Breaking the chain

TL;DR

This paper analyzes the probabilistic behavior of a Brownian particle in a chain with finite-range interactions, focusing on the first chain break location as noise diminishes and the particle moves slowly.

Contribution

It provides a detailed asymptotic analysis of the first chain break location for a Brownian particle with finite-range interactions under slow deterministic motion and small noise.

Findings

Characterizes the asymptotic distribution of the first break point.

Identifies the influence of slow movement and noise intensity on chain stability.

Provides mathematical insights into chain rupture dynamics in stochastic systems.

Abstract

We consider the motion of a Brownian particle in $\mathbb{R}$, moving between a particle fixed at the origin and another moving deterministically away at slow speed $\epsilon>0$. The middle particle interacts with its neighbours via a potential of finite range $b>0$, with a unique minimum at $a>0$, where $b<2a$. We say that the chain of particles breaks on the left- or right-hand side when the middle particle is greater than a distance $b$ from its left or right neighbour, respectively. We study the asymptotic location of the first break of the chain in the limit of small noise, in the case where $\epsilon = \epsilon(\sigma)$ and $\sigma>0$ is the noise intensity.