Phase Diagrams and Current Density Profiles of the Totally Asymmetric Simple Exclusion Process in Two Dimensions, for a Three-Way Junction

This study explores a dynamical model called the totally asymmetric simple exclusion process (TASEP) in two dimensions (2D). An open boundary condition is specified for the model, and sequential updating dynamics are used as the dynamical rule. The system studied is a discrete 2D system of lattice sites, which are modified into a three-way junction. Two cases are considered: a three-way junction with two entrances and one exit, and a three-way junction with one entrance and two exits. The density and current density of the system are determined numerically, such that a phase diagram is obtained. The continuity equation describing the dynamics of particles in the system is solved by using a simple Euler method. The results show that the density and current density profiles, as functions of the lattice sites, are determined by the input and output rates at their boundaries. Moreover, the density phases obtained are combinations of the density phases of the TASEP, which yield a rich phase diagram.