Nonlocal hydrodynamic phonon transport in two-dimensional materials: first-principles validation and experimental observation of the Guyer–Krumhansl phenomenology

In two-dimensional (2D) materials such as graphene, momentum-conserving normal phonon collisions can make heat flow like a viscous fluid, described by a Guyer–Krumhansl-type equation rather than Fourier's law. From the Peierls–Boltzmann equation in the Callaway model, Shang and Lü (2018; published 2020) derived a 2D Guyer–Krumhansl equation including the quadratic flexural mode, predicting Poiseuille flow with $I\propto w^{3}$, temperature-dependent second sound, steady-state heat-current vortices, and negative nonlocal thermal conductance in micron-scale high-quality graphene near 100 K, while flagging the gray scattering approximation, the truncated linearized expansion with $\vec q_1=0$, idealized non-slip boundaries, the unexplored coupled electron–phonon hydrodynamic regime, and the absence of any experimental confirmation in 2D materials. Later work replaced the gray coefficients with first-principles viscous-heat-equation and ab initio Boltzmann treatments, revised the 2D second-sound picture, and observed hydrodynamic phonon transport in bulk graphite and sapphire — but in no genuinely two-dimensional material, and nonlinear transport beyond relaxation-time forms remains untreated. The problem is to complete this program: quantitatively establish the nonlocal hydrodynamic phenomenology for realistic 2D materials, including boundary physics, nonlinear corrections, and electron–phonon coupling, and specify a concrete, falsifiable protocol for observing at least one hallmark signature in a monolayer or few-layer crystal.

Background

Heat in electrically insulating crystals is carried by phonons, the quantized vibrations of the lattice. In the ordinary diffusive regime the heat flux $\vec q$ obeys Fourier's law $\vec q=-\kappa\nabla T$, but when the relevant length scales become comparable to phonon mean free paths, or when collisions that conserve the total crystal momentum of the phonon gas are the fastest microscopic process, this local description fails and the phonon gas behaves like a fluid — the regime of phonon hydrodynamics. The kinetic foundation is the Peierls–Boltzmann equation, the semiclassical balance equation for the occupation of each phonon mode, whose collision integral splits into normal (N) processes, which conserve the gas momentum, and resistive (R) processes — Umklapp events (which transfer momentum to the lattice through a reciprocal-lattice vector), impurity and isotope scattering, and boundary scattering — which dissipate it.

In this regime Fourier's law is replaced, in the standard Guyer–Krumhansl (G–K) equation, by a Navier–Stokes-like law for the heat flux: the flux is driven by the temperature gradient but also experiences a phonon viscosity and nonlocal (spatially averaged) corrections set by the N-scattering time. The G–K framework organizes the classic hydrodynamic signatures of crystal heat transport: second sound, the propagation of temperature as a damped wave rather than a diffusing spread; and Poiseuille flow, a parabolic flux profile across a channel whose total heat current scales as $I\propto w^{3}$ with channel width $w$, by direct analogy with viscous fluid flow through a pipe.

