  We consider the detailed structure of correlations in harmonic chains with
pinning and a bulk velocity flip noise during the heat relaxation phase which
occurs on diffusive time scales, for $t=O(L^2)$ where $L$ is the chain length.
It has been shown earlier that for non-degenerate harmonic interactions these
systems thermalize, and the dominant part of the correlations is given by local
thermal equilibrium determined by a temperature profile which satisfies a
linear heat equation. Here we are concerned with two new aspects about the
thermalization process: the first order corrections in $1/L$ to the local
equilibrium correlations and the applicability of kinetic theory to study the
relaxation process. Employing previously derived explicit uniform estimates for
the temperature profile, we first derive an explicit form for the first order
corrections to the particle position-momentum correlations. By suitably
revising the definition of the Wigner transform and the kinetic scaling limit
we derive a phonon Boltzmann equation whose predictions agree with the explicit
computation. Comparing the two results, the corrections can be understood as
arising from two different sources: a current-related term and a correction to
the position-position correlations related to spatial changes in the phonon
eigenbasis.
