Velocity autocorrelations in a simple model

A one-dimensional Lorentz-type model is studied where a point particle is reflected with some given probability p off randomly located fixed scatterers. The diffusion constant is calculated exactly, and the velocity autocorrelation is shown to decay like $\text{t}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$, for 0<p<1. For finite times, there are oscillations superimposed on this power decay. For p → 1, these oscillations dominate the behaviour for all times.