Ballistic contributions to mesoscale thermal conductivity: Closing a flux-based model with lattice–Boltzmann–Peierls–Callaway simulations

The thermal transport in dielectric and semiconductor solids departs from classical diffusion when characteristic dimensions approach phonon scattering lengths, producing mixed ballistic–diffusive behavior. Although effective conductivity is often correlated with the Knudsen number (the ratio of scattering length to sample size), such formulations do not uniquely capture phonon backscattering or provide a consistent definition for reformulating bulk properties outside the diffusive infinite sample size limit. In the scope of this work, (i) the McKelvey–Shockley Flux Method (MSFM) is extended to decompose heat flux into ballistic and diffusive contributions, and (ii) a mesoscale thermal law is derived to explicitly relate the effective thermal conductivity with ballistic thermal conductance, Knudsen number, effective backscattering length, and bulk-like thermal conductivity. To identify the latter two parameters, we perform systematic steady-state numerical experiments using a D3Q19 Lattice–Boltzmann–Peierls–Callaway (LBPC) model with separate relaxation channels for normal and Umklapp scattering. From LBPC solutions, we extract an effective phonon mean free path (equivalent to backscattering length) and a bulk-like thermal conductivity from thickness sweep regressions and use them to close the MSFM-based constitutive relations. These LBPC-extracted closure parameters are validated against existing theoretical modeling frameworks, as well as experimental data from the literature on a range of different materials. The findings reflect quantitative consistency between MSFM and LBPC as independent descriptors of ballistic–diffusive phonon transport, providing an opportunity to reduce the degrees of freedom in the model. This self-consistent mesoscale description links microscopic scattering to macroscopic behavior and supports incorporating ballistic effects into continuum models relevant to ultrafast heating, thermoelectrics, nanoelectronics, thermophones, and nanoscale information storage.