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  "body": "\n## Summary\n\nThe heat equation $\\partial_t u = \\Delta u$ describes how temperature, or the concentration of a diffusing substance, evolves in time.\nHere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] in the space variables.\nIt is the model [[parabolic partial differential equation|1CCTH8X2FT]].\nIts solutions become smooth at once, spread information at infinite speed, and decay toward equilibrium.\nIt is linked to probability through the [[Wiener process|22735FGJV6]], and to geometry through the spectrum of the Laplacian.\n\n## Explanations\n\n### L1 human\n\nHeat flows from warm places to cold places, at a rate that depends on how uneven the temperature is.\nThe Laplacian measures how much the temperature at a point differs from the average around it.\nThe heat equation says that a point warms up when it is colder than its surroundings and cools down when it is warmer.\nOn a rod, $u(x, t) = e^{-t} \\sin x$ is a solution: a sine-shaped temperature profile that keeps its shape and fades away.\n",
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