Difference of energy density of states in the Wang-Landau algorithm

Paying attention to the difference of density of states, \Delta ln g(E) = ln g(E+\Delta E) - ln g(E), we study the convergence of the Wang-Landau method. We show that this quantity is a good estimator to discuss the errors of convergence, and refer to the $1/t$ algorithm. We also examine the behavior of the 1st-order transition with this difference of density of states in connection with Maxwell’s equal area rule. A general procedure to judge the order of transition is given.

💡 Research Summary

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The paper investigates the convergence properties of the Wang‑Landau (WL) algorithm by focusing on the difference of the logarithmic density of states (DOS), defined as Δ ln g(E) = ln g(E + ΔE) − ln g(E). Because the WL transition probability depends only on the difference of ln g between two energy levels, Δ ln g(E) is a natural, normalization‑independent quantity for assessing the quality of the DOS estimate.

First, the authors apply the method to the two‑dimensional Ising model (L = 32). Using the exact DOS from Beale, they compare Δ ln g(E) obtained after several modification‑factor stages (i = 14, 18, 22, 26). The plots show that as the modification factor f_i approaches unity, the calculated Δ ln g(E) converges toward the exact curve, especially away from the extreme energy edges. To quantify the error they introduce Δ² = ∑_E