Path coalescence transition and its applications

Abstract

We analyze the motion of a system of particles subjected to 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 analyze this transition by mapping it to a Kramers problem which we solve exactly. In the limit of weak random force we characterize 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 realizations of the effect, ranging from trajectories of raindrops on perspex surfaces to animal migration patterns.