Transport properties of stochastic Lorentz models

Diffusion processes are considered for one-dimensional stochastic Lorentz models, consisting of randomly distributed fixed scatterers and one moving light particle. In waiting time Lorentz models the light particle makes instantaneous jumps between scatterers after a stochastically distributed waiting time. In the stochastic Lorentz gas the light particle moves at constant speed and is scattered stochastically at collisions with the scatterers. For the waiting time Lorentz models the Green's function of the diffusion process is calculated exactly. The diffusion coefficient is found to be the same as for a corresponding random walk on a regular lattice, the velocity autocorrelation function exhibits a long-time tail proportional to $t^{-\frac{3}{2}}$ and super Burnett and higher-order transport coefficients are found to diverge. For the stochastic Lorentz gas similar results are found for the diffusion coefficient and the velocity autocorrelation function, but the generalized super Burnett coefficient, as introduced by Alley and Alder, is convergent in this case. For a special case of the waiting time Lorentz models some other aspects are considered, such as periodic boundary conditions, steady-state diffusion and fluctuations of the velocity autocorrelation function about its average value, due to the initial conditions and to the stochastic distribution of scatterers.