Quasi-static limit for the asymmetric simple exclusion

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