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

What does this equation mean?

⟨x2⟩=2Dt,\langle x^{2} \rangle = 2Dt,

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Inputs and operations2Dt
Result or conditionlangle x^2 rangle
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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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x2x^{2}

Symbol x^2

The square of x: multiply x by itself.

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DD

Symbol D

D is an input to the expression that computes the quantity on the left.

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tt

Symbol t

t is an input to the expression that computes the quantity on the left.

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=

=

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

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superscript

superscript

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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How to interpret it

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What the article says around this equation

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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