# 3D rotational diffusion coefficient and angular mean-squared displacement

I would like to calculate the rotational diffusion coefficient of a sphere to verify that my simulation algorithm is working correctly.

I know that for 2D rotation (e.g. about a single axis), the mean-squared angular displacement is given by

$$\langle\theta^2 \rangle = 2D_r t$$

but have not found the 3D equivalent. How does the coefficient change with additional degrees of freedom? Also looking for citations and links.

EDIT: I have found the answer, however if anyone could give me the derivation or the original citation, I will choose that as the best answer.

$$\langle\theta^2\rangle = 4D_Rt$$