Time correlation functions for the one-dimensional Lorentz gas

Abstract

The velocity autocorrelation function and related quantities are investigated for the one-dimensional deterministic Lorentz gas, consisting of randomly distributed fixed scatterers and light particles moving back and forth between two of these at a constant given speed. An expansion for the velocity autocorrelation function given by Grassberger, which is useful for short times, is reconstructed. The long time behavior is investigated by Fourier transform techniques. For large time t the velocity autocorrelation function decays as exp(-ct't1/2) and in addition oscillates with a period increasing as t1/2. A velocity average over a Maxwellian changes this long time behavior to exp(-c't2/3), while the oscillations are removed. The Green's function is also investigated. Its spatial and temporal Fourier transform, the incoherent scattering function, exhibits strongly non-Lorentzian behavior.