Exclusion Process with Slow Boundary

Abstract

Abstract. We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with slow boundary. The term slow boundary means that particles can be born or die at the boundary sites, at a rate proportional to N −θ , where θ > 0 and N is the scaling parameter. In the bulk, the particles exchange rate is equal to 1. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter θ; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on θ. If θ ∈ (0, 1), we get Dirichlet boundary conditions, (which is the same behavior if θ = 0, see [7]); if θ = 1, we get Robin boundary conditions; and, if θ ∈ (1, ∞), we get Neumann boundary conditions.

Non-equilibrium and stationary fluctuations for the SSEP with slow boundary

“…Proof. This result is proved by means of a coupling argument similar to the one presented in Section 3 of [1]. More precisely, we construct another RW…”

“…Note that for θ > 1 we have that µ n → 0. In Subsection 6.4 we prove that if ρ 0 ∈ C 6 then φ n ∈ C 1,4 . Now if γ n and the function G n the explicit expression for T 0 (x, y) .…”

“…Theorem 2.2 (Hydrodynamic Limit, [1,2,9]). Suppose that the sequence {µ n } n∈ is associated to a measurable profile ρ 0 (·) in the sense of Definition 1.…”

“…We observe that the proof of Proposition 2.1 in the case θ > 1 holds for any θ > 0. We decided to present a different proof for the case θ < 1, because there it appears the natural RWs associated with this model, as one can see in (2.3) and (2.10) and in [1] and [8].…”

“…The hydrodynamic limit for this model was analysed in [1]. It is given by the heat equation, but depending on the range of the parameter θ three different types of boundary conditions appear: when θ ∈ [0, 1) the density profile ρ(t, u) satisfies Dirichlet boundary conditions, which means that the density profile is fixed as being α (resp.…”