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 first-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.