The Mean First Rotation Time of a planar polymer

TL;DR

This paper derives an asymptotic formula for the mean first rotation time of a planar polymer modeled as n rods with Brownian angles, revealing dependence on the square root of n and weak dependence on initial configuration.

Contribution

It introduces a novel stochastic equation for the polymer's free end and provides an asymptotic expression for the mean rotation time, confirmed by simulations.

Findings

Mean rotation time scales with the square root of the number of rods.

The initial configuration has a weak influence on the MRT.

Analytical results are validated through Brownian simulations.

Abstract

We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of n rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on the square root of n and, interestingly, depends weakly on the mean initial configuration. Our analytical results are confirmed by Brownian simulations.Our analytical results are confirmed by Brownian simulations.