Understanding ballistic phonon transport at the nanoscale is critical for heat management in modern electronic and photonic devices. While semiclassical Boltzmann transport equations (BTE) successfully describe bulk properties, they often neglect phase coherence, which becomes dominant in heterogeneous nanostructures. In this work, we present a real-space derivation of the Wigner transport equation (WTE) based on a power series expansion of a real-space generator. Starting from a tight-binding-like representation of lattice dynamics, our approach enables the systematic inclusion of phonon coherences through off-diagonal density-matrix elements directly in the spatial domain. Unlike existing formulations limited to homogeneous systems, the proposed framework naturally accounts for coherence-induced phenomena, such as phonon reflections at material interfaces. Benchmarks against full-band BTE models demonstrate that this method accurately reproduces semiclassical limits while providing a physically transparent and computationally flexible bridge between lattice-scale interactions and coherent phase-space transport in heterogeneous material systems.

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• Ballistic transport
• Phonons
• Methods in transport
• Tight-binding model