{
  "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 $\\mathbfκ_{\\mathrm{eff}}(\\mathbf{k},ω)$, 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.23909",
  "author_total": 1,
  "authors": [
    {
      "name": "Luciano C. Lapas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d6b1k2ym9x32x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.23909",
  "fulltext_url": null,
  "id": "d6b1k2ym9x32x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.23909",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mes-hall"
    ],
    [
      "arxiv",
      "cond-mat.stat-mech"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.23909",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d6b1k2ym9x32x.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:30Z",
      "native_id": "2606.23909",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.23909"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mes-hall",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Generalized Fourier's law in mesoscopic systems",
  "title_text": "Generalized Fourier's law in mesoscopic systems",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:33Z",
  "url": "https://research.draconitic.com/paper/d6b1k2ym9x32x",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1103/ggz5-nggb",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}