Steady-State Heat Transport: Ballistic-to-Diffusive with Fourier's Law

It is generally understood that Fourier's law does not describe ballistic phonon transport, which is important when the length of a material is similar to the phonon mean-free-path. Using an approach adapted from electron transport, we demonstrate that Fourier's law and the heat equation do capture ballistic effects, including temperature jumps at ideal contacts, and are thus applicable on all length scales. Local thermal equilibrium is not assumed, because allowing the phonon distribution to be out-of-equilibrium is important for ballistic and quasi-ballistic transport. The key to including the non-equilibrium nature of the phonon population is to apply the proper boundary conditions to the heat equation. Simple analytical solutions are derived, showing that i) the magnitude of the temperature jumps is simply related to the material properties and ii) the observation of reduced apparent thermal conductivity physically stems from a reduction in the temperature gradient and not from a reduction in actual thermal conductivity. We demonstrate how our approach, equivalent to Fourier's law, easily reproduces results of the Boltzmann transport equation, in all transport regimes, even when using a full phonon dispersion and mean-free-path distribution.