- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Time-domain theory of transient heat conduction in the local limit

David E. Crawford1,*, Yi Zeng2,*,†, Judith Vidal2, and Jianjun Dong1,‡

  • *These authors contributed equally to this work.
  • †Contact author: yi.zeng@nrel.gov
  • ‡Contact author: dongjia@auburn.edu
PDF

Phys. Rev. B 112, 115201 – Published 2 September, 2025

DOI: https://doi.org/10.1103/p8wg-p1j3

Abstract

Ultrafast and nanoscale heat conduction demands a unified theoretical framework that rigorously bridges macroscopic transport equations with microscopic material properties derived from statistical physics. Existing empirical generalizations of Fourier's law often lack a solid microscopic foundation, failing to connect observed non-Fourier behavior with underlying atomic-scale mechanisms. In this work, we present a time-domain theory of transient heat conduction rooted in Zwanzig's statistical theory of irreversible processes. Central to this framework is the time-domain transport function Z↔(t) defined through equilibrium time-correlation functions of heat fluxes. This function generalizes the conventional concept of steady-state thermal conductivity, governing the transition of conduction dynamics from onset second sound type wave propagation at finite speeds to diffusion-dominated behavior across broad temporal and spatial scales. Unlike phonon hydrodynamic models that rely on mesoscopic constructs such as phonon drift velocity, our approach provides a quantitative and microscopic description of intrinsic memory effects in transient heat fluxes and applies universally to bulk materials at any temperature or length scale. By integrating atomistic-scale first-principles calculations with continuum-level macroscopic equations, this framework offers a robust foundation for numerical simulations of transient temperature fields. Furthermore, it facilitates the interpretation and design of transient thermal grating experiments using nanometer-scale heat sources and ultrafast laser systems in the extreme ultraviolet and x-ray wavelength ranges, advancing our understanding of heat dissipation dynamics.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (91)

