= Diffusion coefficient
{title2=$D$}
= Diffusivity
{synonym}
The diffusion coefficient $D$ measures the rate at which <diffusion> spreads positions; it has units of squared <length> divided by <time>. For isotropic free <diffusion> in $d$ dimensions, $X_t=X_0+\sqrt{2D}\,W_t$ for standard $d$-dimensional <Brownian motion>, and
$$
\mathbb E|X_t-X_0|^2=2dDt.
$$
Thus one-coordinate <mean squared displacement> is $2Dt$, whereas three-dimensional radial <mean squared displacement> is $6Dt$. The corresponding <diffusion equation> is $\partial_t p=D\Delta p$; for a scalar <Itô diffusion> with constant <diffusion amplitude> $\sigma$, this convention gives $D=\sigma^2/2$.
A constant <drift> $v$ adds $|v|^2t^2$ to the <mean squared displacement> but leaves the displacement <covariance> equal to $2DtI$. Diffusion coefficients inferred from uncentered displacement data therefore require care when there is flow. The <Green-Kubo formula for diffusion> connects $D$ to an equilibrium <velocity autocorrelation function>.
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