ABSTRACT
We compare the convergence of several flat-histogram methods applied to the two-dimensional Ising model, including the recently introduced stochastic approximation with a dynamic update factor (SAD) method. We compare this method to the Wang-Landau (WL) method, the 1/t variant of the WL method, and standard stochastic approximation Monte Carlo (SAMC). In addition, we consider a procedure WL followed by a "production run" with fixed weights that refines the estimation of the entropy. We find that WL followed by a production run does converge to the true density of states, in contrast to pure WL. Three of the methods converge robustly: SAD, 1/t-WL, and WL followed by a production run. Of these, SAD does not require a priori knowledge of the energy range. This work also shows that WL followed by a production run performs superior to other forms of WL while ensuring both ergodicity and detailed balance.
ABSTRACT
We present a Monte Carlo algorithm based on the stochastic approximation Monte Carlo (SAMC) algorithm for directly calculating the density of states. The proposed method is stochastic approximation with a dynamic update factor (SAD), which dynamically adjusts the update factor γ_{t} during the course of the simulation. We test this method on a square-well fluid and a 31-atom Lennard-Jones cluster and compare the convergence behavior of several related Monte Carlo methods. We find that both the SAD and 1/t-Wang-Landau (1/t-WL) methods rapidly converge to the correct density of states without the need for the user to specify an arbitrary tunable parameter t_{0} as in the case of SAMC. SAD requires as input the temperature range of interest, in contrast with 1/t-WL, which requires that the user identify the interesting range of energies. The convergence of the 1/t-WL method is very sensitive to the energy range chosen for the low-temperature heat capacity of the Lennard-Jones cluster. Thus, SAD is more powerful in the common case in which the range of energies is not known in advance.
