Shocks in the asymmetry exclusion model with an impurity

We consider the one-dimensional asymmetric exclusion process with an impurity. This model describes particles hopping in one direction with stochastic dynamics and a hard core exclusion condition. The impurity hops with a rate different from that of the normal particles and can be overtaken by these particles. We solve this model exactly and give its phase diagram. In one of the phases the system presents a shock, i.e. a sharp discontinuity between a region of high density of particles and a region of low density. Density profiles and relevant exponents are explicitly calculated. These exact results for systems of finite size are consistent with anomalous diffusion laws observed in infinite systems.