Equilibrium winding angle of a polymer around a bar

TL;DR

This study investigates the winding angle distribution of a three-dimensional self-avoiding walk around a bar, revealing non-Gaussian behavior and a different scaling of variance than previously predicted by epsilon-expansion theories.

Contribution

The paper provides the first extensive Monte Carlo analysis showing the actual scaling and distribution of winding angles, challenging existing theoretical predictions.

Findings

Variance scales as (ln L)^{1.5} with L.

Distribution tail decreases slower than Gaussian.

Observed ratio gamma indicates non-Gaussian distribution.

Abstract

The winding angle probability distribution of a planar self-avoiding walk has been known exactly since a long time: it has a gaussian shape with a variance growing as $<\theta^2>\sim \ln L$. For the three-dimensional case of a walk winding around a bar, the same scaling is suggested, based on a first-order epsilon-expansion. We tested this three-dimensional case by means of Monte Carlo simulations up to length $L\approx25\,000$ and using exact enumeration data for sizes $L\le20$. We find that the variance of the winding angle scales as $<\theta^2>\sim (\ln L)^{2\alpha}$, with $\alpha=0.75(1)$. The ratio $\gamma = <\theta^4>/<\theta^2>^2=3.74(5)$ is incompatible with the gaussian value $\gamma =3$, but consistent with the observation that the tail of the probability distribution function $p(\theta)$ is found to decrease slower than a gaussian function. These findings are at odds with the existing first-order $\epsilon$-expansion results.