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.

  • $\begingroup$ From looking around for an answer, I think a helpful search term would be the Worm-like chain and freely jointed chain models. $\endgroup$
    – Tyberius
    Commented Jul 4, 2022 at 1:16
  • $\begingroup$ Was that comment by Tyberius helpful? Please let us know! $\endgroup$ Commented Feb 6, 2023 at 18:43


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