Size distribution of ring polymers.

Area of Science:

• Polymer Physics
• Statistical Mechanics
• Stochastic Processes

Background:

• Understanding the spatial distribution of polymer chains is crucial in polymer physics.
• Previous studies have explored monomer distances in linear and branched polymers, but ring polymers present unique topological challenges.
• The connection between polymer conformations and random walk distributions is a well-established concept.

Purpose of the Study:

• To derive an exact solution for the distribution of sample-averaged monomer-to-monomer distances in ring polymers.
• To explore the relationship between these distributions and known stochastic processes like Bessel bridges and excursions.
• To investigate the dimensional dependence and universality of these polymer properties.

Main Methods:

• Analytical solution derivation for non-interacting and local-interaction ring polymer models.
• Mapping polymer distance distributions to area distributions of reflected Bessel bridges and Bessel excursions.
• Numerical analysis for self-avoiding polymers in various dimensions.

Main Results:

• Identical distributions for non-interacting and local-interaction models in dimensions d ≥ 2, with finite size effects at d=2.
• A d ↔ 4-d symmetry was discovered, showing the Airy distribution for 1D Brownian excursions also describes 3D ring polymers.
• Scaled averaged distance fluctuations for self-avoiding polymers in d=2, 3 are nearly identical and approximated by the 5D non-interacting model.

Conclusions:

• The study provides an exact solution for ring polymer distances, revealing universal scaling laws.
• The findings establish a direct link between polymer physics and the mathematical framework of Bessel processes.
• The discovered dimensional symmetry offers new insights into polymer behavior across different spatial dimensions.