---
tag: 7S813PEV69
kind: concept
name: Allen–Cahn equation
msc: [35K58, 35K57]
specializes: [TAAHWE917M]
sources:
  - url: https://en.wikipedia.org/wiki/Allen%E2%80%93Cahn_equation
same_as:
  wikidata: Q4732089
  wikipedia: Allen–Cahn equation
---

## Summary

The Allen–Cahn equation is the [[reaction–diffusion equation|TAAHWE917M]] $u_t = \Delta u - \varepsilon^{-2}(u^3 - u)$.
It models phase separation in a binary alloy, where $u = \pm 1$ are the two pure phases.
It is the gradient flow in $L^2$ of the Ginzburg–Landau energy.
As $\varepsilon \to 0$, the thin interfaces between the phases move approximately by [[mean curvature flow|2YYV01C7QQ]].

## Explanations

### L1 human

Picture a material that wants to be in one of two states, labelled $+1$ and $-1$.
The cubic term pushes each point toward one of the two states, and the Laplacian smooths out the differences between neighbours.
After a short time the material splits into regions of the two phases, separated by thin transition layers.
In one dimension a stationary layer is $u(x) = \tanh\!\big(x / (\sqrt{2}\,\varepsilon)\big)$, which goes from $-1$ to $+1$.