# Statistical properties of semi-flexible polymers

I am reading Statistical Physics of Particles by Kardar. I am struggling with problem 12d, in chapter 2, about semi-flexible polymers in two dimensions. The problem is as follows:

Configurations of a model polymer can be described by either a set of vectors $$\{ t_i \}$$ of length $$a$$ in two dimensions (for $$i=1,\ldots, N$$), or alternatively by angles $$\{ \phi _i \}$$ between successive vectors. The polymer is set at temperature $$T$$, and subject to energy $$H = -k \sum _{i=1}^{N-1} t_i \cdot t_{i+1} = -ka^2 \sum _{i=1}^{N-1} \cos \phi _i$$

The probability of a certain configuration is given by $$\exp (-H/kT)$$.

If the end of the polymer are pulled apart by a force $$F$$, the probabilities for polymer configurations are modified by the Boltzmann weight $$\exp (\mathbf{F}\cdot \mathbf{R}/kT)$$, by expanding this weight, or otherwise, show that $$\langle \mathbf{R} \rangle = K^{-1} \mathbf{F} + O\left( F^3 \right)$$

I know that $$\langle \mathbf{R} \rangle = \frac{\int \mathbf{R} \exp [-\beta H + \beta (\mathbf{F}\cdot \mathbf{R})] d\phi}{\int \exp [-\beta H + \beta (\mathbf{F}\cdot \mathbf{R})] d\phi}$$

From the previous parts of the problem, I know the $$\langle \mathbf{R} \rangle = 0$$ and $$\langle R^2 \rangle = a^2 N \coth \frac{1}{2\xi}$$, where $$\xi$$ is the persistence length.

My question is, how do I expand $$\langle \mathbf{R} \rangle$$? Can anyone give me a mathematically rigorous introduction to expanding such functions? I understand what Taylor expansions are, but how do I apply those principles to an object like $$\langle \mathbf{R} \rangle$$ described above?

I see that $$\begin{equation} \exp(\beta({\bf F}\cdot {\bf R})) = 1 + \beta({\bf F}\cdot {\bf R}) +\frac{1}{2}\left [\beta({\bf F}\cdot {\bf R})\right]^2 +\frac{1}{3!}\left [\beta({\bf F}\cdot {\bf R})\right]^3+... \end{equation}$$

But I do not see how to use this to get the desired result.

I would appreciate any advice you have for me.

• From looking around for an answer, I think a helpful search term would be the Worm-like chain and freely jointed chain models.
– Tyberius
Jul 4, 2022 at 1:16
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