Crystal growth in an undercooled melt is governed by diffusion of heat (or solute). The flat interface is linearly unstable (Mullins-Sekerka 1963): perturbations grow because protrusions see steeper diffusion gradients and grow faster — the tip-splitting instability.
The dendritic (snowflake) shape emerges from competition between this instability and surface tension anisotropy ε (crystal prefers certain facet directions). Higher ε → sharper tips, cleaner dendrite arms; ε=0 → fractal DLA (diffusion-limited aggregation). Undercooling Δ controls growth rate.
This simulation uses the phase-field model — a continuous order parameter φ(r,t) interpolates between solid (φ=1) and liquid (φ=0), avoiding explicit interface tracking. The anisotropy function A(θ) = 1+ε·cos(m·θ) enforces m-fold symmetry.