  1. J. B. J. Fourier, Théorie Analytique de la Chaleur (Gauthier-Villars, 1888), Vol. 1.
  2. H. Carslaw and J. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford University Press, Oxford, 1959).
  3. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, Oxford, 2001).
  4. M. S. Green, Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids, J. Chem. Phys. 22, 398 (1954).
  5. J. Callaway, Model for lattice thermal conductivity at low temperatures, Phys. Rev. 113, 1046 (1959).
  6. R. Zwanzig, Time-correlation functions and transport coefficients in statistical mechanics, Annu. Rev. Phys. Chem. 16, 67 (1965).
  7. R. Guyer and J. Krumhansl, Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals, Phys. Rev. 148, 778 (1966).
  8. A. Majumdar, C. Tien, and F. Gemer, Microscale energy transport in solids, Microscale Energy Transport 3 (1997).
  9. G. Chen, Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons (Oxford University Press, Oxford, 2005).
  10. Y. Dubi and M. Di Ventra, Colloquium: Heat flow and thermoelectricity in atomic and molecular junctions, Rev. Mod. Phys. 83, 131 (2011).
  11. N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, Colloquium: Phononics: Manipulating heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84, 1045 (2012).
  12. R. Biele, R. D'Agosta, and A. Rubio, Time-dependent thermal transport theory, Phys. Rev. Lett. 115, 056801 (2015).
  13. P. B. Allen and J. L. Feldman, Thermal conductivity of disordered harmonic solids, Phys. Rev. B 48, 12581 (1993).
  14. C. Hua, L. Lindsay, X. Chen, and A. J. Minnich, Generalized Fourier's law for nondiffusive thermal transport: Theory and experiment, Phys. Rev. B 100, 085203 (2019).
  15. G. Chen, Non-Fourier phonon heat conduction at the microscale and nanoscale, Nat. Rev. Phys. 3, 555 (2021).
  16. M. Simoncelli, N. Marzari, and F. Mauri, Unified theory of thermal transport in crystals and glasses, Nat. Phys. 15, 809 (2019).
  17. M. Simoncelli, N. Marzari, and A. Cepellotti, Generalization of Fourier's law into viscous heat equations, Phys. Rev. X 10, 011019 (2020).
  18. Y. Zeng, J. T. Avritte, and J. Dong, Generalized Langevin equation theory of thermal conduction across material interfaces, Phys. Status Solidi B 258, 2000454 (2021).
  19. P. B. Allen and N. A. Nghiem, Heat pulse propagation and nonlocal phonon heat transport in one-dimensional harmonic chains, Phys. Rev. B 105, 174302 (2022).
  20. Y. Zeng and J. Khodadadi, Molecular dynamics simulations of the crystallization process of n-alkane mixtures and the resulting thermal conductivity, Energy & Fuels 32, 11253 (2018).
  21. Y. Zeng, J. Dong, and J. Khodadadi, Thermal coupling-decoupling mechanism of heat transfer across van der Waals interfaces in n-eicosane, Int. J. Heat Mass Transf. 164, 120603 (2021).
  22. J. Dong, O. F. Sankey, and C. W. Myles, Theoretical study of the lattice thermal conductivity in Ge framework semiconductors, Phys. Rev. Lett. 86, 2361 (2001).
  23. D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H. J. Maris, R. Merlin, and S. R. Phillpot, Nanoscale thermal transport, J. Appl. Phys. 93, 793 (2003).
  24. D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E. Goodson, P. Keblinski, W. P. King, G. D. Mahan, A. Majumdar et al., Nanoscale thermal transport. II. 2003–2012, Appl. Phys. Rev. 1, 011305 (2014).
  25. Y. K. Koh and D. G. Cahill, Frequency dependence of the thermal conductivity of semiconductor alloys, Phys. Rev. B 76, 075207 (2007).
  26. M. E. Siemens, Q. Li, R. Yang, K. A. Nelson, E. H. Anderson, M. M. Murnane, and H. C. Kapteyn, Quasi-ballistic thermal transport from nanoscale interfaces observed using ultrafast coherent soft x-ray beams, Nat. Mater. 9, 26 (2010).
