  Many complex crystals show a flattening or even increasing lattice thermal
conductivity at high temperatures, which deviates from the traditional 1/T
decay trend given by conventional phonon theory. In this work, we predict the
thermal conductivity of Al2O3 that matches with experimental data from room
temperature to near melting point (2200 K). The lattice thermal conductivity is
found to be composed of contributions of phonon, diffuson, and radiation.
Phonon particle thermal conductivity decays approximately as ~T^-1.14 after
considering four-phonon scattering as well as finite temperature corrections to
lattice constant, harmonic, and anharmonic force constants. Diffuson
(inter-band tunneling) thermal conductivity increases roughly as ~T^0.43.
Radiation thermal conductivity increases as ~T2.51, being slightly smaller than
~T^3 due to the increase of phonon linewidth with temperature, which increases
photon extinction coefficient and reduces photon mean free path. At room
temperature, phonon, diffuson, and radiation contribute about 99%, 1%, and 0,
respectively. At 2200 K, the contributions change to 61%, 20%, and 19%,
respectively. Four-phonon scattering is important at ultra-high temperature,
decreasing the phonon thermal conductivity by a maximum of 24%. The
finite-temperature softening effects of harmonic and anharmonic force constants
increase the phonon thermal conductivity by a maximum of 36% at ultra-high
temperatures. We also verify that Green-Kubo MD can capture phonons' both
particle and wave natures, similar to the Wigner formalism.
