Phonon Boltzmann equation non-local in space and time: the partial failure of the generalized Fourier law

This note has few new results except, at the end, a redefinition of the `thermal distributor. The main purpose of this note is to clarify the solution of the non-local Peierls Boltzmann equation found by Hua and Lindsay (Phys. Rev. B 102, 104310 (2020)). The new, non-Fourier term (B) (in J=-kappa grad T + B) that occurs in non-local situations, gives rise also to a new term in the thermal distributor.