Weakly non-equilibrium properties of symmetric inclusion process with open boundaries

We study close to equilibrium properties of the one-dimensional Symmetric Inclusion Process by coupling it to two particle-reservoirs at the two boundaries with slightly different chemical potentials. The boundaries introduce irreversibility and induce a weak particle current in the system. We calculate the McLennan ensemble for SIP, which corresponds to the entropy production and the first order non-equilibrium correction for the stationary state. We find that the first order correction, while being a product measure, is not consistent with the local equilibrium measure. This provides an interesting counter-example to the more common case, for instance in the boundary driven symmetric exclusion process, where the entropy production accords with the local equilibrium measure.