On Some Classes of Open Two-Species Exclusion Processes

We investigate some properties of the nonequilibrium stationary state (NESS) of a one dimensional open system consisting of first and second class (type 1 and type 2) particles. The dynamics are totally asymmetric but the rates for the different permitted exchanges (10 -> 01, 12 -> 21, and 20 -> 02) need not be equal. The entrance and exit rates of the different species can also be different. We show that for certain classes of rates one can compute the currents and phase diagram, or at least obtain some monotonicity properties. For other classes one can obtain a matrix representation of the NESS; this generalizes previous work in which second class particles can neither enter nor leave the system. We analyze a simple example of this type and establish the existence of a randomly located shock at which the typical density profiles of all three species are discontinuous.