Derivation of the generalized heat transport equation and comparison with existing models

We derive a generalized higher-order heat transport equation from first principles with the closure based on Onsager's symmetry principle consistent second-order single-particle distribution function. The equation contains linear, nonlinear, local, nonlocal, and relaxation terms, and it unifies existing models, such as the Cattaneo-Vernotte model, the dual-phase-lag model, and phonon hydrodynamics, as special cases. The equation is analytically solved for (i) a thin-film semiconductor with diffusive boundaries to obtain effective thermal conductivity (κ) and heat flux (q) as a function of film thickness, where the equation reduces to a similar form as phonon hydrodynamics, and (ii) for flux-limited behavior, where only nonlinear terms survive. The former analytical solution is compared with the in-house Monte Carlo solution of the Boltzmann transport equation using the phonon-dispersion relation and mean free path (λ) in the entire Brillouin zone for varying Knudsen number (Kn), previous experimental data, and other mesoscopic models. For the analytical solution of flux-limited behavior, by comparing the solution of the general equation with previous reports, we highlight the role of nonlinear terms, which have hitherto relied on phenomenological combinations rather than first-principles derivations. We further systematically investigate the Stokes approximation, i.e., qλ²∇²q, on the calculated κ and q values and the use of first- and second-order boundary conditions. In particular, by explicit calculation of q and its higher-order derivatives across the entire simulation domain, we find that the assumption qλ²∇²q, which is often employed for Kn1, is not applicable near boundaries. The study is significant because of the transport equation proposed here, which can be applied to a range of micro- and nanoscale heat transfer problems.