Dynamical transition in inhomogeneous TASEP

Inspired by the huge relevance of simple exclusion processes models as biological traffic models or vehicular traffic as well, we will focus our work on the Totally Asymmetric Simple Exclusion Process (TASEP). The latter is defined on a one dimensional lattice, where each site could be occupied or not by only one particle. The particles can hop from left to right with a given hopping rate provided that the lattice site on the right is empty. There is moreover a specified entry rate in the first lattice site where particles can enter in the system and exit rate from the last one where particles can exit the system. This is a sort of a skeleton of the model that can be filled with particular choices of rates at which hopping processes happen together with suitable boundary conditions. We consider the TASEP with Open Boundary Conditions (OBC) and non uniform, slowly varying, hopping rates with non zero entry rate only at the first site and non zero exit rate only from the last one. In a more general model, one could also consider non zero entry rates and non zero exit rates through all lattice sites giving rise to a generalization of the model called Totally Asymmetric Simple Exclusion Process with Langmuir Kinetics (TASEP - LK). However, we do not focus on this version explicitly, but we make only some comments about it that will be clearly specified at the proper time. The purpose of the present thesis is to investigate the so called dynamical transition making use of several analytical and numerical methods such as Mean Field Theory (MFT), Domain Wall Theory (DWT) and Finite Lattice Extrapolation (FLE).