The path-coalescence transition and its applications

TL;DR

This paper investigates a phase transition in particle systems subjected to random forces and damping, where trajectories coalesce, and explores its theoretical analysis and real-world applications.

Contribution

It introduces a detailed analysis of the path-coalescence transition by mapping it to a solvable Kramers problem and characterizes the dynamics in the weak force limit.

Findings

Exact solution of the Kramers problem for the transition

Characterization of caustic crossing rates in the coalescing phase

Potential applications in raindrop trajectories and animal migration

Abstract

We analyse the motion of a system of particles subjected a random force fluctuating in both space and time, and experiencing viscous damping. When the damping exceeds a certain threshold, the system undergoes a phase transition: the particle trajectories coalesce. We analyse this transition by mapping it to a Kramers problem which we solve exactly. In the limit of weak random force we characterise the dynamics by computing the rate at which caustics are crossed, and the statistics of the particle density in the coalescing phase. Last but not least we describe possible realisations of the effect, ranging from trajectories of raindrops on glass surfaces to animal migration patterns.