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thermodynamic simulations & feynman quantum path integration 403log g(E)403020100-2 -1 0 1 2E/NH(E)12000800040000–2 –1 0 1 2E/NFigure 15.5 Left: Logarithm of the density of states log g(E) ∝ S versus the energy perparticle for a 2-D Ising model on an 8 × 8 lattice. Right: The histogram H(E) showing thenumber of states visited as a function of the energy per particle. The aim of WLS is to makethis function flat.proportional to the (initially unknown) density of states,P(E i )= 1g(E i ) , (15.22)and then build up a histogram of visited states via a random walk.The problem with clever idea number 1 is that g(E i ) is unknown. WLS’s cleveridea 2 is to determine the unknown g(E i ) simultaneously with the constructionof the random walk. This is accomplished by improving the value of g(E i ) viathe multiplication g(E i ) → fg(E i ), where f>1 is an empirical factor. When thisworks, the resulting histogram H(E i ) becomes “flatter” because making the smallg(E i ) values larger makes it more likely to reach states with small g(E i ) values. Asthe histogram gets flatter, we keep decreasing the multiplicative factor f until it issatisfactory close to 1. At that point we have a flat histogram and a determinationof g(E i ).At this point you may be asking yourself, “Why does a flat histogram meanthat we have determined g(E i )?” Flat means that all energies are visited equally,in contrast to the peaked histogram that would be obtained normally without the1/g(E i ) weighting factor. Thus, if by including this weighting factor we producea flat histogram, then we have perfectly counteracted the actual peaking in g(E i ),which means that we have arrived at the correct g(E i ).15.6 Wang–Landau SamplingThe steps in WLS are similar to those in the Metropolis algorithm, but now withuse of the density-of-states function g(E i ) rather than a Boltzmann factor:−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 403