Particle Systems with Stochastic Passing

TL;DR

This paper analyzes a particle system on a line where particles can pass each other, with passing dynamics affecting velocities and leading to different long-term behaviors depending on the velocity distribution.

Contribution

It introduces a model of particle passing with stochastic velocity updates and characterizes the conditions for system stabilization or indefinite evolution.

Findings

System reaches steady state if P_0(v) vanishes at its lower cutoff.

System evolves indefinitely if P_0(v) does not vanish at the lower cutoff.

Passing dynamics depend on the velocity distribution's properties.

Abstract

We study a system of particles moving on a line in the same direction. Passing is allowed and when a fast particle overtakes a slow particle, it acquires a new velocity drawn from a distribution P_0(v), while the slow particle remains unaffected. We show that the system reaches a steady state if P_0(v) vanishes at its lower cutoff; otherwise, the system evolves indefinitely.