Clusters and shocks in a Totally Asymmetric Simple Exclusion Process with interactions.

In this thesis, we study two models related to the Total Asymmetric Simple Exclusion Process(TASEP) with Nearest Neighbor interaction, that are: - Facilitated Model - Antal-Schütz model The TASEP is a 1-D model defined on a linear chain with L sites. Every site can be occupied by almost one particle and the only possible action of our particle is a hop in the subsequent direction if the site is not occupied, with a rate defined by the system. This linear chain can be closed, this implies that the particle in site L hop in site 1 if not occupied, or open, that implies the particle can enter the system at site 1 and exit from the system at site L, with some rates. Here we suppose that our chains are closed, so closed boundary condition. In the first one, the hopping rate depends only on the occupation of site before the site in which takes place the hop. In the second one, the hopping rate depends only on the occupation of the site after the site we reach after the hop. Also the rate depends on the energy of interaction between the particles and based on the value of the energy, we can define 3 different regimes: attractive if the energy is negative, repulsive if the energy is positive and we recover the pure TASEP, so no interaction, if the energy is 0. Once defined the two models, we study both the steady state and the transient behavior, using both a semi-analytical approximation (the pair approximation, PA) and numerical simulations. For the theoretical part, we use the PA theory to understand the behavior of our model at steady state and, integrating time-evolution equations by means of the Euler method, in the transient. For the numerical simulation, we use the Gillespie algorithm to simulate the subsequent transition that happens in the system, and we measure observables like local densities and cluster lengths Following [1] we study the cluster length distribution, which depends on density and interaction energy, and we investigate its transition from a monotonic to a non-monotonic behavior. To study the transient behavior, we consider an initial condition with two shocks in the density profile, so that we can study the dynamics and in particular the stability of a shock. We find that our model reaches a steady state that depends on its density and its current, which, in turn, depends on the regime we are considering through the rate. We also find that the further we are from the maximum of the curve rho-J, the longest is the relaxation time of the model to the steady state. This also affects the behavior of our model, i.e. the further we are from the maximum, the more our density profiles oscillate among the maximal and minimal density. The same is valid for the magnitude of the velocity of the shock. Finally, we observe that the two models are related by a particle-hole symmetry. [1] Hao et al [J. Stat. Mech. (2020) 083302]