I am reading Rubinstein's polymer physics book and on p. 78 it says that if $m$ is the number of monomers within range $r$ of an arbitrary monomer, then $$ m \approx (r/b)^2 $$
where $b$ is the bond length in the chain. The book says that this equation is given by "random walk statistics". I was wondering if someone would be able to provide more information on how this equation is acquired than that.
A very simplified model of a polymer is given as follows:
Start from a monomer, select a random direction in space at a distance roughly $b$ and place a new monomer. From this new monomer, select a random direction and place a new monomer at distance $b$, etc until you have placed $m$ monomers. You have formed a polymer! This of course neglects interaction between monomers as you can have 2 monomers very close to each others which is unphysical. This also neglects preferential angle attachment. But this is a first approximation.
What you have, mathematically, is a random walk here, the position of the $m$th monomers is given by:
$$\boldsymbol R_m=\sum_{i=1}^m b\boldsymbol u_i$$
With $\boldsymbol u_i$ a vector with unit lenght but random direction. From the central limit theorem, you obtain that $\left\langle \left|\sum_i^m\boldsymbol u_i\right|\right\rangle\propto \sqrt m$
From which you find on average the result given above: $b\sqrt m \propto \langle|\boldsymbol R_m| \rangle$.
This is a bit different from the formulation he used, here it is understood as: the average position of the last polymer $\langle|\boldsymbol R_m| \rangle$ grows with the number of monomers squared. Which is not exactly what he wants to measure, which is the radius (of gyration) of the polymer. But it's close enough as it will still be given by, roughly, the variance.
Concerning the average, let's look at $\langle \boldsymbol R_m^2 \rangle$. It's not exactly $\langle |\boldsymbol R_m|\rangle^2$ but it's close. Anyway, we are looking for the gyration radius (squared) technically, which is not really either of these quantities.
$$\langle \boldsymbol R_m^2 \rangle=b^2\left\langle \sum_{i}^m \boldsymbol u_i\cdot\sum_{j}^m \boldsymbol u_j \right\rangle=b^2\left\langle \sum_{i}^m\sum_{j}^m \boldsymbol u_i\cdot \boldsymbol u_j \right\rangle,$$ since $\boldsymbol u_i$ is uncorrelated from $\boldsymbol u_j$ if $i\neq j$, only the terms in $i=j$ will contribute $\langle \boldsymbol u_i\cdot \boldsymbol u_j\rangle=\delta_{ij}$ (recall that $\boldsymbol u_i^2=1$), therefore we get: $$\langle \boldsymbol R_m^2 \rangle=b^2\sum_{i}^m\sum_{j}^m \delta_{ij}=b^2m$$
It tells you that the variance of $\boldsymbol R_m$ is proportional to $m$. This could also be concluded from the central limit theorem. So to conclude, of course, $\langle \boldsymbol R_m\rangle = 0$, but the typical size: $\sqrt{\langle \boldsymbol R_m^2\rangle}\sim \langle |\boldsymbol R_m|\rangle\sim \sqrt{m}$
