# Intrpretting questions on Fickian diffusion

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.

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.