Hydrodynamic limit in a particle system with topological interactions

Abstract

We study a system of particles in the interval $[0,ε^{-1}] \cap \mathbb Z$, $ε^{-1}$ a positive integer. The particles move as symmetric independent random walks (with reflections at the endpoints); simultaneously new particles are injected at site 0 at rate $jε$ ($j>0$) and removed at same rate from the rightmost occupied site. The removal mechanism is therefore of topological rather than metric nature. The determination of the rightmost occupied site requires a knowledge of the entire configuration and prevents from using correlation functions techniques. We prove using stochastic inequalities that the system has a hydrodynamic limit, namely that under suitable assumptions on the initial configurations, the law of the density fields $ε\sum ϕ(εx) ξ_{ε^{-2}t}(x)$ ($ϕ$ a test function, $ξ_t(x)$ the number of particles at site $x$ at time $t$) concentrates in the limit $ε\to 0$ on the deterministic value $\int ϕρ_t$, $ρ_t$ interpreted as the limit density at time $t$. We characterize the limit $ρ_t$ as a weak solution in terms of barriers of a limit free boundary problem.