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Equation 29 · Einstein's Random Walk and the Mathematics of Genetic Drift

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u(p)=p(s→0).u(p) = p \qquad (s \to 0).

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Inputs and operationsp qquad (s to 0)
Result or conditionu(p)
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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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uu

Symbol u

u is part of the quantity the equation computes from the expression on the right.

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pp

Symbol p

the starting from frequency.

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ss

Symbol s

s 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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as verified directly from his derivation [ 7 ] . Let the selective advantage vanish — the allele is neutral, indistinguishable from its alternatives in its effect on survival or reproduction — and the exponentials cancel term by term, leaving u(p)=p(s→0)u(p) = p \qquad (s \to 0). The probability that a neutral allele eventually takes over the population is exactly its own starting frequency, nothing more. No fitness calculation, no bookkeeping of selection coefficients, is needed to answer the single most consequential question about a new mutation’s fate; the diffusion equation alone answers it.

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