Non-equilibrium steady state of the symmetric exclusion process with reservoirs

Consider the open symmetric exclusion process on a connected graph with vertexes in $[N-1]:=\{1,\ldots, N-1\}$ where points $1$ and $N-1$ are connected, respectively, to a left reservoir and a right reservoir with densities $ρ_L,ρ_R\in(0,1)$. We prove that the non-equilibrium steady state of such sy