Particle systems with stochastic passing

We study a system of particles moving on a line in the same direction. Pass ing is allowed and when a fast particle overtakes a slow particle, it acqui res a new velocity drawn from a distribution Pa(upsilon), while the slow pa rticle remains unaffected. We show that the system reaches a steady state i f Po(upsilon) vanishes at its lower cutoff; otherwise, the system evolves i ndefinitely.