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Equation 2 · Part 5 · Einstein's Random Walk and the Mathematics of Genetic Drift

Symbol x^2

∂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}}
x2x^{2}

What this part means

The square of x: multiply x by itself.

Its job in the formula

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

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

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

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Sources cited in the surrounding passage

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