Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics

Enrico Di Lucente; Michele Simoncelli; Nicola Marzari

REVIEW 3 major objections 3 minor 298 references

Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Projecting the Boltzmann equation onto conserved phonon modes yields viscous heat equations that unify Fourier diffusion with second sound, backflow, and vortices.

desk verdict A competent and useful review of phonon hydrodynamics, but the load-bearing spectral-gap assumption in the VHE derivation is asserted rather than proven, and the reviewer should push on it. read the letter →

arxiv 2608.13339 v1 pith:YJCGQU24 submitted 2026-08-13 cond-mat.mtrl-sci cond-mat.otherphysics.app-phphysics.comp-phphysics.flu-dyn

Enrico Di Lucente , Michele Simoncelli , Nicola Marzari This is my paper · ORCID

classification cond-mat.mtrl-scicond-mat.otherphysics.app-phphysics.comp-phphysics.flu-dyn PACS 66.70.-f63.20.kg

keywords phononhydrodynamicsviscousheatequationsthermaltransportrelaxonsBoltzmannequationsecondsoundbackflowKadanoff-Baym

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review tries to establish that the practical levels of thermal transport theory in crystals — from quantum nonequilibrium Green's functions down to Fourier's law — are connected by a chain of controlled approximations, whose mesoscopic endpoint is a single pair of partial differential equations called the viscous heat equations. The authors derive these equations by projecting the linearized phonon Boltzmann transport equation onto the subspace spanned by the collective modes that conserve energy and crystal momentum, which introduces a phonon drift velocity alongside the temperature field. If the derivation holds, one framework reproduces all the familiar limits (Fourier, Cattaneo, dual-phase-lag, Guyer-Krumhansl) and additionally predicts hydrodynamic phenomena Fourier's law cannot see, including steady-state thermal backflow and heat vortices in suitably shaped devices, with those predictions benchmarked against full Boltzmann solutions and recent transient-grating experiments in graphite. The payoff is a continuum model whose coefficients are computable from first principles, so device engineers could predict where heat flows backward or forms vortices without solving the full Boltzmann equation.

What carries the argument

The load-bearing object is the relaxon formalism: the symmetrized scattering matrix of the linearized Boltzmann equation is diagonalized exactly, and its eigenvectors, the relaxons, are collective phonon excitations with definite parity and well-defined relaxation times. Four of them sit at zero eigenvalue: the energy relaxon, an eigenvector of the full scattering operator, and three momentum relaxons, zero-eigenvalue eigenvectors of the normal-scattering operator alone. Projecting the LBTE onto this four-dimensional subspace, then applying a Schrieffer-Wolff transformation to block-diagonalize the collision operator — valid when the strength $\lambda$ of Umklapp scattering is small relative to the spectral gap $g$ separating the momentum modes from all other normal-scattering modes — decomposes the thermal conductivity into a momentum contribution $\kappa^M$ and a diffusion-damped contribution $\kappa^D$, and yields the closed-form coefficients of the viscous heat equations. The new coefficient this machinery produces is the thermal viscosity tensor $\eta$, built from even-parity relaxons, which is what allows the heat flux to carry nonzero curl and thereby exhibit vortices and backflow.

What would settle it

Take a material where hydrodynamic transport is claimed (for example graphite between 70 K and 225 K), compute the full normal-scattering and Umklapp scattering matrices from first principles, and evaluate the spectral gap $g$ of the normal operator and the operator norm $\lambda$ of the Umklapp part; if $\lambda/g$ is not much smaller than one in exactly the temperature and length-scale window where the review predicts backflow, second sound, or lattice cooling, the Schrieffer-Wolff step is invalid and the VHE predictions in that window lack their stated foundation. A complementary experimental check: in the tunnel-chamber graphite device, time-resolved thermal imaging should reveal the predicted steady-state temperature inversion inside the chamber, and its absence at the predicted temperature would directly contradict the framework.

