Transport Properties of the One-Dimensional Stochastic Lorentz Model: I. Velocity Autocorrelation Function

Point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables (stationary renewal process). For a fixed configuration of scatterers a particle performs the following random walk: The particle starts at the point x with velocity v, Iv[ = 1. In between scatterers the particle moves freely. At a scatterer the particle is either transmitted or reflected, both with probability 1/2. For given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations. If this variable is averaged over the distribution of scatterers, it decays not faster than t -3/2.