Hydrodynamic limit in a particle system with topological interactions

Abstract

We study a system of particles in the interval $[0,{\mathit{ϵ}}^{-1}]\cap \mathbb{Z},{\mathit{ϵ}}^{-1}$ 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 $j\mathit{ϵ}$ (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 $\mathit{ϵ}\sum \mathit{\varphi }(\mathit{ϵ}x){\mathit{\xi }}_{{\mathit{ϵ}}^{-2}t}(x)$ (φ a test function, ${\mathit{\xi }}_t(x)$ the number of particles at site x at time t) concentrates in the limit $\mathit{ϵ}\to 0$ on the deterministic value $\int \mathit{\varphi }{\mathit{\rho }}_t$, ${\mathit{\rho }}_t$ interpreted as the limit density at time t. We characterize the limit ${\mathit{\rho }}_t$ as a weak solution in terms of barriers of a limit-free boundary problem.

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