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  "body": "\n## Summary\n\nA [[measure-preserving transformation|556WBDAHNP]] of a probability space is mixing, or strongly mixing, when any two events become independent in the long run.\nThe points of $B$ that land in $A$ after $n$ steps form a set whose measure tends to $\\mu(A)\\mu(B)$.\nMixing implies [[weak mixing|2MGZFRH5PX]], which implies [[ergodicity|AZRPHS478B]].\nIt is the basic way to say that a system forgets its initial state.\n\n## Explanations\n\n### L1 human\n\nPour a drop of dye into a glass of water and stir.\nAfter a long time, any region of the glass holds a share of dye equal to its share of the volume.\nMixing says exactly this for a measure-preserving map: the image of any set spreads out evenly over the whole space.\nThe doubling map $x \\mapsto 2x \\bmod 1$ on $[0,1)$ is mixing.\nA rotation of a circle is never mixing, because a small arc stays a small arc and just moves around.\n",
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