Stochastic transport of interacting particles: a heterogeneous 1-dimensional model

This thesis deals with stochastic transport of interacting particles. To model this kind of phenomena we employ the Totally Asymmetric Simple Exclusion Process (TASEP). TASEP was firstly proposed as a stochastic model for a biological transport process, that is the kinetics of protein synthesis from mRNA via the polyribosome. TASEP model is defined over a 1-dimensional lattice made of a large number of sites in which all particles can move only in one direction and each particle can hop from one site to its adjacent one, provided the latter is empty. In particular this model is characterized by the fact that the hopping rates are inhomogeneous - they are described by a function that varies continuously and slowly from site to site and has a single global minimum - and they depend also on the local configuration due to interactions between particles. A stochastic approach is implemented to compute the particle current and density profiles, and thanks to some approximation methods, the time evolution of the system is obtained. We are interested in the behaviour of the particle current and density in the steady state. Once the steady state is reached it is possible to see that, depending on the average particle density and on the interaction energy, the system is characterized by different density profiles. It can belong to the high or low density phases or it can be defined by a phase with one, two, three shocks. Phases with more than one shock are quite unusual and are a novel feature of this model.