Comparative study of the precision provided by the Wang-Landau Sampling and the Broad Histogram Method

The work presented here investigates the precision provided by two computational techniques in the evaluation of the density of states and canonical averages of the two-dimensional ferromagnetic Ising model. We obtained the microcanonical and canonical averages carrying out a Wang-Landau sampling and performing a statistical study of the convergence of the microcanonical averages and of the temperature related to the peak of the specific heat. In order to use the Broad Histogram Method, besides the computation of the usual microcanonical averages of the powers of the magnetization, we also compute, during the Wang-Landau sampling, the microcanonical averages of the number of possible changes in the system's state with energy $E$ which would increase ($N_{up}^{\Delta E}(E)$) and decrease ($N_{dn}^{\Delta E}(E)$) the energy by an amount $\Delta E$. We performed simulations for several lattice sizes and from each size we took a large number of samples to build a representative sample of the density of states and the canonical averages. We show our findings for the density of states, as well as for the finite-size scaling exponents, and compare it with exact values. Our findings suggest that the Broad Histogram Method provides a slightly better precision on the computation of the density of states, along with the canonical averages, than the one obtained within the Wang-Landau sampling.