Stable configurations of entangled systems with repulsive interactions

TL;DR

This paper analyzes the stable configurations of entangled thread systems with repulsive interactions, revealing how layered structures form and drift apart over time.

Contribution

It introduces a framework for identifying stable states of intertwined threads in three dimensions, including existence, uniqueness, and asymptotic behavior of layers.

Findings

Stable configurations exist and are unique for non-separable weave components.

Two layers drift apart with a growth rate of t^{1/3} as time progresses.

The framework applies to systems with repulsive interactions in three-dimensional space.

Abstract

Entangled systems are prevalent in both biological and synthetic materials. This study examines the stable configurations of weaves consisting of two families of intertwined threads, such as warp and weft threads. By analyzing the steepest descent flow of an energy functional featuring repulsive interactions, we develop a framework for identifying stable states in ${\mathbb R}^3$. Although a weave consists of one-dimensional threads that do not intersect each other, it behaves collectively like a two-dimensional object. To describe this phenomenon, we define a non-separable component of a weave as a ``layer'' and establish the existence and uniqueness of its stable configuration. Furthermore, we show that two distinct layers drift apart with an asymptotic growth rate of $t^{1/3}$ as $t \to \infty$.