Diffusion coefficient ($D$)

The diffusion coefficient sets the linear growth of mean-square displacement. In one dimension the convention $\mathbb{E}[(X_t-X_0{)}^2]=2Dt$ is common.

A mean squared displacement is the expected value of a squared position change over a specified time lag. For isotropic free diffusion in $d$ dimensions, it equals $2dD\tau$. A nonzero mean drift contributes its squared displacement as well as the displacement variance; specifying the tracked coordinates and whether the mean is subtracted is essential.

For isotropic Brownian motion started at zero with independent coordinate variances $2Dt$, the Gaussian distribution has density $(4\pi Dt{)}^{-3/2}{e}^{-|x{|}^2/(4Dt)}$. Integrating the radius with spherical volume element $4\pi r^{2}dr$ gives $\mathbb{E}|X_t|=4\sqrt{Dt/\pi }$. The root mean square displacement is instead $\sqrt{6Dt}$, and the mean vector is zero. None of these is the microscopic path length.

Solutal diffusivity is the diffusion coefficient governing transport of a dissolved constituent down its concentration gradient. For uniform $D$ and a stationary solvent, the diffusion equation is $C_t=D{\nabla }^{2}C$. It need not equal the thermal diffusivity.