I consider a worm-like chain with the Hamiltonian given by: \begin{equation} H(L)=\frac{\kappa}{2}\int_{0}^{L}ds\left(\frac{\partial\vec{t}}{\partial s}\right)^2 \end{equation} where $\kappa$ is the bending module $L$ the length of the polymer and $\vec{t}$ unit vector tangent to the polymer.

I try to calculate its partition function. After some lengthy calculations (that I could add if it is necessary) I got that it is equal to 1.

That result is more than unintuitive for me. It would imply that its free energy is always equal to 0 and therefore its behaviour is purely entropic. But here we indeed have non-zero bending energy so I would expect highly curved polymer to be less probable as it requires some energy to bend it. Therefore I would expect free energy entropy competition determining its behaviour.

Is that result correct? If yes then why and why my intuition is wrong?

  • $\begingroup$ Have you read Marantan and Mahadevan, "Mechanics and statistics of the worm-like chain," Am J Phys (2018)? $\endgroup$ Commented Nov 17, 2022 at 19:11
  • $\begingroup$ no, thank you for the resource! Could you maybe refer to a specific part of the article? $\endgroup$ Commented Nov 17, 2022 at 21:56
  • $\begingroup$ Perhaps the section on calculating the partition function. $\endgroup$ Commented Nov 17, 2022 at 22:36


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