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Extended reading notes

Core claim

The paper argues that projecting the linearized phonon Boltzmann transport equation onto the subspace spanned by the four zero-eigenvalue collective modes of the scattering operator — the energy mode and the three crystal-momentum modes of normal scattering, which in the relaxon formalism are the conserved special relaxons — yields a closed pair of partial differential equations, the viscous heat equations, coupling the temperature field $T(\mathbf{r},t)$ to a phonon drift velocity $\mathbf{u}(\mathbf{r},t)$. The resulting heat flux splits into a drifting component carried by momentum-conserving collisions and a diffusion-damped component proportional to the temperature gradient, and thermal viscosity, defined microscopically from the even-parity relaxons, governs momentum diffusion. On the authors' own terms, this single framework contains Fourier's law, Cattaneo's equation, the dual-phase-lag equation, and the Guyer-Krumhansl equations as special limits, and it predicts Poiseuille heat flow, second sound, negative thermal resistance, steady-state thermal backflow, and heat vortices; the review backs these predictions with solutions of the full LBTE in device geometries, analytical Helmholtz and biharmonic solutions, and agreement with transient-grating experiments in graphite.

Load-bearing premise

Everything rests on the premise that momentum-destroying Umklapp scattering is weak compared with the gap separating the three momentum-conserving modes from all other collision modes ($\lambda/g \ll 1$); if that separation is not large, the block diagonalization, and with it the closed-form coefficients of the viscous heat equations, is not mathematically guaranteed.

Editorial extensions

If this is right

For device design, the Fourier deviation number, a dimensionless combination of the VHE coefficients, device size, and applied gradients, tells engineers when Fourier's law fails, so that hot spots and temperature inversions can be predicted from the VHE alone.
For theory, the dual-phase-lag and Guyer-Krumhansl equations are demoted from independent models to limiting cases of the VHE (inviscid, and linear-isotropic drift-dominated steady state, respectively), so disagreements between those models are resolved inside one framework.
For experiments, the VHE predict resonant temperature-wave amplification, steady-state temperature inversion in tunnel-chamber geometries, and lattice cooling after pulsed heating, at temperatures and length scales where the full Boltzmann equation agrees, giving observable signatures for thermal-imaging and pump-probe setups.
For computation, all VHE coefficients come from ab initio solutions of the LBTE with no fitting, so the same framework scales to screening materials and device geometries for hydrodynamic heat transport.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The review leaves implicit that the spectral-gap condition ($\lambda/g \ll 1$) is itself a screening descriptor: materials with a wide separation between slow momentum-conserving and fast diffusive collision modes should display hydrodynamic heat transport, so the ratio could be computed and tabulated for candidate crystals before any device simulation.
Because the analytical solutions decompose the temperature field into compressibility and vorticity contributions, phonon devices could plausibly be designed as thermal analogs of fluidic circuits, with chambers and openings acting as vortex generators; testing this would require extending the analytical biharmonic solutions beyond the graphite strip studied here, which the paper does not do.
The review states that current phonon hydrodynamics is confined to the laminar, zero-Reynolds-number regime and that phonon turbulence has not been modeled; a natural open problem this implies is a nonlinear extension of the VHE that restores the advective term, which would predict whether and when collective heat transport becomes turbulent.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The short version: this is a genuine review, not a disguised research paper, and it does its job well. The authors trace a coherent line from phonon Dyson and Kadanoff-Baym equations through the linearized Boltzmann equation to the viscous heat equations (VHE) and their limiting cases (DPLE, Guyer-Krumhansl, Fourier). The derivations are internally consistent, and the paper includes a new analytical recasting of the steady-state VHE as decoupled Helmholtz and biharmonic equations, which is a useful addition. The benchmarks against first-principles LBTE solutions and experimental second-sound data in graphite give the framework credible empirical grounding.

Credit where due: the review is clearly written, the hierarchy of models is illuminating, and the discussion of when Fourier fails is concrete. The explicit limitations section is honest about linear-response and gradient assumptions, though it does not address the deeper spectral-gap question.

The soft spot, as you would expect, is the projection that produces the VHE. The review assumes a finite spectral gap g separating the three momentum relaxons from all other modes of the normal-scattering operator, and that Umklapp strength lambda satisfies lambda/g << 1. This is asserted rather than established. The stress-test raises a fair mathematical point: for acoustic phonons, normal-scattering eigenvalues can accumulate at zero as q approaches zero because long-wavelength lifetimes diverge, so the gap can vanish in the thermodynamic limit without any extra conserved quantity. That means the Schrieffer-Wolff block diagonalization and the clean split into momentum and diffusion-damped conductivities are not guaranteed beyond the specific benchmarked windows. The paper's material-specific benchmarks (graphite, around 70 K) are suggestive but not a general proof.

