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  • Open Access

Unified statistical theory of heat conduction in nonuniform media

Yi Zeng1,* and Jianjun Dong2,†

  • *Contact author: yi.zeng@nlr.gov
  • †Contact author: dongjia@auburn.edu
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Phys. Rev. B 113, 235204 – Published 8 June, 2026

DOI: https://doi.org/10.1103/hx28-4ksc

Abstract

Using the Zwanzig projection-operator formalism, we derive a causal two-point spatiotemporal kernel for heat conduction, defined microscopically as a space-resolved equilibrium heat-flux time-correlation function, that encodes temporal memory, spatial nonlocality, and material heterogeneity on equal footing. Classical diffusion, nonlocal transport, and hydrodynamic models emerge as controlled asymptotic limits of this kernel, providing a unified constitutive description across diffusive, quasiballistic, and hydrodynamic regimes. Interfacial heat transfer is incorporated through a spatially resolved kernel formulation, in which the conventional Kapitza resistance arises as a coarse-grained limit. The kernel admits a spatiotemporal Green–Kubo representation and can, in principle, be evaluated from atomistic simulations for bulk media, providing a direct connection between microscopic dynamics and continuum transport without empirical closure. For crystalline solids, we derive explicit kernel forms in the hydrodynamic and attenuated-streaming limits and introduce a hybrid reduction that captures the coexistence of collective and quasiballistic transport. For disordered harmonic solids, the framework recovers a spatial diffusion kernel consistent with the Allen–Feldman limit. To illustrate the theory, we construct the kernel for silicon at room temperature within the relaxation-time approximation and apply it to transient thermal-grating configurations. Spatial nonlocality associated with the phonon mean-free-path distribution is the primary source of deviation from Fourier transport under these conditions, while temporal memory mainly influences short-time dynamics. These findings identify the spatiotemporal kernel as a unifying constitutive descriptor whose coarse-grained limits recover conventional transport coefficients.

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Phys. Rev. B 113, 235204– Published 8 June, 2026
  • Received 8 July 2025
  • Revised 14 April 2026
  • Accepted 26 May 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. Open access publication funded by the National Laboratory of the Rockies (NLR), part of a national laboratory of the U.S. Department of Energy.

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