Non-local Hydrodynamics as a slow manifold for the one-dimensional kinetic equation

We prove an explicit, non-local hydrodynamic closure for the linear one-dimensional kinetic equation independent on the size of the relaxation time. We compare this dynamical equation to the local approximations obtained from the Chapman–Enskog expansion for small relaxation times. Our results rely on the spectral theory of Jacobi operators with rank-one perturbations.