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Equation 13 · Part 4 · Einstein's Random Walk and the Mathematics of Genetic Drift

Symbol mu

∂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],
μ\mu

What this part means

mu is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

mu is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

…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 right-hand term carries the probability density along a deterministic drift μ(x,t)\mu(x,t) and the second spreads it by diffusion at rate D(x,t) [ 3 ] . Andrei Kolmogorov arrived at the identical equation independently in 1931, from pure probability theory, as the forward equation governing how the probability distribution of any continuous-time Markov process evolves [ 3 ] .…

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A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the surrounding passage

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