Fourier's law fails when the mean free path of the energy carriers becomes comparable to the length and time scales over which the temperature field varies. We derive a thermodynamically consistent generalization in which the conductivity becomes a nonlocal memory operator $\mathbfκ_{\mathrm{eff}}(\mathbf{k},ω)$, by combining mesoscopic nonequilibrium thermodynamics with the Mori--Kubo--Zwanzig projection-operator formalism. The Onsager kernel is a tensor sum over vibrational normal modes weighted by their Bose heat capacities and relaxation functions; it satisfies the second law for any closure consistent with the fluctuation--dissipation theorem and reduces, in the relaxation-time closure, to the nonlocal conductivity of the phonon Boltzmann equation. Two consequences follow. First, direction-dependent sampling of this tensor in a pump--probe experiment makes a nominally isotropic crystal exhibit apparent conductivities $Λ_z\neqΛ_r$. Second, the modulation frequency does not introduce temporal memory but sets the probed wavevector through the thermal penetration depth, so that the frequency-dependent suppression of the apparent conductivity measured by time-domain thermoreflectance on Ge and SiGe alloys is a spatial-nonlocality effect, with effective lengths of $0.25$--$0.4~μ$m. With published mean-free-path spectra and no parameter adjusted to the thermoreflectance data, the kernel reproduces the spot-size dependence of Si, the frequency dependence of the apparent conductivity of Ge and Si$_{0.2}$Ge$_{0.8}$ and the rise of the apparent interface conductance of Al/Si$_{0.99}$Ge$_{0.01}$, and contains the two-channel ballistic/diffusive model as a two-sector projection.
Spectroscopy of Heat Transport and Violation of the Wiedemann--Franz Law in a GaAs Hydrodynamic Mesoscopic Channel
São Carlos Institute of Physics, University of São Paulo（圣保罗大学圣卡洛斯物理研究所）；Physics Department, Federal University of São Carlos（圣卡洛斯联邦大学物理系）；Ioffe Institute（约费研究所）；Institute of Physics, University of São Paulo（圣保罗大学物理研究所）；Institute of Semiconductor Physics（半导体物理研究所）