  27. A. J. Minnich, J. A. Johnson, A. J. Schmidt, K. Esfarjani, M. S. Dresselhaus, K. A. Nelson, and G. Chen, Thermal conductivity spectroscopy technique to measure phonon mean free paths, Phys. Rev. Lett. 107, 095901 (2011).
  28. J. A. Johnson, A. A. Maznev, J. Cuffe, J. K. Eliason, A. J. Minnich, T. Kehoe, C. M. S. Torres, G. Chen, and K. A. Nelson, Direct measurement of room-temperature nondiffusive thermal transport over micron distances in a silicon membrane, Phys. Rev. Lett. 110, 025901 (2013).
  29. Y. Hu, L. Zeng, A. J. Minnich, M. S. Dresselhaus, and G. Chen, Spectral mapping of thermal conductivity through nanoscale ballistic transport, Nat. Nanotechnol. 10, 701 (2015).
  30. K. Hoogeboom-Pot, J. Hernandez-Charpak, T. Frazer, X. Gu, E. Turgut, E. Anderson, W. Chao, J. Shaw, R. Yang, M. Murnane et al., Mechanical and thermal properties of nanomaterials at sub-50 nm dimensions characterized using coherent euv beams, in Metrology, Inspection, and Process Control for Microlithography XXIX (SPIE, 2015), Vol. 9424, pp. 401–408.
  31. T. D. Frazer, J. L. Knobloch, K. M. Hoogeboom-Pot, D. Nardi, W. Chao, R. W. Falcone, M. M. Murnane, H. C. Kapteyn, and J. N. Hernandez-Charpak, Engineering nanoscale thermal transport: Size- and spacing-dependent cooling of nanostructures, Phys. Rev. Appl. 11, 024042 (2019).
  32. F. Bencivenga, R. Mincigrucci, F. Capotondi, L. Foglia, D. Naumenko, A. Maznev, E. Pedersoli, A. Simoncig, F. Caporaletti, V. Chiloyan et al., Nanoscale transient gratings excited and probed by extreme ultraviolet femtosecond pulses, Sci. Adv. 5, eaaw5805 (2019).
  33. A. Beardo, J. L. Knobloch, L. Sendra, J. Bafaluy, T. D. Frazer, W. Chao, J. N. Hernandez-Charpak, H. C. Kapteyn, B. Abad, M. M. Murnane et al., A general and predictive understanding of thermal transport from 1D- and 2D-confined nanostructures: Theory and experiment, ACS Nano 15, 13019 (2021).
  34. J. Jeong, X. Li, S. Lee, L. Shi, and Y. Wang, Transient hydrodynamic lattice cooling by picosecond laser irradiation of graphite, Phys. Rev. Lett. 127, 085901 (2021).
  35. B. McBennett, A. Beardo, E. E. Nelson, B. Abad, T. D. Frazer, A. Adak, Y. Esashi, B. Li, H. C. Kapteyn, M. M. Murnane et al., Universal behavior of highly confined heat flow in semiconductor nanosystems: From nanomeshes to metalattices, Nano Lett. 23, 2129 (2023).
  36. V. Peshkov, Determination of the velocity of propagation of the second sound in helium II, J. Phys. USSR 10, 389 (1946).
  37. C. C. Ackerman, B. Bertman, H. A. Fairbank, and R. Guyer, Second sound in solid helium, Phys. Rev. Lett. 16, 789 (1966).
  38. H. E. Jackson, C. T. Walker, and T. F. McNelly, Second sound in NaF, Phys. Rev. Lett. 25, 26 (1970).
  39. V. Narayanamurti and R. Dynes, Observation of second sound in Bismuth, Phys. Rev. Lett. 28, 1461 (1972).
  40. A. Koreeda, R. Takano, and S. Saikan, Second sound in SrTiO3, Phys. Rev. Lett. 99, 265502 (2007).
  41. A. Beardo, M. López-Suárez, L. A. Pérez, L. Sendra, M. I. Alonso, C. Melis, J. Bafaluy, J. Camacho, L. Colombo, R. Rurali et al., Observation of second sound in a rapidly varying temperature field in Ge, Sci. Adv. 7, eabg4677 (2021).
  42. S. Huberman, R. A. Duncan, K. Chen, B. Song, V. Chiloyan, Z. Ding, A. A. Maznev, G. Chen, and K. A. Nelson, Observation of second sound in graphite at temperatures above 100 K, Science 364, 375 (2019).
  43. Z. Ding, K. Chen, B. Song, J. Shin, A. A. Maznev, K. A. Nelson, and G. Chen, Observation of second sound in graphite over 200 K, Nat. Commun. 13, 285 (2022).
  44. D. D. Joseph and L. Preziosi, Heat waves, Rev. Mod. Phys. 61, 41 (1989).
  45. J. C. Maxwell, On the dynamical theory of gases, Philos. Trans. Roy. Soc. London 157, 49 (1867).
  46. C. Cattaneo, Sur une forme de l'equation de la chaleur eliminant la paradoxe d'une propagation instantantee, Compt. Rendu 247, 431 (1958).
