Quasi-static limit for the asymmetric simple exclusion

Abstract

We study the one-dimensional asymmetric simple exclusion process on the lattice $\{1, \dots ,N\}$ 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.

Appendix A. Logarithmic Sobolev inequalities

In this appendix we fix a box of length k. For $\rho \in (0,1)$, let $\nu _\rho$ be the product Bernoulli measure on ${\Omega }_k=\{0,1\}^K$ with density $\rho$. For $h = 0, 1, \dots , k$, let $\nu _\rho (\eta |h) ={{\tilde{\nu }}}(\eta |h)$ be the uniform distribution on

$$\begin{aligned} {\Omega }_{k,h} := \left\{ \eta \in {\Omega }_k~\bigg |~\sum _{i=1}^k \eta _j =h \right\} , \end{aligned}$$

(A.1)

and ${\bar{\nu }}_\rho (h)$ be the Binomial distribution ${\mathcal {B}}(k,\rho )$.

The log-Sobolev inequality for the simple exclusion ([16]) yields that there exists a universal constant $C_{\mathrm {LS}}$ such that

$$\begin{aligned} \begin{aligned} \sum _{\eta \in {\Omega }_{k,h}} f(\eta )\log f(\eta ) {{\tilde{\nu }}}(\eta |h) \le \frac{C_{\mathrm {LS}}k^2}{2} \sum _{\eta \in {\Omega }_{k,h}} \sum _{i=1}^{k-1} \left( \sqrt{f} (\eta ^{i,i+1}) - \sqrt{f} (\eta ) \right) ^2 {{\tilde{\nu }}}(\eta |h). \end{aligned} \end{aligned}$$

(A.2)

for any $f\ge 0$ on ${\Omega }_{k,h}$ such that $\sum _{\eta \in {\Omega }_{k,h}} f{{\tilde{\nu }}}(\eta |h) = 1$.

In the following we extend (A.2) to a log-Sobolev inequality associated to the product measure $\nu _\rho$ with boundaries. The result is necessary for the boundary block estimates in Sects. 8.3 and 8.4.

Proposition A.1

There exists a constant $C_\rho$ such that

$$\begin{aligned} \begin{aligned} \sum _{\eta \in {\Omega }_k} f(\eta )\log f(\eta ) \nu _\rho (\eta ) \le&C_\rho k^2 \sum _{\eta \in {\Omega }_k} \sum _{i=1}^{k-1} \left( \sqrt{f} (\eta ^{i,i+1}) - \sqrt{f} (\eta ) \right) ^2 \nu _\rho (\eta ) \\&+ C_\rho k \sum _{\eta \in {\Omega }_k} \rho ^{1-\eta _1}(1\!-\!\rho )^{\eta _1} \left( \sqrt{f} (\eta ^1) - \sqrt{f} (\eta ) \right) ^2 \nu _\rho (\eta ). \end{aligned} \end{aligned}$$

(A.3)

for any $f\ge 0$ on ${\Omega }_k$ such that $\sum _{\eta \in {\Omega }_k} f\nu _\rho = 1$.

Proof

As the reference measures $\nu _\rho$ are equivalent for $0<\rho <1$, without loss of generality we can fix $\rho =1/2$ and thus $\nu _\rho \equiv 2^{-k}$. Consider the log-Sobolev inequality for the dynamics where particles are created and destroyed at each site with intensity 1/2. Since this is a product dynamics, the log-Sobolev constant is uniform in k:

$$\begin{aligned} \begin{aligned} \frac{1}{2^k}\sum _{\eta \in {\Omega }_k} f(\eta )\log f(\eta ) \le \frac{C}{2^{k+1}} \sum _{i=1}^k \sum _{\eta \in {\Omega }_k} \left[ \sqrt{f}(\eta ^i) - \sqrt{f}(\eta ) \right] ^2. \end{aligned} \end{aligned}$$

(A.4)

We apply a telescopic argument on (A.4). For $\eta \in \Omega _k$ and $1 \le i \le k$, let

$$\begin{aligned} \begin{aligned} \tau _0 := \eta , \quad \tau _j := {\left\{ \begin{array}{ll} (\tau _{j-1})^{i-j,i-j+1}, &{}1 \le j \le i-1 \\ (\tau _{j-1})^1, &{}j=i \\ (\tau _{j-1})^{j-i,j-i+1}, &{}i+1 \le j \le 2i-1. \end{array}\right. } \end{aligned} \end{aligned}$$

(A.5)

Observing that $\tau _{2i-1}=\eta ^i$, therefore

$$\begin{aligned} \sqrt{f}(\eta ^i) - \sqrt{f}(\eta ) = \sum _{j=0}^{2i-2} \left[ \sqrt{f}(\tau _{j+1}) - \sqrt{f}(\tau _j) \right] , \end{aligned}$$

(A.6)

and elementary computation then gives

$$\begin{aligned} \left[ \sqrt{f}(\eta ^i) - \sqrt{f}(\eta )\right] ^2 \le&\;4 (i-1) \sum _{0 \le j \le 2(i-1), j\not =i-1} \left[ \sqrt{f}(\tau _{j+1}) - \sqrt{f}(\tau _j) \right] ^2 \\&+2 \left[ \sqrt{f}(\tau _i) - \sqrt{f}(\tau _{i-1}) \right] ^2. \end{aligned}$$

Noting that as $\rho =1/2$, $\nu _\rho$ is invariant with respect to the exchange, creation as well as elimination of particles, we obtain by summing up in $\eta$ that

$$\begin{aligned} \begin{aligned} \sum _{\eta \in {\Omega }_k} \left[ \sqrt{f}(\eta ^i) \!-\! \sqrt{f}(\eta ) \right] ^2\nu _\rho (\eta ) \le&\;8(i-1) \sum _{\eta \in {\Omega }_k} \sum _{j=1}^{i-1} \left[ \sqrt{f}(\eta ^{j,j+1}) \!-\! \sqrt{f}(\eta ) \right] ^2\nu _\rho (\eta ) \\&+ 2\sum _{\eta \in {\Omega }_k} \left[ \sqrt{f}(\eta ^1) - \sqrt{f}(\eta ) \right] ^2\nu _\rho (\eta ). \end{aligned} \end{aligned}$$

Summing up in i we get the required inequality. $\square$