Kinetics of Ring Formation
E. Ben-Naim and P. L. Krapivsky
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.
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