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  • Letter
  • Open Access

Generalized Fourier’s law in mesoscopic systems

Luciano C. Lapas*

  • *Contact author: luciano.lapas@unila.edu.br
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APS Open Sci. 1, L000216 – Published 13 November, 2026

DOI: https://doi.org/10.1103/ggz5-nggb

Abstract

Fourier’s law fails when the mean free path of the energy carriers becomes comparable to the length scales and timescales 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 wave vector 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 0.25–0.4μm. 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.

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APS Open Sci. 1, L000216– Published 13 November, 2026
Vol. 1
  • Received 21 June 2026
  • Accepted 4 November 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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