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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.

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In 3D it is

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

as given for example in this paper.

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