Equilibrium winding angle of a polymer around a bar

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 $<θ^2>\sim \ln L$. 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 $<θ^2>\sim (\ln L)^{2α}$, with $α=0.75(1)$. The ratio $γ= <θ^4>/<θ^2>^2=3.74(5)$ is incompatible with the gaussian value $γ=3$, but consistent with the observation that the tail of the probability distribution function $p(θ)$ is found to decrease slower than a gaussian function. These findings are at odds with the existing first-order $ε$-expansion results.