Kinetics of Ring Formation

TL;DR

This paper investigates the reversible polymerization of rings, revealing a phase transition between microscopic and macroscopic ring structures, supported by theoretical analysis and efficient simulations.

Contribution

It introduces a comprehensive stochastic model for ring formation, analyzes the phase transition, and develops a novel card-shuffling algorithm for simulations.

Findings

Identification of a percolation transition with distinct phases

Universal size distribution of macroscopic rings

Logarithmic scaling of the number of giant rings with system size

Abstract

We study reversible polymerization of rings. In this stochastic process, two monomers bond and as a consequence, two disjoint rings may merge into a compound ring, or, a single ring may split into two fragment rings. This aggregation-fragmentation process exhibits a percolation transition with a finite-ring phase in which all rings have microscopic length and a giant-ring phase where macroscopic rings account for a finite fraction of the entire mass. Interestingly, while the total mass of the giant rings is a deterministic quantity, their total number and their sizes are stochastic quantities. The size distribution of the macroscopic rings is universal, although the span of this distribution increases with time. Moreover, the average number of giant rings scales logarithmically with system size. We introduce a card-shuffling algorithm for efficient simulation of the ring formation process, and present numerical verification of the theoretical predictions.