Another caveat: the abstract says 'we demonstrate' for effects that mostly come from earlier papers by the same group. The review is not a new result; it is a synthesis. That is fine for a review, but readers should not mistake the exotic predictions (backflow, vortices, negative thermal resistance) for experimentally confirmed phenomena. The steady-state backflow is a VHE prediction, backed by LBTE comparison in a specific device geometry, but no experiment has seen it yet.

Who benefits: newcomers and practitioners who want a map from the many-body formalism to continuum hydrodynamics, and a clear statement of where each model lives. The analytical section is handy for device modeling. I would assign it to a referee with a background in kinetic theory; that referee should push on the gap condition or at least ask for a supplementary analysis of its range of validity.

Verdict: accept for peer review, with revisions expected. The paper is a useful reference and deserves publication after the gap issue is addressed, even if only by sharpening the caveats and supplying more evidence of where the condition holds.

Referee Report

3 major / 3 minor

Summary. This review article aims to present a unified theoretical description of thermal transport in insulating crystals from the quantum regime to mesoscopic device modeling. It derives the phonon Boltzmann transport equation from the Kadanoff-Baym equation using the Wigner representation and the generalized Kadanoff-Baym ansatz, introduces the relaxon formalism for the linearized Boltzmann equation, and then constructs the viscous heat equations (VHE) by projecting the LBTE onto the subspace of energy and momentum relaxons. The central claim is that the VHE constitute a general mesoscopic framework that reduces to Fourier's law, the dual-phase-lag equation, and the Guyer-Krumhansl equations in appropriate limits, and that they predict Poiseuille heat flow, second sound, steady-state heat backflow, vortices, and transient lattice cooling. The parameters entering the VHE are computed from first principles, and predictions are benchmarked against full LBTE solutions and against transient-thermal-grating and pump-probe experiments in graphite.

Significance. If the central claims hold, this review would provide a valuable bridge between nonequilibrium Green's function theory and practical mesoscopic modeling of phonon hydrodynamics. The manuscript is strongest where it makes the approximation chain explicit: from the Kadanoff-Baym equation to the BTE, from the BTE to the relaxon picture, and from the relaxon picture to the VHE. The introduction of thermal viscosity from even relaxons and the analytical reduction of steady-state VHE to modified Helmholtz and biharmonic equations are genuinely useful contributions. The benchmarks against spatially and temporally resolved LBTE solutions and against experimental data in graphite are also a clear strength, as is the explicit discussion of boundary conditions and finite-size corrections. The main caveat is that the 'unified framework' claim rests on a spectral-gap separation that is asserted rather than established, which limits the demonstrated generality of the VHE beyond the specific benchmarks presented.

major comments (3)

[Section 4.3.1, Eqs. (136)-(138)] The assertion that the projected normal-scattering operator has a finite spectral gap g because 'otherwise additional conserved quantities would exist' is not a valid proof in the infinite-dimensional phonon space. Eigenvalues of the normal-scattering operator can accumulate at zero as q approaches zero for acoustic phonons with divergent long-wavelength lifetimes, without any additional exactly conserved mode. Since the Schrieffer-Wolff block diagonalization and the decomposition kappa = kappa_MM + kappa_DD in Eq. (137) rely on a positive gap, the authors should either prove the existence of the gap from the structure of the normal-scattering operator, provide numerical spectra for representative materials, or explicitly restrict the VHE derivation to regimes or geometries where a minigap is known to exist.
[Section 4.3.1, after Eq. (138)] The smallness condition lambda/g << 1, with lambda = ||Omega~_U||, is stated but not quantified. In the high-temperature diffusive regime, where the VHE are claimed to reduce exactly to Fourier's law following Eq. (143), Umklapp scattering is typically not a small perturbation to the normal-scattering spectrum. In that regime the mixed M-D blocks omitted in Eq. (137) can contribute at order one to the conductivity and, more importantly, can modify the transient and backflow predictions that are central to the review's claims. The authors should report numerical estimates of lambda/g across the temperature range for the materials considered, or qualify the generality of the VHE framework accordingly.
[Section 4, Figs. 13, 18, 19] The supporting benchmarks are limited to graphite in a relatively narrow temperature window around 70-120 K. The claim that the VHE constitute a general mesoscopic framework for phonon hydrodynamics should be supported by evidence that the spectral-gap condition and the smallness of lambda/g also hold for other materials discussed in the text, such as diamond, silicon, and hBN, or by providing explicit criteria for when the projection fails. Without such evidence, the 'unified' status of the VHE remains a conjecture whose demonstrated range is narrower than the review's title and abstract suggest.

