Deviations from diffusive heat transport in high thermal conductivity crystalline insulators are generally understood within the framework of the phonon Boltzmann Transport Equation. However, for low thermal conductivity materials with large primitive cells or strong anharmonicity, the recently developed Wigner Transport Equation is more appropriate as it includes tunneling between overlapping phonon bands. In this work, via solutions to the Wigner Transport Equation, we develop a scheme to obtain the dynamics of the phonon populations and coherences as a function of an arbitrary heat source. The approach is applied to predict size effects and dynamical thermal conductivities in ${\mathrm{CsPbBr}}_3$ and ${\mathrm{La}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ using first-principles data as input. We predict significant deviations from the bulk thermal conductivity in these materials at length scales on the order of hundreds of nanometers to a few microns at room temperature, well within the reach of direct observation using current experimental techniques.

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• Heat transfer
• Phonons
• Thermal conductivity
• Boltzmann theory