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 ($\kappa$) and heat flux ($\mathbf{q}$) 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 ($\lambda$) 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., $\mathbf{q}\ll {\lambda }^2{\mathrm{\nabla }}^2\mathbf{q}$, on the calculated $\kappa$ and $\mathbf{q}$ values and the use of first- and second-order boundary conditions. In particular, by explicit calculation of $\mathbf{q}$ and its higher-order derivatives across the entire simulation domain, we find that the assumption $\mathbf{q}\ll {\lambda }^2{\mathrm{\nabla }}^2\mathbf{q}$, which is often employed for $\mathrm{Kn}\gg 1$, 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.

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• Heat transfer
• Lattice thermal conductivity
• Phonons
• Transport phenomena
• Boltzmann theory
• Finite-element method
• Hydrodynamics
• Monte Carlo methods