minor comments (3)

[Section 4.10, text after Eq. (185)] The text states 'sigma_x = 2 micron and sigma_x = 2.8 micron'; the second occurrence should presumably be sigma_y, since the Gaussian heating profile is defined with widths in both the x and y directions.
[Section 4.3 and Section 4.10.1] The claim that 'all transport coefficients ... without relying on experimental inputs or fitting procedures' is slightly overstated, because the finite slip length b = 0.4 micron used in Section 4.10.1 is obtained by matching VHE results to LBTE solutions with diffusive boundaries. The text should distinguish between first-principles transport coefficients and fitted boundary parameters.
[Eqs. (138) and (140)] Equation (138) reports off-diagonal corrections of order O(lambda/g^2) and diagonal corrections O(lambda^2/g^4) in the inverse of the transformed operator, while Eq. (140) states kappa_MM = kappa_M + O(lambda/g). These order-of-magnitude notations should be harmonized so that the size of the neglected terms is unambiguous.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the VHE are a coarse-grained projection of the LBTE with first-principles transport coefficients, and the central predictions are benchmarked against an external LBTE solver and independent experiments.

full rationale

The review's central derivation (Section 4.3) is a mathematical projection of the linearized Boltzmann transport equation onto the subspace spanned by the energy and momentum eigenvectors of the normal-scattering operator. The VHE coefficients (kappa_D, eta, D_U, W) are stated to be computed from first-principles LBTE relaxon calculations (Refs. [18,89]), not fitted to the phenomena they predict. The 'predictions' of Poiseuille flow, second sound, and thermal backflow are consequences of the structure of the VHE, and the review validates them against an independent full LBTE solver (BTE-Barna, Ref. [181]) and against external experimental data (Huberman et al., Ding et al.). Thus, although the review leans heavily on the authors' own prior work for presentation, the load-bearing content has independent support. The spectral-gap and lambda/g << 1 assumption used to justify the Schrieffer-Wolff block diagonalization is an approximation whose general validity is not established; this is a correctness/domain-of-validity concern, not circularity. The identity kappa = kappa_M + kappa_D used to recover Fourier's law in the strong-Umklapp limit is definitional (Eq. 142), but it is a consistency relation, not a fitted input disguised as a prediction. The only parameter adjusted to an external calculation (slip length b = 0.4 micron) calibrates boundary conditions to LBTE boundary-scattering models and does not control the existence of the central backflow phenomenon, which appears for no-slip and frictionless limits as well. Score 2 reflects the presence of self-citation in the exposition without circularity in the derivation or predictions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The VHE framework introduces no fitted parameters for its central claim; all transport coefficients are computed from first principles. The derivation rests on standard semiclassical and many-body assumptions (quasiparticle picture, gradient expansion, GKB ansatz, linear response, lowest-order three-phonon scattering) plus a material-specific spectral-gap assumption for the Schrieffer-Wolff decomposition. The latter is the most fragile premise and is not established in this review beyond the cited benchmarks.

