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

Abstract

Consider the open symmetric exclusion process on a connected graph with vertexes in $[N-1]:=\{1,\dots ,N-1\}$ where points $1$ and $N-1$ are connected, respectively, to a left reservoir and a right reservoir with densities ${\rho }_L,{\rho }_R\in (0,1)$. We prove that the non-equilibrium steady state of such system is $${\mu }_{\text{stat}}=\sum _{I\subset \mathcal{P}([N-1])}F(I)({\otimes }_{x\in I}\mathrm{Bernoulli}({\rho }_{\mathrm{R}}){\otimes }_{\mathrm{y}\in [\mathrm{N}-1]\setminus \mathrm{I}}\mathrm{Bernoulli}({\rho }_{\mathrm{L}})).$$ In the formula above $\mathcal{P}([N-1])$ denotes the power set of $[N-1]$ while the numbers $F(I)>0$ are such that ${\sum }_{I\subset \mathcal{P}([N-1])}F(I)=1$ and given in terms of absorption probabilities of the absorbing stochastic dual process. Via probabilistic arguments we compute explicitly the factors $F(I)$ when the graph is a homogeneous segment.