Two-dimensional materials add a specific ingredient: the out-of-plane flexural acoustic mode (ZA mode), whose dispersion $\omega=ak^{2}$ is quadratic rather than linear at long wavelength, with a diverging equilibrium occupation. Shang and Lü (arXiv:1803.08372, 2018; published as Shang, Zhang, Guo, and Lü, Scientific Reports, 2020) derived the 2D analogue of the G–K equation. They started from the Peierls–Boltzmann equation in the Callaway model — in which the full collision integral is replaced by just two relaxation times, $\tau_N$ and $\tau_R$, taken to be the same for all phonon branches and wavevectors, the so-called gray approximation — and treated the phonon branches as isotropic linear modes plus one quadratic flexural mode. A multiscale expansion in the natural small parameter $\varepsilon=\tau_N/\tau_R$ (the source separately introduces the dimensionless diagnostic $\mathcal{K}=w^{2}/(\eta\tau_R)$, with $\eta$ the phonon viscosity, to characterize the relative contribution of diffusive and hydrodynamic transport), truncated at first order with the closure $\vec q_1=0$ for the first-order flux correction and with all nonlinear couplings of flux and temperature gradient discarded, yielded the 2D G–K equation. Solving it, they predicted Poiseuille flow with $I\propto w^{3}$ under a non-slip boundary condition, a temperature-dependent second-sound velocity and viscosity, steady-state vortices of the heat current around a point heat source, and negative nonlocal thermal conductance — heat locally flowing from colder to hotter regions — signatures familiar from classical fluids and from electron hydrodynamics in graphene, but at that time unobserved for phonons in 2D. With graphene parameters from density-functional theory, they estimated a hydrodynamic window around 100 K in micron-scale, high-quality (suspended) ribbons. The authors themselves listed what the predictions rest on: gray isotropic scattering rates; a truncated, linearized expansion; idealized non-slip boundaries without edge-specific phonon physics; neglect of electron–phonon coupling; and the absence of any experimental observation in 2D materials — while noting an encouragingly large parameter-space overlap between electron and phonon hydrodynamics in graphene.

A later theoretical framework recast this phenomenology on first-principles footing: the viscous heat equations of Simoncelli, Marzari, and Cepellotti (Physical Review X, 2020) coarse-grain the full linearized Boltzmann equation, so that thermal conductivity and viscosity tensors are computed ab initio and the G–K equation emerges as an isotropic, linear-band limit. This is the standard against which 2D hydrodynamic predictions are now benchmarked.

Problem Statement

Establish the nonlocal hydrodynamic phonon transport regime predicted by the two-dimensional Guyer–Krumhansl equation of Shang and Lü in real two-dimensional materials, in quantitatively validated form and with a concrete, falsifiable protocol for its experimental observation. Concretely, determine and demonstrate:

1. Realistic-scattering prediction. From mode-resolved, first-principles scattering — relaxing the gray Callaway approximation of branch- and wavevector-independent $\tau_N$ and $\tau_R$ and the isotropic linear-mode assumption — the transport coefficients and the nonlocal signatures (Poiseuille flow and its $I\propto w^{3}$ width scaling, second sound, steady-state heat-current vortices, and negative nonlocal thermal conductance) of high-quality two-dimensional materials, including the role of the quadratic flexural mode and of membrane elasticity.

2. Boundaries and nonlinearity. The robustness of these signatures under realistic boundary conditions (slip, edge roughness, finite size) and the corrections arising from retaining the higher-order flux term $\vec q_1\neq0$ and the nonlinear couplings between heat flux and temperature gradient, including a nonlinear hydrodynamic treatment that retains the full collision operator rather than relaxation-time forms.

3. Experimental observation protocol. A concrete, falsifiable protocol by which at least one hallmark nonlocal hydrodynamic signature could be observed in a genuinely two-dimensional material — a monolayer or few-layer crystal, such as suspended isotopically purified graphene — specifying the disorder level, temperature window, and sample dimensions, predicting the discriminating observables and quantitative signatures in agreement with the first-principles predictions of items 1 and 2, and with the mechanism (normal-scattering-driven viscosity rather than ballistic or boundary-limited transport) identified.

4. Coupled electron–phonon hydrodynamics. In the regime where the electron and phonon fluids of a two-dimensional material are simultaneously hydrodynamic, how electron–phonon momentum exchange modifies the phonon hydrodynamic equation and its nonlocal signatures, and how this coupled regime can be detected experimentally. Here the cited coupled electron–phonon hydrodynamic theories leave untreated the nonlocal signatures specific to a genuinely two-dimensional material and a concrete detection scheme.

