc(L), approaches the thermodynamic transition temperature, . This asymptotic behavior can be predicted by finite size scaling analysis61 as
(14) |
in which the critical temperature, βc(L), was determined as the temperature at the peak in Cv(T).
As seen in Fig. 10b, the critical temperatures determined by the gREMII and the WL-MUCA are in quantitative agreement. Furthermore, plotting βc(L) versus L−2 yields a good straight line, extrapolating to , close to the exact .56 On the other hand, a noticeable deviation from a linear fit shows up in the scaling behavior of βc(L) in the original WL. To eliminate finite size effects at a smaller L, we used the data set between L=50 and L=120 for a linear fit.
CONCLUSIONS
The generalized Replica Exchange Method (gREM) has been developed to facilitate effective configurational sampling in systems exhibiting backbending, or an S-loop, in the statistical temperature, TS(E), characteristic of first-order phase transitions in finite systems. By combining optimally parametrized, generalized ensemble samplings with replica exchanges, our method enables a comprehensive sampling for phase transition regions with successive unimodal energy distributions, by transforming metastable or unstable energy states of canonical ensembles into stable ones. Exploiting the one-to-one correspondence between the sampling weight and the effective temperature, we also present an inverse mapping strategy, which determines optimal sampling weights from tailored effective temperatures, avoiding an intrinsic instability of the canonical ensemble to the negative slope region of TS(E).
The effectiveness of our method has been explicitly demonstrated for Potts spin systems, for various simulations conditions, as a function of the system size, L. The quantitative comparison between the gREM and WL sampling reveals that the gREM provides an order of magnitude acceleration of tunneling transitions over the WL, while maintaining a faithful sampling for the phase transition region as in flat histogram methods. The underlying mechanism for accelerated tunneling transitions is the capacity of the gREM to preserve mixed-phase configurations in intermediate replicas located at the transition region. Finite size scaling analysis shows that in the optimum case of the gREM, the scaling of the mean tunneling time, τE, approaches ∝L2d, corresponding to the ideal limit of flat histogram methods. It is also shown that the gREM provides a correct canonical thermodynamics via the reweighting with a much smaller simulation time.
Finally, we would like to emphasize that the inverse mapping strategy is a general framework that can be applied to the combination of any generalized ensemble sampling and the replica exchange method. We anticipate that a hybrid extension augmented by the inverse mapping, applying the gREM to selective transition regions while retaining the tREM for other energy regions, will allow the widespread use of our method to various systems with first-order transitions.
ACKNOWLEDGMENTS
We are grateful to the National Science Foundation (Grant Nos. CHE-0750309 and CHE-0848427) and the National Institutes of Health (Grant No. RO1 GM076688) for the generous support of our research.
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