  We show that vertex corrections to the quasi-harmonic Green-Kubo theory of
heat transport in insulators naturally lead to a generalisation of the
expression for the conductivity that could be derived from the linearized
Boltzmann equation, when the effects of the full scattering matrix are
accounted for. Our results, which are obtained from the Mori-Zwanzig
memory-function formalism, provide a fully ab initio derivation of the
linearized Boltzmann transport equation and establish a connection between two
recently proposed unified approaches to heat transport in insulating crystals
and glasses.
