Determining efficient temperature sets for the simulated tempering method

Abstract

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.

Efficient simulated tempering with approximated weights: Applications to first-order phase transitions

“…The BC model corresponds to the K ≡ K/J = 0 case. In order to compare with previous results [14,31,33] we also consider the K = 3 case, in which the system displays ferromagnetic and paramagnetic phases. All phases can be characterized in terms of two orderparameters q = σ 2 i and m = σ i .…”

“…Henceforth, the analysis of all cases will show that the approximated weights work properly (and hence lead to correct results) in the regime of large replica numbers R for an appropriate choice of temperature sets [33]. On the other hand, for sufficiently small R its performance is strongly reduced and the system does not visit properly the distinct coexisting regions.…”

“…In order to scrutinize them, we compare numerical results at the phase coexistence points for distinct R's, with temperature schedules estimated as proposed in Ref. [33] and described as follows: starting from a fixed T 1 we choose the next R − 1 temperatures T 2 < T 3 < . .…”

“…To answer the aforementioned points, we have analyzed, under the ST with approximated weights, four distinct lattice models, namely, Blume-Capel (BC) and Blume-Emery-Griffiths (BEG) [26], Bell-Lavis (BL) [27,28] and associating lattice gas (ALG) water models [29,30]. The former two are interesting tests, due to the existence of very precise results available from cluster algorithms [31], Wang-Landau [32], PT and ST with free-energy weights [14,18,19,33]. Therefore, they constitute relevant benchmarks for our purposes.…”

“…On the other hand, for sufficiently small R its performance is strongly reduced and the system does not visit properly the distinct coexisting regions. Finally we extend for the approximated weights, a simple protocol for obtaining proper temperature sets initially proposed for the freeenergy weights [33].…”