assumptions (6)

domain assumption Phonon quasiparticle approximation: the spectral function is replaced by Dirac deltas (Eq. 55).
Used in Section 3.2 to reduce the Kadanoff-Baym equation to the Boltzmann equation; requires well-defined phonon quasiparticles with negligible broadening and frequency shifts.
domain assumption Gradient approximation: center-of-mass variations are slow compared with relative-coordinate variations, so only leading-order gradients are retained (Section 3.1).
Needed to convert the nonlocal KBE into a local BTE; fails for fast temporal or sharp spatial variations.
domain assumption Generalized Kadanoff-Baym ansatz: the lesser/greater Green's functions are expressed in terms of a Wigner distribution (Section 3.2).
This ansatz connects Green's functions to the phonon distribution; its validity out of equilibrium is assumed following Ref. [135].
domain assumption Linear response: phonon deviations from the local drifting equilibrium are small, and only terms linear in temperature and drift-velocity gradients are kept (Section 4.1).
The VHE are derived in this regime; they cannot describe strongly nonlinear transport or large temperature gradients.
ad hoc to paper Spectral-gap separation: the Umklapp scattering strength is much smaller than the spectral gap of the normal-scattering operator, lambda/g << 1 (Section 4.3.1).
This justifies the Schrieffer-Wolff block diagonalization that separates momentum and diffusion-damped subspaces; without it, the closed-form VHE coefficients are not guaranteed. No general proof of this condition is provided in the review.
domain assumption Bubble self-energy with bare vertex: only leading three-phonon scattering at lowest order is included; vertex corrections and frequency shifts are neglected (Sections 3.4 and 3.5).
Standard for first-principles LBTE calculations; limits accuracy for strongly anharmonic or high-temperature systems.

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Cite this review

Pith. "Pith review of Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics." pith.science (2026). https://pith.science/paper/YJCGQU24

@misc{pith2026260813339,
  author       = {Pith},
  title        = {Pith review of: Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YJCGQU24}},
  note         = {Machine review of arXiv:2608.13339}
}

read the original abstract

Thermal transport in dielectric, non-magnetic crystals is mediated by quantized lattice vibrations, which drift and interact when driven out of equilibrium by a temperature gradient. This phenomenon can be described at multiple theoretical levels, ranging from fully quantum descriptions to semiclassical and mesoscopic continuum approaches. This review rigorously discusses the theoretical steps and approximations connecting these levels, bridging quantum phonon Dyson and Kadanoff-Baym equations and semiclassical Boltzmann transport formalism, and discussing the coarse-graining procedures that yield mesoscopic viscous heat equations for non-diffusive, hydrodynamic heat transport in devices. We show how the Guyer-Krumhansl and dual-phase-lag equations emerge as special linear-isotropic-band and inviscid limits of the viscous heat equations, respectively; most importantly, we demonstrate that these equations predict not only Poiseuille flow and second sound, but also more exotic effects such as negative thermal resistance, steady-state thermal backflow and vortices. We highlight how combining these frameworks with first-principles simulations connects microscopic phonon physics to observable non-diffusive heat-transport phenomena and guides their detection, amplification, and control. We recast the viscous heat equations in terms of Helmholtz and biharmonic equations solved analytically, and use this to discuss similarities and differences between the macroscopic behavior of the phonon fluid and other hydrodynamic systems, such as classical and electron fluids, focusing on compressibility, vorticity, and their influence on phonon hydrodynamics. We conclude with a roadmap to generalize the tools used to describe phonon hydrodynamics to other quasiparticles, motivating future advances in collective quantum transport phenomena in solids.

Figures

Figures reproduced from arXiv: 2608.13339 by the authors.

**Figure 1.** Qualitative difference between diffusive and hydrodynamic thermal transport. (Left panel) Schematic representation of the temporal evolution of a temperature perturbation in the diffusive regime, where Fourier’s equation is accurate. (Right panel) Schematic representation of heat propagation in the form of “second sound” in the hydrodynamic regime of thermal transport, where Fourier’s equation fails. The color here … view at source ↗

**Figure 2.** Diagrammatic representation of the leading contributions to the phonon selfenergy expressed through Feynman diagrams. The two lowest-order terms proportional to the two-point correlation function G correspond to the standard “loop” (ψ4G) and “bubble” (ψ3GΓ3G) diagrams. The meaning of the symbols and line styles are also clarified: the symbol Γ denotes interaction vertices that should, in principle, be renormalized … view at source ↗

**Figure 3.** Diagrammatic illustration of the nonlinear integral (Bethe-Salpeter–type) equation (66)) for the vertex function. The diagram emphasizes the ladder corrections that renormalize, or “dress” the vertex Γ3 appearing in the bubble contribution of [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

