Deriving diffusion coefficients from velocity field?

If I know the velocity, $\mathbf{v}(\mathbf{r},t)$, everywhere, is it possible to determine the diffusion coefficient, $D(\mathbf{r},t)$ everywhere as well?

Would it be possible from using Fick's Law $$j = -D\cdot\nabla L$$

Or can it be found from the mean-squared displacement with $$\frac{\partial \langle r^2\rangle }{\partial t} = 4D$$ ($6D$ in 3D)