Non-equilibrium unidimensional traffic model with pausing particles: a biological-motivated extension of the exclusion process

This master thesis examines nonequilibrium, one-dimensional traffic phenomena modeled as lattice gas systems, where particles move on a discrete lattice. Movement follows a continuous time Poisson process. The focus is on the Totally Asymmetric Simple Exclusion Process (TASEP), where particles hop in one direction only if the landing site is empty, and a derived model, pausing-TASEP (pTASEP), where particles can stochastically pause and resume motion. The work is divided into two parts: approximate mean-field theories and exact combinatorial results. The thesis first explores pTASEP as a model for translationally-inhibited protein synthesis, where ribosomes on mRNA (the lattice) face pauses due to antibiotics. Unlike TASEP, pTASEP lacks exact solutions for key observables, with existing literature offering limited mean-field theories in restrictive geometries. Here recent advances in expanding said mean-field theory to more realisitc open boundaries geometry are summarized. This work also includes numerical analysis of biologically relevant parameter spaces, revealing significant finite size effects. The focus was therefore set on developing minimal models to predict finite size phenomena, finding good agreement with simulations. These models were based on the time scale separation between particle dynamics and pauses, enabling insights into the system's behavior under both periodic and open boundaries. A simple relation between particle current and pausing rate was derived, offering improvements over traditional mean-field theory. The second part of this thesis establishes a novel link between the algebraic properties of the transition rate matrix in the master equation and the closed-form expressions of physical observables in standard TASEP. Using the Faddeev-Leverrier algorithm, it shows that all observables can be expressed as functions of the trace of powers of the transition rate matrix. By decomposing the matrix through the quantum Hamiltonian formalism, the values of these traces are shown to depend on the commutation of terms within the Hamiltonian. This approach aims to recover known analytical results for TASEP without relying on the current ad hoc Ansatz and seeks to develop a more general solution applicable to models like pTASEP, which lack an exact solution.