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  "body": "\n## Summary\n\nThe Allen–Cahn equation is the [[reaction–diffusion equation|TAAHWE917M]] $u_t = \\Delta u - \\varepsilon^{-2}(u^3 - u)$.\nIt models phase separation in a binary alloy, where $u = \\pm 1$ are the two pure phases.\nIt is the gradient flow in $L^2$ of the Ginzburg–Landau energy.\nAs $\\varepsilon \\to 0$, the thin interfaces between the phases move approximately by [[mean curvature flow|2YYV01C7QQ]].\n\n## Explanations\n\n### L1 human\n\nPicture a material that wants to be in one of two states, labelled $+1$ and $-1$.\nThe cubic term pushes each point toward one of the two states, and the Laplacian smooths out the differences between neighbours.\nAfter a short time the material splits into regions of the two phases, separated by thin transition layers.\nIn one dimension a stationary layer is $u(x) = \\tanh\\!\\big(x / (\\sqrt{2}\\,\\varepsilon)\\big)$, which goes from $-1$ to $+1$.\n",
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