Generalized Fourier's law in mesoscopic systems
Abstract
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 κeff(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 Λz≠Λr. 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 Si0.2Ge0.8 and the rise of the apparent interface conductance of Al/Si0.99Ge0.01, and contains the two-channel ballistic/diffusive model as a two-sector projection.
Versions
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BibTeX
@misc{lapas2026generalized,
title = {{Generalized Fourier's law in mesoscopic systems}},
author = {Luciano C. Lapas},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.23909},
eprint = {2606.23909},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.23909},
}
RIS
TY - UNPB
TI - Generalized Fourier's law in mesoscopic systems
AU - Luciano C. Lapas
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2606.23909
UR - https://arxiv.org/abs/2606.23909
AB - 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 κ_eff(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≠Λ_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_0.2Ge_0.8 and the rise of the apparent interface conductance of Al/Si_0.99Ge_0.01, and contains the two-channel ballistic/diffusive model as a two-sector projection.
ER -