Phase-plane analysis of driven multi-lane exclusion models

We show how a fixed-point-based boundary layer analysis technique can be used to obtain the steady-state particle density profiles of driven exclusion processes on two-lane systems with open boundaries. We have considered two distinct two-lane systems. In the first, particles hop on the lanes in one direction obeying the exclusion principle and there is no exchange of particles between the lanes. The hopping on one lane is affected by the particle occupancies on the other, which thereby introduces an indirect interaction among the lanes. Through a phase-plane analysis of the boundary layer equation, we show why the bulk density undergoes a sharp change as the interaction between the lanes is increased. The second system involves one lane with a driven exclusion process and the other with biased diffusion of particles. In contrast to the previous model, here there is a direct interaction between the lanes due to particle exchange between them. In this model, we have looked at two possible scenarios with constant (flat) and non-constant bulk profiles. The fixed-point-based boundary layer method provides a new perspective on several aspects including those related to maximal/minimal current phases, the possibilities of shocks under very restricted boundary conditions for the flat profile but over a wide range of boundary conditions for the non-constant profile.