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Equation 2 · Part 6 · Einstein's Random Walk and the Mathematics of Genetic Drift

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∂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}}
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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

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 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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