Diffusion coefficient ($D$)

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.

For a stationary zero-mean velocity with an absolutely integrable velocity autocorrelation function $\chi$, the diffusion coefficient is its time integral. Indeed, differentiation of the mean squared displacement identity gives $\frac{1}{2}\frac{d}{dt}\mathbb{E}[(X(t)-X(0){)}^2]={\int }_0^t\chi (u)du$, and the right side tends to the displayed integral. In the equilibrium generalized Langevin equation, this gives $D=\Theta /\tilde{\kappa }(0)$ whenever stable relaxation and the required integrability hold. The formula is named after Melville S. Green and Ryogo Kubo.

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.

The unnormalized autocorrelation function of a stationary velocity. For zero mean, $\chi (0)$ is the velocity variance. It is different from the normalized autocorrelation function $\chi (u)/\chi (0)$ often used in time series.

If $X(t)-X(0)={\int }_0^{t}V(s)ds$, integrating the two-time covariance over the square $[0,t{]}^2$ givesThis identity connects a mean squared displacement to the Green-Kubo formula for diffusion.

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.