The four parts are jointly posed, and within each item the target is the full conjunction of the ingredients enumerated in it. Item 1 comprises all four nonlocal signatures — Poiseuille flow with its $I\propto w^{3}$ width scaling, second sound, heat-current vortices, and negative nonlocal thermal conductance — obtained for a genuinely two-dimensional crystal from mode-resolved, non-gray first-principles scattering, with the quadratic flexural-mode and membrane-elasticity effects included. Item 2 comprises all of the listed boundary conditions (slip, edge roughness, finite size) and the nonlinear corrections: the retained flux term $\vec q_1\neq0$, the flux–temperature-gradient couplings, and a treatment that retains the full collision operator rather than relaxation-time forms. Item 3 is the specification of a concrete, quantitative experimental protocol with its falsifiable predictions (no physical measurement is required). Item 4 comprises both the nonlocal signatures specific to a genuinely two-dimensional material and the concrete detection scheme named above.

References

1. Man-Yu Shang and Jing-Tao Lü, 'Nonlocal hydrodynamic phonon transport in two-dimensional materials' (2018), arXiv:1803.08372, DOI 10.48550/arXiv.1803.08372, https://arxiv.org/abs/1803.08372
2. Man-Yu Shang, Chuang Zhang, Zhaoli Guo, and Jing-Tao Lü, 'Heat vortex in hydrodynamic phonon transport of two-dimensional materials', Scientific Reports 10, 8272 (2020), DOI 10.1038/s41598-020-65221-8, https://doi.org/10.1038/s41598-020-65221-8 (journal version of arXiv:1803.08372)
3. Michele Simoncelli, Nicola Marzari, and Andrea Cepellotti, 'Generalization of Fourier's Law into Viscous Heat Equations', Physical Review X 10, 011019 (2020), DOI 10.1103/PhysRevX.10.011019, https://doi.org/10.1103/PhysRevX.10.011019
4. Martí Raya-Moreno, Jesús Carrete, and Xavier Cartoixà, 'Hydrodynamic signatures in thermal transport in devices based on two-dimensional materials: An ab initio study', Physical Review B 106, 014308 (2022), DOI 10.1103/PhysRevB.106.014308, https://doi.org/10.1103/PhysRevB.106.014308 (arXiv:2203.05293, whose preprint title abbreviates 'two-dimensional' as '2D')
5. Yangyu Guo, Zhongwei Zhang, Masahiro Nomura, Sebastian Volz, and Moran Wang, 'Phonon vortex dynamics in graphene ribbon by solving Boltzmann transport equation with ab initio scattering rates', International Journal of Heat and Mass Transfer 169, 120981 (2021), DOI 10.1016/j.ijheatmasstransfer.2021.120981, https://doi.org/10.1016/j.ijheatmasstransfer.2021.120981
6. Chuang Zhang, Songze Chen, and Zhaoli Guo, 'Heat vortices of ballistic and hydrodynamic phonon transport in two-dimensional materials', International Journal of Heat and Mass Transfer 176, 121282 (2021), DOI 10.1016/j.ijheatmasstransfer.2021.121282, https://doi.org/10.1016/j.ijheatmasstransfer.2021.121282 (arXiv:2009.12587, preprint title 'Heat vortexes of ballistic, diffusive and hydrodynamic phonon transport in two-dimensional materials')
7. Man-Yu Shang, Wen-Hao Mao, Nuo Yang, Baowen Li, and Jing-Tao Lü, 'Unified theory of second sound in two dimensional materials', Physical Review B 105, 165423 (2022), DOI 10.1103/PhysRevB.105.165423, https://doi.org/10.1103/PhysRevB.105.165423 (arXiv:2204.04852)
8. 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, 375–379 (2019), DOI 10.1126/science.aav3548, https://doi.org/10.1126/science.aav3548
