Mean squared displacement ($⟨|X(t+\tau )-X(t){|}^2⟩$)

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.