  This article provides non-trivial technical ingredients for the article "The
quantitative hydrodynamic limit of the Kawasaki dynamics" by the same authors.
In that work a quantitative version of the hydrodynamic limit is deduced using
a refinement of the two-scale approach. In this work, we deduce the strict
convexity of the coarse-grained Hamiltonian, a uniform logarithmic Sobolev
inequality for the canonical ensemble and the convergence of the coarse-grained
Hamiltonian to the macroscopic free energy of the system. We deduce those
results following an approach developed by Grunewald, Otto, Villani and
Westdickenberg. Because in our setting the associated coarse-graining operator
is non-local, the arguments are much more subtle and need additional
ingredients like the Brascamp-Lieb inequality and a multivariate local
central-limit theorem.
