Hydrodynamic limit for asymmetric simple exclusion with accelerated boundaries

Abstract

We consider the asymmetric simple exclusion process (ASEP) on the one-dimensional lattice {1, 2, . . . , N }. The particles can be created/annihilated at the boundaries with timedependent rate. These boundary dynamics are properly accelerated. We prove the hydrodynamic limit of the particle density profile, under the hyperbolic space-time rescaling, evolves with the entropy solution to Burgers equation with Dirichlet boundary conditions. The boundary conditions are characterised by boundary entropy flux pair.

Hydrodynamics for one-dimensional ASEP in contact with a class of reservoirs

“…The macroscopic time evolution is governed by the entropy solution to a nonlinear scalar conservation law [19]. When reservoirs are attached, the solution is constrained by corresponding boundary conditions [1,21]. Different from the classical case, these boundary conditions do not fix the boundary value of the solution.…”

“…It turns out to be the main issue in proving hydrodynamic limit, since it restricts us from employing replacement lemma for small macroscopic blocks. When θ > 0, the reservoirs dominate the drift at the boundaries, thus the boundary data are set to be the reversible densities of the reservoirs, see [21] or Theorem 2.7. The proof in [21] exploits a grading scheme to control the formulation of boundary layers on a mesoscopic level.…”

“…When θ > 0, the reservoirs dominate the drift at the boundaries, thus the boundary data are set to be the reversible densities of the reservoirs, see [21] or Theorem 2.7. The proof in [21] exploits a grading scheme to control the formulation of boundary layers on a mesoscopic level. In [16], the time scales longer than the hyperbolic one are considered, under which the density converges to the quasi-stationary solution, that is, the stationary solution associated to dynamical boundary data.…”

“…We also prove in Theorem 2.11 that these conditions are in accordance with the vanishing viscosity limit of the diffusive hydrodynamic equation for the corresponding weakly asymmetric simple exclusion. The bulk equation is proved in Section 4 employing the logarithmic Sobolev inequality [22], similarly to [21]. The boundary conditions are proved through a strategy different from existing works [1,21].…”

“…The bulk equation is proved in Section 4 employing the logarithmic Sobolev inequality [22], similarly to [21]. The boundary conditions are proved through a strategy different from existing works [1,21]. We apply Vasseur's result on the existence of strong boundary traces [20] to reduce the boundary conditions to equivalent integral formulas.…”