Ising Model

DEFINITION

E = −Σ⟨ij⟩ sᵢ·sⱼ − B·Σ sᵢ,  s = ±1
flip with probability min(1, e^(−ΔE/T))
T_c = 2 / ln(1 + √2) ≈ 2.269 at B = 0

NOTES

Ising solved the one-dimensional chain for his 1925 thesis, found no phase transition, and concluded the model explained nothing; Onsager proved in 1944 that two dimensions transition sharply at T_c. This plate rides the temperature slowly through that point, both directions, forever: above it the spins are noise, below it domains freeze out and coarsen, and the crossing is announced each way. The sampling is Metropolis 1953 — the algorithm that named Monte Carlo methods.

PROVENANCE

Origin

E. Ising, Zeitschrift für Physik 31, 1925; N. Metropolis, A. & M. Rosenbluth, A. & E. Teller, J. Chemical Physics 21, 1953; T_c from L. Onsager, Physical Review 65, 1944

Standing

Public domain — statistical mechanics

Constants

Span sets how far the temperature rides either side of critical

Source

doi:10.1007/BF02980577