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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)

What would be the best way to go about this?

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I've made some significant changes to make the question more understandable/clear. Please edit it if you feel that I have incorrectly modified the question. –  Kyle Kanos Mar 20 at 12:51
    
What is being diffused here? A single gas in ambient? A few different gases in ambient? Something else? –  Kyle Kanos Mar 20 at 12:55

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