the Allen-Cahn equation
/ AL-en KAHN /
Mix two phases of a material — say two crystal orientations in an alloy, or oil and water — and watch the boundary between them. The interface does not sit still; it tightens and moves to reduce its area, much like a soap film pulling itself flat. The Allen-Cahn equation is the simplest PDE that models exactly this: a field that wants to be in one of two states almost everywhere, with thin transition layers between them that evolve so as to smooth and shrink.
The equation is u_t = epsilon^2 Laplacian u + u - u^3, a reaction-diffusion equation whose reaction f(u) = u - u^3 = -W'(u) comes from the double-well potential W(u) = (1/4)(1 - u^2)^2. That potential has two equal minima at u = +1 and u = -1 (the two phases) and an unstable hump at u = 0. So the dynamics drive u toward +1 or -1 almost everywhere, while diffusion forbids a jump, forcing a smooth transition layer of width about epsilon. Where two phases meet, this layer is the diffuse interface. The beautiful result is that, as epsilon goes to 0, the interface moves by mean-curvature flow: each point of the boundary moves inward proportional to how curved it is. A bump flattens, a small circular domain shrinks and disappears in finite time. This is a gradient flow: it decreases the Ginzburg-Landau energy integral of (epsilon^2/2)|grad u|^2 + W(u).
Why care? Allen-Cahn is the model problem for interface dynamics and a cornerstone of the 'phase-field' method used to simulate crystal growth, solidification and image processing. It is the natural sibling of Fisher-KPP — same reaction-diffusion frame — but with a BISTABLE reaction (two stable states), so instead of one state invading another you get domains of each phase competing and coarsening. It is also the cleanest analytic bridge between a PDE and a geometric flow: solving Allen-Cahn approximately solves mean-curvature flow.
Start with u = +1 inside a circle of radius R and u = -1 outside, with a thin diffuse layer between. The circle shrinks: its radius obeys roughly dR/dt = -1/R, so R(t)^2 = R(0)^2 - 2 t and the domain vanishes at time t = R(0)^2 / 2. Smaller bubbles die first — exactly the coarsening you see when a foam ages.
A double-well reaction makes interfaces move by their curvature.
Despite the resemblance, Allen-Cahn does NOT conserve the total of u — bubbles simply vanish. The conservative cousin, where the total amount of each phase is preserved, is the Cahn-Hilliard equation, a fourth-order PDE with quite different dynamics.