2D Ising Model - Statistical Mechanics

Physical Model

The Ising model describes ferromagnetism in statistical mechanics. Each spin on a lattice can be in one of two states (↑ or ↓), representing microscopic magnetic moments.

H = -J Σ s_i·s_j - h Σ s_i

Metropolis Algorithm: Accept spin flip if ΔE < 0, or with probability exp(-ΔE/(k_BT))

Critical Temperature: T_c ≈ 2.27 J/k_B (exact: 2/(ln(1+√2)) ≈ 2.269)

Spin Lattice

White = Spin Up (+1) | Black = Spin Down (-1)

Magnetization vs Time

Temperature (T): 2.27 J/k_B

⚠️ Near Critical Temperature

External Field (h): 0.00 J

Lattice Size: 100 × 100

Simulation Speed: 10 steps/frame

Magnetization

0.00

Energy/Spin

0.00

Steps