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We propose a general mechanism to rigorously derive an exact, closed systemfor the hydrodynamic variables from linear kinetic models. Our approach, basedon spectral theory, structural properties of eigenvectors and the theory ofslow manifolds, allows us to define a unique, optimal reduction in phase spaceclose to the global equilibrium distribution. The hydrodynamically constrainedsystem induces a modification of entropy that ensures pure viscous dissipationon the hydrodynamic manifold, which is interpreted as a non-local variant ofKorteweg's theory of viscosity-capillarity balance.

Rigorous Hydrodynamics for Linear Boltzmann Equations and Viscosity-Capillarity