# LisMath Seminar

### Hydrodynamics for SSEP with non-reversible slow boundary dynamics: the critical regime and beyond

In this talk we present the Law of Large numbers for three quantities (local density, current and mass) for the Symmetric Simple Exclusion Process (SSEP) on the lattice $\{1, . . . , N − 1\}$ with “nonlinear” boundary dynamics. Informally, we let a particle jump only to its neighbor site if such site is empty. Then, we let the system be in contact with two reservoirs, which inject/remove particles from a window of size $K$ from the boundaries, at rates depending on the site of injection/removal. We let a particle enter to the first free site, and leave from the first occupied site. This boundary dynamics impose strong correlations between particles, which leads to the sudy of most physical quantities of the system being a challenge. Multiplying the boundary rates by $N^{-\theta}$, one observes macroscopically phase transitions on those quantities in the following way. Under a $N^2$ time-scale, macroscopically the local density behaves as a weak solution to the heat equation. For $\theta \in [0, 1)$ we have Dirichlet B.C., nonlinear Robin for $\theta = 1$, and Neumann for $\theta \gt 1$. For the current, we microscopically derive Fick’s Law, which depends on the B.C. for the density, while for the mass, we see that instead the time scale $N^{1+ \theta}$ is the most natural one, and obtain an ODE. We present only results for $\theta \geq 1$. We then show that starting from the stationary measure, we obtain steady state solutions of the aforementioned equations.

Bibliography:

[1] Gonçalves, P., Erignoux, C., Nahum, G.: Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part I, the critical regime and beyond, arXiv:1912.09841 (2019).