Mean Squared Displacement

The mean-squared dislacement of a particle is defined simply as, $$\rho(t) = \langle r^2(t) \rangle = \int c(r) p(r,t) dr,$$ where $c(r) = 1$, $2 \pi r$, and $4 \pi r^2$ in 1, 2, and 3 dimensions, respectively. This evaluates to $\langle r^2(t) \rangle = 2dDt$, where $d$ is the dimension (1, 2, or 3).

One can also compute the variance of the MSD, as $$\text{var}(\rho) = \langle \rho^2(t) \rangle - \left(\langle \rho(t) \rangle\right)^2.$$ This can be evaluated as,
$$\begin{align} 1D:& 2\,(2Dt)^2 = 8D^2t^2\\ 2D:& \dfrac{2}{2} (4Dt)^2 = 16 D^2 t^2\\ 3D:& \dfrac{2}{3} (6Dt)^2 = 24 D^2 t^2 \end{align}$$
This can be simplified into a common expression as, $$\text{var}(\rho) = \dfrac{2}{d} \rho^2$$