Seminars and short courses

Exclusion processes with non-reversible boundary dynamics are one-dimensional interacting particle systems evolving on a finite lattice. These systems arise from the superposition of two different types of dynamics. In the bulk, particles evolve according to the usual symmetric simple exclusion dynamics, that is, at most one particle is allowed per site and jumps occur through nearest-neighbor interactions. Near the boundary, within a window of fixed size $l\geq 1$, particles may be created or annihilated according to rates depending on the local configuration in a finite neighborhood of the boundary. This choice of boundary dynamics is non-conservative and is considered under very general rates, allowing, in particular, the simultaneous creation of more than one particle. This flexibility creates a crucial distinction between our model and the simplified version first introduced in 2011 to describe boundary current fluctuations, where a very specific choice of boundary rates was considered. In this sense, our model generalizes earlier works by capturing not only non-linear evolutions of the particle density, but also, and more importantly, the surprising emergence of multiple stationary solutions to the hydrodynamic equation describing the density profile. In this talk, we define the model, discuss its hydrodynamic limit, and analyze several properties of the stationary solutions of the associated hydrodynamic equation. This presentation is based on joint works with Claudio Landim and João Pedro Mangi.