Proof — The Wulff construction theorem
Statement
The Wulff construction predicts the equilibrium shape of a crystal or macroscopic droplet from anisotropic surface tension (or free energy per unit interface area) γ on the unit sphere: the stable shape is the convex body W_γ = ⋂_{u∈S^{d−1}} { x ∈ ℝ^d : x·u ≤ γ(u) } (equivalently, the convex hull of polar data). In rigorous statistical mechanics—especially for the two-dimensional Ising ferromagnet at low temperature—large microscopic droplets and phase-separation contours converge (in the appropriate scaling and sense) to this Wulff shape. Thus a macroscopic variational principle is recovered from local lattice interactions. Theorems of this type are standard rigorous results, formalizable in ZFC.
Formula
W_γ = ⋂_{||u||=1} { x : x·u ≤ γ(u) } (Wulff crystal / Winterbottom variant with substrate)
Equation
Ising droplet → Wulff shape (2D low T; Dobrushin–Kotecký–Shlosman program)
Proof sketch
Dobrushin, Kotecký, and Shlosman developed a complete microscopic justification for the 2D Ising model: surface tension, large deviations for contours, and convergence to the Wulff minimizer. Related work addresses other models and dimensions.
Source
Wulff, G. Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Krystallflächen. Z. Kristallogr. 34 (1901), 449–530. Dobrushin, R. L.; Kotecký, R.; Shlosman, S. Wulff Construction: A Global Shape from Local Interaction. American Mathematical Society Translations of Mathematical Monographs, vol. 104, AMS, Providence, RI, 1992.
Domains
mathematics, physics