Figures from the paper (31 more)

**Figure 4.** Figure 4: Normal and Umklapp phonon scattering processes. Normal and Umklapp processes during a coalescence event involving two phonons with wave vectors q ′ and q ′′ merging into a phonon with wave vector q. In a normal process (left), the sum of the initial momenta lies inside…

**Figure 5.** Figure 5: Relaxon picture of lattice thermal transport. (Left panel) Schematic of lattice-vibration equilibration following a thermal perturbation. Each relaxon is a collective phonon excitation that evolves independently and relaxes exponentially toward equilibrium with a chara…

**Figure 6.** Figure 6: Thermal transport regimes in a generic three-dimensional crystal as a function of temperature. At low temperatures, transport is ballistic, with extrinsic scattering much weaker than both normal and resistive processes. At intermediate temperatures, the hydrodynamic re…

**Figure 7.** Figure 7: Influence of boundary conditions and Poiseuille heat profile of the in-plane heatflow obtained by solving the VHE. The panels show the x-y heat flux in graphite, computed by solving the VHE with fixed temperatures of 80 K at x = 0µm and 60 K at x = 15µm, while all rem…

**Figure 8.** Figure 8: Comparison of the heat-flux predictions obtained from different theoretical models as discussed in Ref. [217]. (Left panel) Schematic of the silicon thin-film geometry considered in Ref. [217]. The top and bottom surfaces (the x–z planes at y = ±h/2) are modeled as dif…

**Figure 9.** Figure 9: Temperature and transport length-scale evolution of transient thermal grating signals. (Left panel) Time evolution of the transient thermal grating (TTG) temperature signal at several temperatures. Experimental measurements (orange) and corresponding theoretical predic…

**Figure 10.** Figure 10: Temperature and characteristic length-scale window for phonon hydrodynamics in graphite. (Left panel) Hydrodynamic regime of heat transport in graphite with natural isotope composition. The color map quantifies the relative strength of the temperature waves, defined a…

**Figure 11.** Figure 11: Viscous heat backflow and temperature inversion in a graphite tunnel-chamber geometry. In-plane (x–y) heat-flow streamlines and temperature field in a graphite tunnel-chamber device. The left and central panels display, respectively, the solutions of Fourier’s law and…

**Figure 12.** Figure 12: Tunnel-chamber geometry enabling viscous heat backflow. The region in yellow represents the physical simulation domain in which all VHE coefficients take their actual material values, whereas the green region denotes the lubrication layer, where the viscosity is artif…

**Figure 13.** Figure 13: Viscous heat backflow and temperature inversion from VHE and LBTE, and sensitivity to boundary conditions. Top panels display the in-plane (x–y) heat-flow streamlines together with the temperature field, while the bottom panels show the corresponding vorticity. Each c…

**Figure 14.** Figure 14: Time-domain evolution of transient thermal grating signals. Time-domain transient thermal grating (TTG) response in diamond at 100 K (left panel) and in silicon at 50 K (right panel) for different grating periods. Insets show the corresponding temperature response amp…

**Figure 15.** Figure 15: Transient viscous heat backflow. Each row displays the system at successive times after the heating pulse is switched off at t = 0.4 ns (see main text). From left to right, we report the temperature field, the heat flux driven by the temperature gradient (Qδ ), the dr…

**Figure 16.** Figure 16: Sigmoid function for realistic thermalization. The smooth-step function of Eq. (187) is shown for Rin = 10 µm and several choices of Rout. A value Rout = 11 µm (red) corresponds to an almost instantaneous thermalization occurring over a very narrow region, Rout = 15 µ…

**Figure 17.** Figure 17: Realistic thermalization in a smoothed rectangular domain. The green region (outside the dashed lime contour) corresponds to the fully thermalized portion of the device, where the sigmoid function f[d(˜r)] = 1 enforces a fixed temperature and vanishing drift velocity…

**Figure 18.** Figure 18: Lattice cooling strength as a function of system size, temperature, and isotopic composition. For graphite with natural isotope concentrations (left panel), the lattice cooling strength (LCS, Eq. (194)) is noticeably reduced compared to isotopically enriched samples (…

