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Suppose we have a 2D polymer model described by a set of 2D vectors {$\mathbf{t}_i$} ($i=1,2,\dots N$) of length $a$.

The energy of the polymer is given by: $$ \mathcal{H}~=~-k\sum^N_{i=1}\mathbf{t}_i\cdot\mathbf{t}_{i+1} ~=~-ka^2\sum^N_{i=1}\cos\phi_i. $$

The constant $k$ is a measure of bending rigidity so that a probability of finding any configuration of the polymer is proportional to $e^{-\frac{\mathcal{H}}{k_BT}}$.

Essentially, this is a discrete version of the wormlike-chain model by Kratky and Porod.

  1. What is the expression for propability density $p(\mathbf{R})$, where $\mathbf{R}$ is an end-to-end vector $\sum_i\mathbf{t_i}$, for this model? Let's assume that $N \rightarrow \infty$.

  2. How do I find linear force extension relation,when a force $\mathbf{F}$ is applied to the ends of the chain? Let's assume that the force is weak.

  3. Can anyone point out useful literature?

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  • $\begingroup$ Cool question, +1. I don't know the answer but it looks like the probability distribution should be a function of $|\mathbf{R}|^2$ only, since there are no external fields to break rotational invariance. Maybe try to relate the Hamiltonian to $|\mathbf{R}|^2$. For part 2), once you have the probability distribution p($\mathbf{R}$) you can find the expectation value of the energy $<U(\mathbf{R})>$. This should have a minimum at the equilibrium position. Expand about the minimum to second order; the spring constant is $\partial_R^2<U(\mathbf{R}_0)>$. I look forward to seeing the answer! $\endgroup$ Feb 6, 2013 at 23:36
  • $\begingroup$ Correction: you want to find the minimum of the free energy $F(\mathbf{R}) = <U(\mathbf{R})> - T S(\mathbf{R})$, since your chain is in contact with a heat bath. You can calculate this from the partition function by the usual procedure. $\endgroup$ Feb 7, 2013 at 0:30
  • $\begingroup$ I think $\mathbf{R}$ follows Gaussian distribution with <$\mathbf{R}$> =0 and <$\mathbf{R}^2$> to be determined. Am I right? $\endgroup$
    – molkee
    Feb 7, 2013 at 3:56
  • $\begingroup$ Seems equivalent to asking the magnetization of a classical XY chain. This is certainly well known (and probably amenable to a transfer matrix computation). Or am I missing something? $\endgroup$ Feb 7, 2013 at 13:01
  • $\begingroup$ A good book is Statistical Physics of macromolecules, by Grosberg and Khokhlov, amazon.com/… Already the first chapter addresses the question of the end-to-end distance, although from a more general prospective. $\endgroup$
    – Roger V.
    Apr 13, 2020 at 12:31

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This question has been worked out in the first chapter of "Statistical Physics of DNA: An Introduction to Melting, Unzipping and Flexibility of the Double Helix", by Nikos Theodorakopoulos. The first chapter is freely available and can be found here: https://www.worldscientific.com/doi/pdf/10.1142/9789811209543_0001. As can be seen, there is no closed form for this problem. Nevertheless, since you are pointing out the limit of $N \rightarrow \infty$, a good approximation is to rescale your average intermonomer distance $a$ to the value of the persistence length $l_p = \kappa/(k_BT)$ and then use it as a freely jointed chain. The bending stiffness will then not be important anymore making this a valid approximation. For the freely jointed chain, the exact expression is also given in the link. Notice also that the freely jointed chain for $N \rightarrow \infty$ becomes a Gaussian.

So to wrap up and give a short answer: For your problem, a Gaussian distribution with an intermonomer distance of $a = \kappa/(k_BT)$ would be a good approximation for $N \rightarrow \infty$. Hope this answers your question :)

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