# Stochastic curvature of enclosed semiflexible polymers

## TL;DR

This paper models the stochastic conformations of enclosed semiflexible polymers using curvature-based equations, deriving a probabilistic framework and algorithms to analyze shape transitions within compact domains.

## Contribution

It introduces a stochastic curvature model for semiflexible polymers, derives the Hermans-Ullman and Telegrapher's equations, and develops a Monte Carlo simulation method for confined polymer conformations.

## Key findings

- Positional probability density follows a Telegrapher's equation when 2a/ℓp > 1.
- Identifies a shape transition in polymers within square domains at critical persistence length.
- Shows oscillating end-to-end distance behavior for ℓp > a/8 in square confinement.

## Abstract

The conformational states of a semiflexible polymer enclosed in a compact domain of typical size $a$ are studied as stochastic realizations of paths defined by the Frenet equations under the assumption that stochastic "curvature" satisfies a white noise fluctuation theorem. This approach allows us to derive the Hermans-Ullman equation, where we exploit a multipolar decomposition that allows us to show that the positional probability density function is well described by a Telegrapher's equation whenever $2a/\ell_{p}>1$, where $\ell_{p}$ is the persistence length. We also develop a Monte Carlo algorithm for use in computer simulations in order to study the conformational states in a compact domain. In addition, the case of a semiflexible polymer enclosed in a square domain of side $a$ is presented as an explicit example of the formulated theory and algorithm. In this case, we show the existence of a polymer shape transition similar to the one found by Spakowitz and Wang [Phys. Rev. Lett. {\bf 91}, 2 (2003)] where in this case the critical persistence length is $\ell^{*}_{p}\simeq a/8$ such that the mean-square end-to-end distance exhibits an oscillating behavior for values $\ell_{p}>\ell^{*}_{p}$, whereas for $\ell_{p}<\ell^{*}_{p}$ it behaves monotonically increasing.