On the Limits of Analogy Between Self-Avoidance and Topology-Driven Swelling of Polymer Loops

TL;DR

This paper explores the analogy between knotting and swelling in polymer loops, providing a simple swelling expression, a new loop generation method, and analyzing the analogy's validity through simulations.

Contribution

It introduces a simple swelling expression for trivially knotted loops, a new bias-free loop generation method, and assesses the analogy between knotting and excluded volume effects.

Findings

Swelling expression agrees with simulation data

Graphical mapping of excluded volume to trivial knots is developed

New computational method for loop generation without bias

Abstract

The work addresses the analogy between trivial knotting and excluded volume in looped polymer chains of moderate length, $N<N_0$, where the effects of knotting are small. A simple expression for the swelling seen in trivially knotted loops is described and shown to agree with simulation data. Contrast between this expression and the well known expression for excluded volume polymers leads to a graphical mapping of excluded volume to trivial knots, which may be useful for understanding where the analogy between the two physical forms is valid. The work also includes description of a new method for the computational generation of polymer loops via conditional probability. Although computationally intensive, this method generates loops without statistical bias, and thus is preferable to other loop generation routines in the region $N<N_0$.