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.
