Hydrodynamic limit in a particle system with topological interactions

Abstract

We study a system of particles in the interval [0, −1 ] ∩ Z, −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 ( 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 φ( x)ξ −2 t (x) (φ a test function, ξ t (x) the number of particles at site x at time t) concentrates in the limit → 0 on the deterministic value 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.

Non local branching Brownian motions with annihilation and free boundary problems

“…Let ψ ′′ and ψ be the coarse grained versions of dλ ′′ and dµ; φ ′′ and φ those relative to dν ′′ and dν, see (6.2) for notation. Let R ν and R µ be such that Since the cutting operation does not increase the L 1 norm, see Proposition 5.2 in [5],…”

Super-Hydrodynamic Limit in Interacting Particle Systems

“…In this paper we continue the analysis of the stochastic process introduced in [3]. This is a particles process in the interval Λ ǫ := [0, ǫ −1 ]∩Z, ǫ −1 a positive integer.…”

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This paper is a follow-up of the work initiated in [3], where it has been investigated the hydrodynamic limit of symmetric independent random walkers with birth at the origin and death at the rightmost occupied site. Here we obtain two further results: first we characterize the stationary states on the hydrodynamic time scale and show that they are given by a family of linear macroscopic profiles whose parameters are determined by the current reservoirs and the system mass.

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A Free Boundary Problem with Non Local Interaction

“…In fact the notion of leftmost particle is highly non local: one needs to know the positions of all the particles to determine which is the leftmost one. This is therefore a "topological" interaction which cannot be treated with the usual methods of interacting particle systems, it is the analogue in PDE's of free boundary problems in which the domain where the PDE's are defined is itself one of the unknowns, see for instance the survey by Carinci, De Masi, Giardinà and Presutti, [6], on topological interactions and their relation in the "hydrodynamic limit" with free boundary problems.…”