Asymptotics of particle trajectories in infinite one‐dimensional systems with collisions

Abstract

AbstractWe investigate the asymptotic behavior of the trajectories of a tagged particle (tp) in an infinite one‐dimensional system of point particles. The particles move independently when not in contact: the only interactions are Harris type generalized elastic collisions which prevent crossings. This is achieved by relabeling the independent trajectories when they cross. When these trajectories are differentiable, as in particles with velocities undergoing Ornstein‐Uhlenbeck processes, collisions correspond to exchange of velocities.We prove very generally that the suitably scaled tp trajectory converges (weakly) to a simple Gaussian process. This extends the results of Spitzer for New tonian particles to very general non‐crossing processes. The proof is based on the consideration of the simpler process which counts the crossings of the origin by the independent trajectories.