  We propose a general mechanism to rigorously derive an exact, closed system
for the hydrodynamic variables from linear kinetic models. Our approach, based
on spectral theory, structural properties of eigenvectors and the theory of
slow manifolds, allows us to define a unique, optimal reduction in phase space
close to the global equilibrium distribution. The hydrodynamically constrained
system induces a modification of entropy that ensures pure viscous dissipation
on the hydrodynamic manifold, which is interpreted as a non-local variant of
Korteweg's theory of viscosity-capillarity balance.