  47. P. Vernotte, Les paradoxes de la theorie continue de l'equation de la chaleur, Compt. Rendu 246, 3154 (1958).
  48. J. Dong, O. F. Sankey, and G. Kern, Theoretical study of the vibrational modes and their pressure dependence in the pure clathrate-II silicon framework, Phys. Rev. B 60, 950 (1999).
  49. X. Tang, J. Dong, P. Hutchins, O. Shebanova, J. Gryko, P. Barnes, J. K. Cockroft, M. Vickers, and P. F. McMillan, Thermal properties of Si136: Theoretical and experimental study of the type-II clathrate polymorph of Si, Phys. Rev. B 74, 014109 (2006).
  50. A. Ward, D. A. Broido, D. A. Stewart, and G. Deinzer, Ab initio theory of the lattice thermal conductivity in diamond, Phys. Rev. B 80, 125203 (2009).
  51. X. Tang and J. Dong, Pressure dependence of harmonic and anharmonic lattice dynamics in MgO: A first-principles calculation and implications for lattice thermal conductivity, Phys. Earth Planet. Inter. 174, 33 (2009).
  52. A. Togo and I. Tanaka, First principles phonon calculations in materials science, Scr. Mater. 108, 1 (2015).
  53. W. Li, J. Carrete, N. A. Katcho, and N. Mingo, Shengbte: A solver of the Boltzmann transport equation for phonons, Comput. Phys. Commun. 185, 1747 (2014).
  54. T. Feng and X. Ruan, Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids, Phys. Rev. B 93, 045202 (2016).
  55. N. K. Ravichandran and D. Broido, Phonon-phonon interactions in strongly bonded solids: Selection rules and higher-order processes, Phys. Rev. X 10, 021063 (2020).
  56. X. Tang and J. Dong, Lattice thermal conductivity of MgO at conditions of earth's interior, Proc. Natl. Acad. Sci. USA 107, 4539 (2010).
  57. X. Tang, M. C. Ntam, J. Dong, E. S. Rainey, and A. Kavner, The thermal conductivity of earth's lower mantle, Geophys. Res. Lett. 41, 2746 (2014).
  58. A. Togo, L. Chaput, and I. Tanaka, Distributions of phonon lifetimes in Brillouin zones, Phys. Rev. B 91, 094306 (2015).
  59. C. Hua and A. J. Minnich, Analytical Green's function of the multidimensional frequency-dependent phonon Boltzmann equation, Phys. Rev. B 90, 214306 (2014).
  60. C. de Tomas, A. Cantarero, A. Lopeandia, and F. Alvarez, Thermal conductivity of group-IV semiconductors from a kinetic- collective model, Proc. Roy. Soc. A 470, 20140371 (2014).
  61. S. Lee, D. Broido, K. Esfarjani, and G. Chen, Hydrodynamic phonon transport in suspended graphene, Nat. Commun. 6, 6290 (2015).
  62. A. Cepellotti, G. Fugallo, L. Paulatto, M. Lazzeri, F. Mauri, and N. Marzari, Phonon hydrodynamics in two-dimensional materials, Nat. Commun. 6, 6400 (2015).
  63. A. Cepellotti and N. Marzari, Transport waves as crystal excitations, Phys. Rev. Mater. 1, 045406 (2017).
  64. P. T. Alvarez, Thermal Transport in Semiconductors: First Principles and Phonon Hydrodynamics (Springer, Berlin, 2018).
  65. V. Chiloyan, S. Huberman, Z. Ding, J. Mendoza, A. A. Maznev, K. A. Nelson, and G. Chen, Green's functions of the boltzmann transport equation with the full scattering matrix for phonon nanoscale transport beyond the relaxation-time approximation, Phys. Rev. B 104, 245424 (2021).
  66. H. Honarvar, J. L. Knobloch, T. D. Frazer, B. Abad, B. McBennett, M. I. Hussein, H. C. Kapteyn, M. M. Murnane, and J. N. Hernandez-Charpak, Directional thermal channeling: A phenomenon triggered by tight packing of heat sources, Proc. Natl. Acad. Sci. USA 118, e2109056118 (2021).
  67. J. Ward and J. Wilks, Second sound and the thermo-mechanical effect at very low temperatures, London, Edinburgh, Dublin Philos. Mag. J. Sci. 43, 48 (1952).
  68. M. Chester, Second sound in solids, Phys. Rev. 131, 2013 (1963).
  69. J. Sussmann and A. Thellung, Thermal conductivity of perfect dielectric crystals in the absence of Umklapp processes, Proc. Phys. Soc. 81, 1122 (1963).
  70. C. P. Enz, One-particle densities, thermal propagation, and second sound in dielectric crystals, Ann. Phys. 46, 114 (1968).
