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Equation 3 · Part 5 · Einstein's Random Walk and the Mathematics of Genetic Drift

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⟨x2⟩=2Dt,\langle x^{2} \rangle = 2Dt,
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What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

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A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

the diffusion equation, in a form still written the same way today [ 1 ] . Solving it for particles released from a point gives a spreading distribution whose variance grows linearly, not with the square root of time as a naive guess might suggest but as ⟨x2⟩=2Dt\langle x^{2} \rangle = 2Dt. with the diffusion constant D fixed by measurable quantities. Einstein derived it explicitly, from the balance between the drag on a sphere of radius P in a fluid of viscosity k and the osmotic pressure driving diffusion, as

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An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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