On some generalisations of the Car-Oriented approach to the Totally Asymmetric Simple Exclusion Process

The goal of this thesis is to develop generalisations of the Car-Oriented Mean-Field approximation for the Totally Asymmetric Simple Exclusion Process (TASEP) in the inhomogeneous case, that is, under Open Boundary Conditions (OBC) and inhomogeneous hopping rates. We investigate the non-equilibrium stationary-state probability distribution of the model through its marginals in various settings. Since the main source of inspiration for these generalisations is the Cluster approximation, we first study several Cluster approximations (namely, the Mean-Field, Pair, and Triplet approximations) under OBC. We compare them with some kinetic Monte-Carlo simulations both in the case of homogeneous inner hopping rates and in the case of sitewise disorder. We then combine the two frameworks --- the Cluster approximations and the Car-Oriented Mean-Field approximation --- and present our new approach, which mixes aspects of both. This leads to the formulation of the inhomogeneous versions of the Car-Oriented Mean-Field, Car-Oriented Pair, and Car-Oriented Triplet approximations. Among these, the most promising is the Car-Oriented Triplet approximation, to which we devote particular attention, especially by comparing it to its Cluster-approximation counterpart (the traditional Triplet approximation) in different scenarios. Finally, we propose a further generalisation of the Car-Oriented variables by introducing a new set of variables --- the 'Queue-Oriented' variables --- along with analogous approximation schemes, and compare the results across these different approaches.