  71. R. A. Guyer and J. Krumhansl, Solution of the linearized phonon Boltzmann equation, Phys. Rev. 148, 766 (1966).
  72. A. Beardo, S. Alajlouni, L. Sendra, J. Bafaluy, A. Ziabari, Y. Xuan, J. Camacho, A. Shakouri, and F. X. Alvarez, Hydrodynamic thermal transport in silicon at temperatures ranging from 100 to 300 K, Phys. Rev. B 105, 165303 (2022).
  73. L. Sendra, A. Beardo, P. Torres, J. Bafaluy, F. X. Alvarez, and J. Camacho, Derivation of a hydrodynamic heat equation from the phonon Boltzmann equation for general semiconductors, Phys. Rev. B 103, L140301 (2021).
  74. Y. Guo and M. Wang, Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures, Phys. Rev. B 97, 035421 (2018).
  75. L. Sendra, A. Beardo, J. Bafaluy, P. Torres, F. X. Alvarez, and J. Camacho, Hydrodynamic heat transport in dielectric crystals in the collective limit and the drifting/driftless velocity conundrum, Phys. Rev. B 106, 155301 (2022).
  76. Y. Zeng and J. Dong, Fokker-planck equation for lattice vibration: Stochastic dynamics and thermal conductivity, Phys. Rev. B 99, 014306 (2019).
  77. C. Hua and L. Lindsay, Space-time dependent thermal conductivity in nonlocal thermal transport, Phys. Rev. B 102, 104310 (2020).
  78. M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Ration. Mech. Anal. 31, 113 (1968).
  79. J. W. Nunziato, On heat conduction in materials with memory, Q. Appl. Math. 29, 187 (1971).
  80. F. Alvarez and D. Jou, Memory and nonlocal effects in heat transport: From diffusive to ballistic regimes, Appl. Phys. Lett. 90, 083109 (2007).
  81. R. Zwanzig, Memory effects in irreversible thermodynamics, Phys. Rev. 124, 983 (1961).
  82. R. Zwanzig, Frequency-dependent transport coefficients in fluid mechanics, J. Chem. Phys. 43, 714 (1965).
  83. H. R. Shanks, P. D. Maycock, P. H. Sidles, and G. C. Danielson, Thermal conductivity of silicon from 300 to 1400∘K, Phys. Rev. 130, 1743 (1963).
  84. C. J. Glassbrenner and G. A. Slack, Thermal conductivity of silicon and germanium from 3∘K to the melting point, Phys. Rev. 134, A1058 (1964).
  85. W. Fulkerson, J. P. Moore, R. K. Williams, R. S. Graves, and D. L. McElroy, Thermal conductivity, electrical resistivity, and seebeck coefficient of silicon from 100to1300∘K, Phys. Rev. 167, 765 (1968).
  86. P. Torres, A. Ziabari, A. Torelló, J. Bafaluy, J. Camacho, X. Cartoixà, A. Shakouri, and F. X. Alvarez, Emergence of hydrodynamic heat transport in semiconductors at the nanoscale, Phys. Rev. Mater. 2, 076001 (2018).
  87. J. Tur-Prats, Z. Han, A. Beardo, X. Ruan, and F. X. Alvarez, High-order anharmonicities shape phonon hydrodynamic effects in graphene, Nano Lett. 2511203 (2025).
  88. Y. Zeng and J. Dong, Unified statistical theory of heat conduction in nonuniform media, arXiv:2507.05682.
  89. A. Vega-Flick, R. A. Duncan, J. K. Eliason, J. Cuffe, J. A. Johnson, J.-P. Peraud, L. Zeng, Z. Lu, A. A. Maznev, E. N. Wang et al., Thermal transport in suspended silicon membranes measured by laser-induced transient gratings, AIP Adv. 6, 121903 (2016).
  90. D. G. Cahill, Analysis of heat flow in layered structures for time-domain thermoreflectance, Rev. Sci. Instrum. 75, 5119 (2004).
  91. G. Kresse and J. Furthmüller, Efficient ab initio calculations of lattice dynamics for crystals with a large number of atoms, Phys. Rev. B 54, 11169 (1996).

Outline

Information

Phys. Rev. B 112, 115201– Published 2 September, 2025
  • Received 31 December 2024
  • Revised 19 July 2025
  • Accepted 13 August 2025
Crossmark - Check for updates button
Creative Commons Logo

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 National Renewable Energy Laboratory (NREL) Library, part of a national laboratory of the U.S. Department of Energy.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation