Determining efficient temperature sets for the simulated tempering method

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 (ST) 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 a fixed exchange frequency scheme is considered 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 towards 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.