Stationary state and diffusion for a charged particle in a one dimensional medium with lifetimes

We study a one-dimensional semi-infinite system of identical particles with random lifetimes, interacting with a charged particle (the leftmost) which is driven by a constant positive force $F$. Particles interact through elastic collisions and at the initial time all particles are at rest, and the interparticle distances are independent identically distributed positive random variables. Each neutral particle has an exponentially distributed lifetime, which starts counting as soon as the particle moves, and which is independent and identically distributed. Under suitable conditions we prove a strong cluster property, convergence to a limiting measure for the law of the system as seen from a charged particle, and a central limit theorem for the motion of the charged particle.