  We develop a Boltzmann transport theory of coupled magnon-phonon transport in
ferromagnetic insulators. The explicit treatment of the magnon-phonon coupling
within the Boltzmann approach allows us to calculate the low-temperature
magnetic-field dependence of the spin-Seebeck voltage. Within the Boltzmann
theory we find that this magnetic field dependence shows similar features as
found by Flebus et al. [Phys. Rev. B 95, 144420 (2017)] for a strongly coupled
magnon phonon system that forms magnon-polarons, and consistent with
experimental findings in yttrium iron garnet by Kikkawa et al. [Phys. Rev.
Lett. 117, 207203 (2016)]. In addition to the anomalous magnetic-field
dependence of the spin Seebeck effect, we also predict a dependence on the
system size.
