Totally Asymmetric Generalized Exclusion Processes

The aim of this thesis is to expand the study of totally asymmetric models. The paradigm of these models is the TASEP (Totally Asymmetric Simple Exclusion Process), which consists of a chain of "L" sites, each of which can be occupied by at most one particle. These particles can move from one site to another in only one direction, say to the right, provided that the next site is unoccupied. This relatively simple model exhibits a series of extremely interesting characteristics. Under open boundary conditions, meaning when the number of particles on the chain is determined by injection and extraction rates, the occupation density profile of the chain, in the steady state, is described by a phase diagram divided into regions of high density, low density, and maximal current. The parameters that describe the phase diagram are the injection and extraction rates. The model also describes a type of transition that does not involve a phase change in the steady state but only the relaxation times toward it. This type of transition is called a “dynamic transition.” Furthermore, although all of these results are known exactly, the mean-field treatment of the equations yields equally exact results concerning the study of the steady state and phase diagram, and qualitatively correct results regarding the dynamic transition. This thesis aims to generalize this model to cases where the maximum site occupation number is greater than one, under a mean-field approximation. Specifically, the study will focus on the "Site Uniform" and "Particle Uniform" hopping rates, originally proposed by C. Arita, P.L. Krapivsky, and K. Mallick. In the first part of this thesis, the model is defined and the necessary equations to describe its dynamics are derived. The steady state is then studied both under periodic boundary conditions, where the number of particles on the chain is constant, and under open boundary conditions, where the number of particles is controlled by the injection and extraction rates. We will then outline the phase diagram for both hopping rates and for a generic occupation number. We will also study the relaxation toward the steady state in the specific case of a maximum occupation number of two particles, again under a mean-field approximation. The study, conducted for both hopping rates, demonstrates the presence of a dynamic transition in both cases. Finally, we will compare the steady-state density profiles obtained with the mean-field approximation with the solution obtained through simulations using the Gillespie algorithm.