Sharp Convergence to Equilibrium for the SSEP with Reservoirs

We consider the symmetric simple exclusion process evolving on the path of length n − 1 in contact with reservoirs of density ρ ∈ (0, 1) at the boundary. We use Yau’s relative entropy method to show that if the initial measure is associated with a profile u0 : [0, 1] → (0, 1), then at explicit times t (b) that depend on u0, the distance to equilibrium, in total variation distance, converges, as n → ∞, to a profile G(γe). The parameter γ also depends on the initial profile u0 and G(m) stands for the total variation distance ‖N (m, 1)−N (0, 1)‖TV .

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