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Equation 13 · Part 12 · Einstein's Random Walk and the Mathematics of Genetic Drift

Numerator: partial p(x,t)

∂p(x,t)∂t=−∂∂x[μ(x,t) p(x,t)]+∂2∂x2[D(x,t) p(x,t)],\frac{\partial p(x,t)}{\partial t} = -\frac{\partial}{\partial x}\big[\mu(x,t)\,p(x,t)\big] + \frac{\partial^{2}}{\partial x^{2}}\big[D(x,t)\,p(x,t)\big],
∂p(x,t)\partial p(x,t)

What this part means

The complete quantity above the fraction bar.

Its job in the formula

partial p(x,t) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

Einstein’s equation described one physical system: particles buffeted by a fluid held at a fixed temperature, governed by a single constant D . Within a decade, physicists working on an unrelated problem — how the energy of a rotating electric dipole settles under both random radiative kicks and a systematic damping force in a radiation field — found they needed the same mathematical object with one addition: a drift term standing alongside the diffusion term. Adriaan Fokker wrote down that generalization in 1914, and Max Planck rederived and extended it in 1917; the combined equation has carried both names since [ 3 ] . In one dimension it reads ∂p(x,t)∂t=−∂∂x[μ(x,t) p(x,t)]+∂2∂x2[D(x,t) p(x,t)]\frac{\partial p(x,t)}{\partial t} = -\frac{\partial}{\partial x}\big[\mu(x,t)\,p(x,t)\big] + \frac{\partial^{2}}{\partial x^{2}}\big[D(x,t)\,p(x,t)\big]. where the first…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

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