One-dimensional asymmetric exclusion model with open boundaries

One-dimensional asymmetric exclusion model, in which the probabilities of hopping to the left and right are in general different, is studied. The boundaries are open; a particle is added at the left end and is removed at the right end. First, the mean field theory is applied to discuss the case of arbitrary hopping probabilities. Second, exact solution is obtained for the case of equal hopping probability. Third, the properties of the model near the equal hopping probability case are investigated by the perturbation theory. It is found that the mean field theory does not give the correct density profile near the boundaries.