Effects of multiple cycles on the resistance distance of a strand in a homogeneous polymer network

TL;DR

This study investigates how the presence and density of short cycles in a homogeneous polymer network influence the resistance distance between adjacent vertices, revealing that cycles reduce resistance distance and act independently.

Contribution

The paper introduces a Monte Carlo method to analyze how multiple short cycles affect resistance distance, demonstrating their independent effects in polymer networks.

Findings

Resistance distance decreases when a strand is part of a short cycle.

Multiple cycles influence resistance distance linearly and independently.

Resistance distance is approximately 2/f when no short cycles are present.

Abstract

We show that the resistance distance between a pair of adjacent vertices in a phantom network generated randomly by a Monte-Carlo method depends on the existence of short cycles around it. Here we assume that phantom networks have no fixed points but their centers of mass are located at a point. The resistance distance corresponds to the mean-square deviation of the end-to-end vector along the strand connecting the adjacent vertices. We generate random networks with fixed valency $f$ but different densities of short cycles via a Metropolis method that rewires edges among four vertices chosen randomly. In the process the cycle rank is conserved. However, the densities of short cycles are determined by the rate of randomization $kT$ which appears in the acceptance ratio $\exp(-\Delta U/kT)$ of rewiring. If a strand has few short cycles around itself, the mean squared deviation of the strand is equal to $2/f$. If it is part of a short cycle, i.e., the network has a short loop which consists of a sequence of strands including the given strand itself, its resistance distance is smaller than $2/f$, while if it is not included in a cycle but adjacent to cycles, its resistance distance is larger than $2/f$. We show it via an electrical circuit analogy of the network. Moreover, we numerically show that the effect of multiple cycles on the resistance distance is expressed as a linear combination of the effects of isolated single cycles. It follows that cycles independently have an effect on the fluctuation properties of a strand in a polymer network.