Abstract
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 j 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 ϵ ∑ ϕ ( ϵ x ) 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 → 0 on the deterministic value ∫ ϕ ρ 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.
Highlights
This is a first in a series of three papers where we study a particle system whose hydrodynamic limit is described by a free boundary problem
Our system is made of particles confined to the lattice [0, −1] ∩ Z, for brevity in the sequel we shall just write [0, −1]. In this notation, −1 is a positive integer denoting the system size and we will be eventually interested in the asymptotics as → 0
The main ingredient in the proof is established here and it is based on the notion of upper and lower barriers. These are “approximate solutions” of (2.11) which bound from below and from above the hydrodynamic limit ρ(r, t), the inequalities being in the sense of mass transport
Summary
The interaction described by La is highly non-local as Rξ depends on the positions of all the particles. This spoils any attempt to use the BBGKY hierarchy of equations for the correlation functions, as customary in perturbations of the independent system, see for instance [7]. The action of Lb and La is to add from the left and, respectively, remove from the right particles at rate j They act, as “current reservoirs” [8,9,10], because they are imposing a current j (recall that for density reservoirs [4,6] the particles current scales by ). We shall give answers to most of the above issues, with our main results being stated
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