Wang-Landau Density of States

Lattice

Size L×L 8×8

Wang-Landau Parameters

Flatness criterion 85%

MC steps / frame 300

stage 1 — building histogram…

Live Readouts

ln f (mod. factor)

1.000000

Histogram flatness

Current energy E

MC sweeps done

How it works

The 2D Ising model on an L×L periodic lattice has Hamiltonian

E = -J Σ<i,j> s_i s_j ,  s_i = ±1,  J = 1

The density of states g(E) — how many of the 2^N spin configurations share a given energy — determines every thermodynamic quantity at every temperature at once, via Z(T) = Σ_E g(E) e^-E/k_BT. Wang & Landau (2001) estimate ln g(E) directly with a single random walk:

repeat:
  flip a random spin  (E_old → E_new)
  accept with P = min(1, g(E_old)/g(E_new))
  ln g(E_current) += ln f
  H(E_current) += 1
when H(E) is flat (min/mean > criterion):
  H(E) ← 0,  ln f ← ln f / 2
stop when ln f < 1e-8  (f → 1)

Flat-histogram trick — visiting a rarely-reached energy repeatedly makes it "expensive" to revisit (g(E) grows), which biases the walk toward unexplored energies until every bin has been sampled roughly equally often.
ln f halving — each time the histogram is flat, the modification factor is squared down (ln f → ln f/2), so late visits refine g(E) ever more gently; the run is done once ln f is negligible.
One run, every temperature — unlike Metropolis Monte Carlo, which must be re-run at each temperature separately, a converged g(E) gives the free energy, specific heat and magnetisation curves at any T by direct summation.

Left grid: the current spin configuration (red = +1, blue = −1). Right bars: the running estimate of ln g(E) as a function of energy — the tall gold bar marks the walker's current energy bin.

Spin +1 Spin −1 ln g(E) bars Current E

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