Winding Angle Distributions for Directed Polymers

TL;DR

This paper analyzes the distribution of winding angles for directed polymers, revealing exponential decay in unconfined cases, Gaussian behavior under confinement, and reduced entanglements in random media, with detailed analytical and numerical results.

Contribution

It provides a comprehensive analytical and numerical study of winding angle distributions for directed polymers, including effects of confinement, chirality, and randomness, highlighting new universal and non-universal behaviors.

Findings

Unconfined polymers have exponentially decaying winding angle distributions.

Confined polymers exhibit Gaussian winding angle distributions with variance proportional to polymer length.

Randomness in the system reduces entanglements, resulting in narrower, Gaussian winding angle distributions.

Abstract

We study analytically and numerically the winding of directed polymers of length $t$ around each other or around a rod. Unconfined polymers in pure media have exponentially decaying winding angle distributions, the decay constant depending on whether the interaction is repulsive or neutral, but not on microscopic details. In the presence of a chiral asymmetry, the exponential tails become non universal. In all these cases the mean winding angle is proportional to $\ln t$. When the polymer is confined to a finite region around the winding center, e.g. due to an attractive interaction, the winding angle distribution is Gaussian, with a variance proportional to $t$. We also examine the windings of polymers in random systems. Our results suggest that randomness reduces entanglements, leading to a narrow (Gaussian) distribution with a mean winding angle of the order of $\sqrt{\ln t}$.