9. Xin Huang, Yangyu Guo, Yunhui Wu, Satoru Masubuchi, Kenji Watanabe, Takashi Taniguchi, Zhongwei Zhang, Sebastian Volz, Tomoki Machida, and Masahiro Nomura, 'Observation of phonon Poiseuille flow in isotopically purified graphite ribbons', Nature Communications 14, 2044 (2023), DOI 10.1038/s41467-023-37380-5, https://doi.org/10.1038/s41467-023-37380-5 (arXiv:2207.01469)
10. Xin Huang, Roman Anufriev, Laurent Jalabert, Kenji Watanabe, Takashi Taniguchi, Yangyu Guo, Yuxiang Ni, Sebastian Volz, and Masahiro Nomura, 'A graphite thermal Tesla valve driven by hydrodynamic phonon transport', Nature 634, 1086–1090 (2024), DOI 10.1038/s41586-024-08052-1, https://doi.org/10.1038/s41586-024-08052-1
11. Yujie Quan and Bolin Liao, 'Coupled Electron-Phonon Hydrodynamics in Two-Dimensional Semiconductors', Physical Review Letters 134, 226301 (2025), DOI 10.1103/PhysRevLett.134.226301, https://doi.org/10.1103/PhysRevLett.134.226301 (arXiv:2411.14649)
12. Jennifer Coulter, Bogdan Rajkov, and Michele Simoncelli, 'Coupled electron-phonon hydrodynamics and viscous thermoelectric equations' (2025), arXiv:2503.07560, https://arxiv.org/abs/2503.07560
13. Enrico Di Lucente, Francesco Libbi, and Nicola Marzari, 'Vortices and Backflow in Hydrodynamic Heat Transport', Physical Review Letters 136, 056307 (2026), DOI 10.1103/g9dx-hjyn, https://doi.org/10.1103/g9dx-hjyn (arXiv:2501.16580)
14. Chuang Zhang and Lei Wu, 'Transient hydrodynamic phonon transport in two-dimensional disk geometry' (2022), arXiv:2203.12958, https://arxiv.org/abs/2203.12958
15. Wanying Liu and Yangyu Guo, 'Effects of nonlinearity on phonon hydrodynamics in graphite', Physical Review B 111, 125417 (2025), DOI 10.1103/PhysRevB.111.125417, https://doi.org/10.1103/PhysRevB.111.125417
16. Takuya Kawabata, Kosuke Shimura, Yuto Ishii, Minatsu Koike, Kentaro Yoshida, Shu Yonehara, Kohei Yokoi, Alaska Subedi, Kamran Behnia, and Yo Machida, 'Phonon hydrodynamic regimes in sapphire', Physical Review Research 7, 033017 (2025), DOI 10.1103/ptds-chrz, https://doi.org/10.1103/ptds-chrz (arXiv:2407.16196)
17. Jan Dragašević, Bogdan Rajkov, and Michele Simoncelli, 'Viscous Heat Backflow and Temperature Resonances in Extreme Thermal Conductors', Physical Review Letters 136, 186302 (2026), DOI 10.1103/nbbn-56hr, https://doi.org/10.1103/nbbn-56hr
18. Mingtian Xu, 'Boundary effect and heat vortices of hydrodynamic heat conduction in graphene', Physics Letters A 492, 129231 (2023), DOI 10.1016/j.physleta.2023.129231, https://doi.org/10.1016/j.physleta.2023.129231
19. Navaneetha K. Ravichandran, 'Elasticity reshapes heat flow in graphene' (2026), arXiv:2604.03910, https://arxiv.org/abs/2604.03910
20. Houssem Rezgui, 'Phonon Hydrodynamic Transport: Observation of Thermal Wave-Like Flow and Second Sound Propagation in Graphene at 100 K', ACS Omega 8, 23964–23974 (2023), DOI 10.1021/acsomega.3c02558, https://doi.org/10.1021/acsomega.3c02558 (finite-element simulation study, not an experimental observation)
21. Enrico Di Lucente, Michele Simoncelli, and Nicola Marzari, 'Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics', Advances in Physics 75, 83–222 (2026), DOI 10.1080/00018732.2026.2685376, https://doi.org/10.1080/00018732.2026.2685376 (arXiv:2608.13339)
22. Cuiqian Yu, Yulou Ouyang, and Jie Chen, 'A perspective on the hydrodynamic phonon transport in two-dimensional materials', Journal of Applied Physics 130, 010902 (2021), DOI 10.1063/5.0056315, https://doi.org/10.1063/5.0056315