Exclusion processes in one dimension first appeared in the 70s and have since dragged much attention from communities in different domains: stochastic processes, out of equilibriums statistical physics, and more recently integrable systems. While the state of the art for a single species totally asymmetric simple exclusion process (TASEP) can be described, from different aspects as mature, much less is known when multiple interacting species are present. Using tools from integrable systems and hydrodynamics in the first place and stochastic processes in the second place, this work attempts to study the behavior of a novel version of the model with different species of particles having hierarchical dynamics that depend on arbitrary parameters. While Burger’s equation famously represents the hydrodynamic limit of TASEP with a single species, we present a counterpart coupled system of PDE representing the hydrodynamic limit for a model with two species. The solutions of these PDEs display a rich phenomenology of solutions best characterized through the underlying normal modes. We discuss the associated Riemann problem and validate our results with numerical simulations. This system with two species can be used as a toy model for studying driven diffusive systems with open boundaries. Using heuristics, we present results suggesting a general principle governing the boundary induced phase diagram of systems with multiple coupled driven conserved quantities, generalizing thus the extremal current principle known for the case of a single driven quantity. The integrability side of our study is mainly concerned with developing a formalism allowing the computation of the finite-time probability distribution of particle positions on the 1D lattice, generalizing therefore known results for TASEP and other multi-species models. We finally study the behavior and the impact of a single second class impurity initially located at the interface separating two regions of different densities of first class particles. Different limit shapes are deduced and observed. Using tools from probability theory, we generalize the asymptotic speed properties of the impurity for a regime of the hopping parameters.

Contact: Jérôme Dubail