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Equation 3 · Part 4 · 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

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

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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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

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Sources cited in the surrounding passage

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