Fluctuations of ring polymers

TL;DR

This paper provides an exact analytical solution for the distribution of average monomer distances in ring polymers, revealing dimension symmetries and connections to Bessel processes, with implications for understanding polymer fluctuations.

Contribution

It introduces an exact solution linking ring polymer distance distributions to Bessel bridge and excursion models, uncovering a symmetry between dimensions and extending understanding of polymer fluctuations.

Findings

Distribution matches Bessel bridge and excursion models in certain dimensions

Dimension symmetry relates d and 4 - d, linking 1D and 3D cases

Fluctuations in 2D and 3D are similar, approximated by a 5D non-interacting model

Abstract

We present an exact solution for the distribution of sample averaged monomer to monomer distance of ring polymers. For non-interacting and weakly-interacting models these distributions correspond to the distribution of the area under the reflected Bessel bridge and the Bessel excursion respectively, and are shown to be identical in dimension d greater or equal 2. A symmetry of the problem reveals that dimension d and 4 minus d are equivalent, thus the celebrated Airy distribution describing the areal distribution of the one dimensional Brownian excursion describes also a polymer in three dimensions. For a self-avoiding polymer in dimension d we find numerically that the fluctuations of the scaled averaged distance are nearly identical in dimensions 2 and 3, and are well described to a first approximation by the non-interacting excursion model in dimension 5.