Wang-Landau Density of States (2D)

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

Read Off Thermodynamics

Temperature T / J 2.30

Specific heat C(T)/N

—

Free energy F(T)/N

—

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 — the specific-heat and free-energy panel below sums the running g(E) estimate directly against e^-E/T (in log-space, for numerical stability) to read off C(T) and F(T) at whatever T the slider picks — no re-simulation, even mid-run.

Top grid: the current spin configuration (red = +1, blue = −1). Middle bars: the running estimate of ln g(E) — the gold bar marks the walker's current energy bin. Bottom curve: C(T)/N (teal) and F(T)/N (violet) computed from the g(E) accumulated so far; the vertical marker tracks the temperature slider.

Spin configuration

Spin +1 Spin −1

ln g(E) — density of states

C(T)/N and F(T)/N from g(E)