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Equation 2 · Part 12 · Einstein's Random Walk and the Mathematics of Genetic Drift

Numerator: partial^2 f(x,t)

∂f(x,t)∂t=D ∂2f(x,t)∂x2\frac{\partial f(x,t)}{\partial t} = D\,\frac{\partial^{2} f(x,t)}{\partial x^{2}}
∂2f(x,t)\partial^{2} f(x,t)

What this part means

The complete quantity above the fraction bar.

Its job in the formula

partial2l^2 f(x,t) occurs above the fraction bar. The numerator is divided by the entire denominator below it.

The passage around this formula

Working through the statistical mechanics of a wall permeable to solvent but not to suspended particles, Einstein arrived at a differential equation for how the concentration f(x,t) of particles along one axis changes over time: ∂f(x,t)∂t=D ∂2f(x,t)∂x2\frac{\partial f(x,t)}{\partial t} = D\,\frac{\partial^{2} f(x,t)}{\partial x^{2}}. 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

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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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