**Figure 19.** Figure 19 [PITH_FULL_IMAGE:figures/full_fig_p069_19.png]

**Figure 20.** Figure 20: Geometry of the strip where the modified biharmonic equations (216) are solved. The strip extends infinitely along x and has width h along y. Point-like injection of drift velocity and temperature gradient is imposed at x = 0 as specified in Eqs. (218)–(220). In numer…

**Figure 21.** Figure 21: Experimental setup for phonon drift injection and thermal backflow in a 2D strip device. Schematic of a two-channel thermal device in which vertically offset hot and cold reservoirs impose laterally displaced temperature gradients. This configuration generates a contr…

**Figure 22.** Figure 22: Temperature distribution in the 2D strip device obtained by analytically solving the modified biharmonic equation in both the viscous and diffusive regimes of thermal transport. Temperature distribution in a two-dimensional graphite strip of height h, together with th…

**Figure 23.** Figure 23: Non-local thermal response in viscous and diffusive regimes. The temperature difference T(x, 0) − T(x, h) is shown as a function of the horizontal coordinate x for several values of ξ, with ϵ = 0.1 fixed (see Eq. (248)). Far from the contacts (large |x|), the response…

**Figure 24.** Figure 24: Comparison between compressible and incompressible thermal responses. The temperature difference T(x, 0)−T(x, h) is shown as a function of x for several injected drift velocities U. Solid curves correspond to the general compressible case, while dashed curves represen…

**Figure 25.** Figure 25: Numerical validation of thermal backflow in 2D strip devices. (Left panel) Steadystate temperature distributions (top) and corresponding heat-flux patterns (bottom) obtained with beyond-RTA simulations as implemented in the BTE-Barna package [181]. The system dimensi…

**Figure 26.** Figure 26: Transient viscous heat vortices. Time evolution of the temperature-driven flow field. Transient vortex-like structures may appear at intermediate times due to the different decay rates of Fourier modes, but they do not correspond to any growing instability and progres…

**Figure 27.** Figure 27: Phonon hydrodynamic regimes in sapphire. (Left panel) Schematics of the experimental configurations performed in Ref. [275] and used to measure thermal conductivity are illustrated for the steady-state method (top) and the 3ω method (bottom). (Right panel) Temperature…

**Figure 28.** Figure 28: Graphite thermal Tesla valve. Thermal transport in a graphite Tesla valve. (Left panel) Optical micrographs of the graphite Tesla valve illustrating the forward (top) and reverse (bottom) configurations, respectively. The arrows denote the imposed directions of heat f…

**Figure 29.** Figure 29 [PITH_FULL_IMAGE:figures/full_fig_p113_29.png]

**Figure 30.** Figure 30: Diagrammatic representation of the phonon Dyson equation and self-energy. (Upper row) Feynman-diagram interpretation of the phonon Dyson equation given in Eq. (381), illustrating the renormalization of the phonon propagator. (Lower row) Diagrammatic representation of…

**Figure 31.** Figure 31: Comparison between vertex-correction and self-energy-insertion contributions to the three-phonon bubble self-energy involving quartic interactions. (Left) Self-energy diagram given by Eq. (390) obtained by accounting for vertex correction to the three-phonon bubble se…

**Figure 32.** Figure 32: Comparison between vertex-correction and self-energy-insertion contributions to the three-phonon bubble self-energy involving cubic interactions. (Left) Self-energy diagram given by Eq. (396) obtained by accounting for vertex corrections to the three-phonon bubble sel…

**Figure 33.** Figure 33: Analytical structure of the drift velocity and temperature gradient contribution to the temperature profile. 3D plot of x 2−y 2 (x2+y2) 2 appearing in the solution of T U ξ→0 (x, y) (upper-left panel) and ∆T π y x2+y2 appearing in the solution of T ∆T ξ→0 (x, y) (uppe…

**Figure 34.** Figure 34: Integrands governing the temperature difference across the strip. Plots of Eqs. (479) (green), (450) (red), together with their |k| asymptotic limit. In this example h = 2. − k tanh k h 2 sinh(kh) kh − sinh(kh) −−−→ k→∞ |k|, (479) 135 [PITH_FULL_IMAGE:figures/full_…

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