Hydrodynamic limit in a particle system with topological interactions

We study a system of particles in the interval [0 , ϵ- 1] ∩ 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∑ϕ(ϵx)ξϵ-2t(x) (φ a test function, ξt(x) the number of particles at site x at time t) concentrates in the limit ϵ→ 0 on the deterministic value ∫ ϕρt, ρt interpreted as the limit density at time t. We characterize the limit ρt as a weak solution in terms of barriers of a limit-free boundary problem. [MediaObject not available: see fulltext.] © 2014, The Author(s).