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I am considering the Zimm model for polymer dynamics, and have come across a question

Find an expression for the time it takes for the polymer to diffuse a distance equal to its contour length $L=Nb$, if the drag coefficient for the polymer is $\gamma = N\beta b$ where the polymer consists of $N$ segments of Kuhn length $b$.

My thoughts on this question were:

  • For Fickian diffusion $\langle R^2 \rangle = 2Dt$
  • So if we plug in $D=\frac{k_BT}{\gamma}$ and $R=L$, we should get the right answer?

For some reason, I am not convinced that "diffusing a distance $L$" translates to $\langle R^2 \rangle = \langle L^2 \rangle$.

I'm not sure if this should be obvious or not. Unfortunately, I do not have answers to compare with.

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I guess that should be that easy; you may swap $L$ with any other length... the only thing that could be bit fishy for me is that $\sqrt {\langle R^2 \rangle}$ (the position variance) is not necessarily equal to $\langle \vert R \vert \rangle$ (the average diffusion distance). But usually I use them interchangeably.

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  • $\begingroup$ Yes, that was my concern! $\endgroup$
    – Meep
    Commented Apr 30, 2019 at 14:08
  • $\begingroup$ Ok! you may try to see how much they differ for Gaussian distribution $\endgroup$
    – patta
    Commented Apr 30, 2019 at 14:42

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