= Diffusive broadening of electrophoretic bands
{title2=$\sigma^2(t)=\sigma_0^2+2Dt$}
A narrow population band undergoing uniform <electrophoresis> and ordinary <diffusion> has mean position $x_0+\mu Et$ and <variance> $\sigma_0^2+2Dt$. This follows from the <advection-diffusion equation> by changing to the coordinate $x-\mu Et$. For two species with mobility difference $\Delta\mu$, their centre separation is $|\Delta\mu|Et$, whereas diffusion widths grow only as $\sqrt t$. Starting from point-like bands with common <diffusion coefficient> $D$, a required separation of $q$ standard deviations needs
$$
t\geq\frac{2q^2D}{(\Delta\mu E)^2}.
$$
A prescribed distance between centres alone is a resolution convention; finite loading widths and other dispersion mechanisms must be included when interpreting actual <electrophoresis> data.
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