Sharp Convergence to Equilibrium for the SSEP with Reservoirs

Abstract

We consider the symmetric simple exclusion process evolving on the interval of length $n-1$ in contact with reservoirs of density $\rho \in (0,1)$ at the boundary. We use Yau's relative entropy method to show that if the initial measure is associated with a profile $u_0:[0,1]\to (0,1)$, then at explicit times $t^n(b)$ that depend on $u_0$, the distance to equilibrium, in total variation distance, converges, as $n\to \infty$, to a profile $\mathcal{G}(\gamma e^{-b})$. The parameter $\gamma$ also depends on the initial profile $u_0$ and $\mathcal{G}(m)$ stands for the total variation distance $\Vert \mathcal{N}(m,1)-\mathcal{N}(0,1){\Vert }_{\mathrm{TV}}$.

16 pages