Hydrodynamic limit for conservative spin systems with super-quadratic, partially inhomogeneous single-site potential

We consider an interacting unbounded spin system, with conservation of the mean spin. We derive quantitative rates of convergence to the hydrodynamic limit provided the single-site potential is a bounded perturbation of a strictly convex function with polynomial growth, and with an additional random inhomogeneous linear term. This additional linear term models the impact of a random chemical potential. The argument adapts the two-scale approach of Grunewald, Otto, Villani and Westdickenberg from the quadratic to the general case. The main ingredient is the derivation of a covariance estimate that is uniform in the system size. We also show that this covariance estimate can be used to change the iterative argument of [MO] for deducing the optimal scaling LSI for the canonical ensemble into a two-scale argument in the sense of [GOVW]. We also prove the LSI for canonical ensembles with an inhomogeneous linear term.