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Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

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Published on: September 26, 2016

Distribution functions in percolation problems.

Hans-Karl Janssen¹, Olaf Stenull

¹Institut für Theoretische Physik III, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|March 5, 2009

Summary

This summary is machine-generated.

This study analyzes probability distributions in percolation clusters, revealing their asymptotic forms using renormalized field theory. These findings apply to various fractal properties and network resistances, regardless of theoretical approximation order.

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Area of Science:

Statistical Physics
Complex Systems
Fractal Geometry

Background:

Percolation clusters are random fractals with complex geometrical and transport properties.
Probability distribution functions are crucial for characterizing these properties.

Purpose of the Study:

To determine the asymptotic forms of various probability distribution functions for percolation clusters.
To analyze scaling behaviors in small and large variable limits.

Main Methods:

Renormalized field theory was employed to analyze the system.
The study focused on general, structural features of diagrammatic perturbation theory.

Main Results:

Asymptotic forms of key distribution functions were derived.
This includes pair-connection probability, fractal masses (backbone, red bonds), and self-avoiding walk distributions.
The distribution of total resistance in random resistor networks was also analyzed.

Conclusions:

The derived results are valid to arbitrary loop order due to the method's generality.
This provides a robust framework for understanding percolation cluster properties.