Inhomogeneous asymmetric exclusion processes: steady-state and relaxation transitions.

The Asymmetric Exclusion Process represents a fundamental family of models in out of equilibrium statistical physics, for which applications are found in a large variety of contexts, ranging from biology (e.g. the motion of ribosomes along mRNA or molecular motors on intracellular filaments) to electron hopping transport with applied voltage and vehicular highway traffic. The specific model we want to consider for this work is the Totally Asymmetric Simple Exclusion Process (TASEP). It is defined on a one dimensional lattice, where particles can hop only rightward, from each node to the next one, provided the latter is empty. Despite its simplicity, it can offer a very interesting series of results, depending on the characteristics with which we equip it. We work with open boundary conditions (OBC) and inhomogeneous hopping rates. Indeed, in real life we expect to find conditions that alter the ideal condition of constant rate. The most interesting aspect is that in this regime all the steady-state transitions are first order. The purpose of this work is not tied to a specific application of this model, but rather to a theoretical exploration; in particular, we aim to find: the steady-state transitions both analytically, using Fokker-Planck (FP) equation, and numerically, using Monte Carlo simulations (MCS); the relaxation transitions in Mean Field Theory (MFT) and Pair Approximation (PA); the Domain Wall Theory (DWT) picture.