Diffusion length ($\sqrt{D/\mu }$)

For diffusion coefficient $D>0$ and first-order removal rate $\mu >0$, the diffusion length isIt measures the spatial range reached when diffusion competes with removal on the mean timescale $\tau =1/\mu$.

For example, a one-dimensional reaction-diffusion equation with constant coefficients isA steady concentration maintained at $c(0)=c_0$ in $x>0$, decaying at infinity, satisfiesSubstitution gives $D/{\ell }^2=\mu$. Thus $\ell$ is an exact decay length for this geometry.

A root-mean-square displacement uses a different numerical convention. For ordinary diffusion in $d$ dimensions and an independent exponential removal time with mean $\tau$, the mean squared displacement at removal is $2dD\tau$; its square root is $\sqrt{2d}\ell$. Calling $\ell$ a characteristic travel distance does not equate these two definitions.