Kinetics of Ring Formation

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.

6 pages, 7 figures

References in corpus (15)

• The spread of epidemic disease on networks
• Scaling Theory and Exactly Solved Models In the Kinetics of Irreversible Aggregation
• The Scaling Window of the 2-SAT Transition
• Nonequilibrium Phase Transitions in Models of Aggregation, Adsorption, and Dissociation
• Strongly discontinuous explosive percolation with multiple giant components
• On the critical behavior of the Susceptible-Infected-Recovered (SIR) model on a square lattice
• Borderline Aggregation Kinetics in ``Dry'' and ``Wet'' Environments
• Kinetic Theory of Random Graphs: from Paths to Cycles
• Velocity Distributions of Granular Gases with Drag and with Long-Range Interactions
• Hot String Soup
• Percolation with Multiple Giant Clusters
• Phase Transition with Non-Thermodynamic States in Reversible Polymerization
• Unicyclic Components in Random Graphs
• Polymerization with Freezing
• Dynamics of Triangulations

Cited by in corpus (9)

• Short Cyclic Structures in Polymer Model Networks: A Test of Mean Field Approximation by Monte Carlo Simulations
• Oscillations in aggregation-shattering processes
• Emergence of the giant weak-component in directed random graphs with arbitrary degree distributions
• Random graph approach to multifunctional molecular networks
• Native ultrametricity of sparse random ensembles
• Dynamics of Random Graphs with Bounded Degrees
• Correspondence between noisy sample space reducing process and records in correlated random events
• Mathematical model of alternative mechanism of telomere length maintenance
• Relaxation dynamics of maximally clustered networks