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  "body": "\n## Summary\n\nThe Kardar–Parisi–Zhang (KPZ) equation is the [[stochastic partial differential equation|PRCQG82QTK]] $h_t = \\nu h_{xx} + \\frac{\\lambda}{2} (h_x)^2 + \\xi$, where $\\xi$ is space-time white noise.\nIt was proposed by Kardar, Parisi and Zhang in 1986 as a model for a randomly growing interface.\nIts solutions are so rough that the square $(h_x)^2$ has no classical meaning, and giving the equation a meaning is a hard problem.\nIt is the centre of the KPZ universality class, which contains many growth and particle models.\nThe large-time fluctuations in this class are described by distributions from random matrix theory, such as the Tracy–Widom distribution.\n\n## Explanations\n\n### L1 human\n\nPicture the edge of a burning sheet of paper or a growing bacterial colony.\nThe height $h$ of the edge grows faster where it is tilted, is smoothed by the second derivative, and is kicked by random noise.\nThe fluctuations of such an interface grow like $t^{1/3}$, not like $t^{1/4}$ as for the linear equation without the square term.\nThe [[Cole–Hopf transformation|VBMQ9SJ3DH]] $Z = e^{\\lambda h/(2\\nu)}$ turns the equation formally into a linear heat equation with random multiplicative noise.\n",
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