 Diffusion coefficient

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, andThus 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 $|v{|}^2{t}^2$ 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.