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Equation 13 · Part 7 · Einstein's Random Walk and the Mathematics of Genetic Drift

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∂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],
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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

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

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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