Quasi-static limit for the asymmetric simple exclusion

Abstract

We study the one-dimensional asymmetric simple exclusion process on the lattice {1, . . . , N } with creation/annihilation at the boundaries. The boundary rates are time dependent and change on a slow time scale N −a with a > 0. We prove that at the time scale N 1+a the system evolves quasi-statically with a macroscopic density profile given by the entropy solution of the stationary Burgers equation with boundary densities changing in time, determined by the corresponding microscopic boundary rates. We consider two different types of boundary rates: the "Liggett boundaries" that correspond to the projection of the infinite dynamics, and the reversible boundaries, that correspond to the contact with particle reservoirs in equilibrium. The proof is based on the control of the Lax boundary entropy-entropy flux pairs and a coupling argument.

Hydrodynamic limit for asymmetric simple exclusion with accelerated boundaries

“…Remark 2.3 (Quasi-static limit). By the method used in this article one can extend the quasistatic limit [12,Theorem 3.3] to L ∞ boundary data ρ ± = ρ ± (t). Precisely speaking, for a > 0 let N 1+a L N,t generate the process η ′ (t) ∈ Ω N .…”

“…De Masi etc. [12] studied the quasi-static transformation: if the boundary rates change sufficiently slow in time, the macroscopic density profile is determined by the entropy solution to quasi-static Burgers equation. The phase diagram and the variational property were also reproduced.…”

“…The main ingredients of our approach are the microscopic (boundary) entropy production (cf. [15,12]) and the stochastic compensated compactness developed in [7,8]. The key idea is the carefully chosen auxiliary functions in (6.1) to analyse the singular boundary term in the time evolution.…”

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

“…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. This result generalises the usual hydrostatic scenario, where only stationary dynamics is evolved.…”

Quasi-static limit for a hyperbolic conservation law

“…In a companion article [4] we study the quasi-static limit for the one-dimensional open asymmetric simple exclusion process (ASEP). This is a dynamics where the stationary non-equilibrium states are well studied [5,8,9].…”