Two Kinds of Local Temperature in Boltzmann Theory of Inhomogeneous Vibrational Heat Transport

Heat in insulators, carried by phonons, is described by the Peierls Boltzmann equation (PBE). In nanoscale situations, the heat current is partly diffusive (from phonons with mean free paths (mfp) less than sample size L), and partly ballistic (mfp > L). We find that, when the temperature gradient varies spatially, the local temperature T(r) appearing in the PBE differs from the temperature associated with the total non-equilibrium energy. This latter temperature is probably measured by most external probes. Our relaxation time approximation (RTA) solutions of the PBE quantify the difference, which is typically 5—20%. The difference depends on how much dispersion there is in phonon scattering rates 1/τ_Q, and disappears if 1/τ_Q=constant. The temperature difference does not seem to be an artifact of the RTA.