Allen-Cahn Phase Field Equation
∂φ/∂t = ε²Δφ − W'(φ) — interface motion and phase coarsening
Parameters
Interface width ε
0.05
Time step dt
0.1
Speed
3
x
Initial Conditions
Random
Bubbles
Stripes
Checkers
Maze-like
Draw
Stop Draw
Observables
Step
0
Phase + (φ>0)
–
Energy F
–
Physics
The Allen-Cahn equation models a
diffuse interface
between two phases (φ = ±1).
W(φ) = ¼(φ²−1)² is the double-well potential. The equation is the
L² gradient flow
of the Ginzburg-Landau energy:
F[φ] = ∫ [ε²/2|∇φ|² + W(φ)] dx
Key behaviors:
• Interfaces sharpen (ε controls width)
• Small bubbles shrink (curvature-driven)
• Ostwald ripening: big domains grow
• Mean curvature flow in sharp-interface limit (ε→0)