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Equation 13 · Part 17 · Einstein's Random Walk and the Mathematics of Genetic Drift

Denominator: partial x^2

∂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],
∂x2\partial x^{2}

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

partial x2x^2 occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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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