In statistical physics, the efficiency of tempering approaches strongly depends on ingredients such as the number of replicas R, reliable determination of weight factors and the set of used temperatures, TR = {T1, T2, . . . , TR}. For the simulated tempering (SP) in particular -useful due to its generality and conceptual simplicity -the latter aspect (closely related to the actual R) may be a key issue in problems displaying metastability and trapping in certain regions of the phase space. To determine TR's leading to accurate thermodynamics estimates and still trying to minimize the simulation computational time, here it is considered a fixed exchange frequency scheme for the ST. From the temperature of interest T1, successive T 's are chosen so that the exchange frequency between any adjacent pair Tr and Tr+1 has a same value f . By varying the f 's and analyzing the TR's through relatively inexpensive tests (e.g., time decay toward the steady regime), an optimal situation in which the simulations visit much faster and more uniformly the relevant portions of the phase space is determined. As illustrations, the proposal is applied to three lattice models, BEG, Bell-Lavis, and Potts, in the hard case of extreme first-order phase transitions, always giving very good results, even for R = 3. Also, comparisons with other protocols (constant entropy and arithmetic progression) to choose the set TR are undertaken. The fixed exchange frequency method is found to be consistently superior, specially for small R's. Finally, distinct instances where the prescription could be helpful (in second-order transitions and for the parallel tempering approach) are briefly discussed.
We considered a higher-dimensional extension for the replica-exchange Wang-Landau algorithm to perform a random walk in the energy and magnetization space of the two-dimensional Ising model. This hybrid scheme combines the advantages of Wang-Landau and Replica-Exchange algorithms, and the one-dimensional version of this approach has been shown to be very efficient and to scale well, up to several thousands of computing cores. This approach allows us to split the parameter space of the system to be simulated into several pieces and still perform a random walk over the entire parameter range, ensuring the ergodicity of the simulation. Previous work, in which a similar scheme of parallel simulation was implemented without using replica exchange and with a different way to combine the result from the pieces, led to discontinuities in the final density of states over the entire range of parameters. From our simulations, it appears that the replica-exchange Wang-Landau algorithm is able to overcome this difficulty, allowing exploration of higher parameter phase space by keeping track of the joint density of states.
Simulated tempering (ST) has attracted a great deal of attention in the last years, due to its capability to allow systems with complex dynamics to escape from regions separated by large entropic barriers. However its performance is strongly dependent on basic ingredients, such as the choice of the set of temperatures and their associated weights. Since the weight evaluations are not trivial tasks, an alternative approximated approach was proposed by Park and Pande (Phys. Rev. E 76, 016703 (2007)) to circumvent this difficulty. Here we present a detailed study about this procedure by comparing its performance with exact (free-energy) weights and other methods, its dependence on the total replica number R and on the temperature set. The ideas above are analyzed in four distinct lattice models presenting strong first-order phase transitions, hence constituting ideal examples in which the performance of algorithm is fundamental. In all cases, our results reveal that approximated weights work properly in the regime of larger R's. On the other hand, for sufficiently small R its performance is reduced and the systems do not cross properly the free-energy barriers. Finally, for estimating reliable temperature sets, we consider a simple protocol proposed at Comp. Phys. Comm. 128, 2046Comm. 128, (2014.
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