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

This review bridges quantum phonon dynamics and mesoscopic hydrodynamics by rigorously deriving viscous heat equations from first principles to explain and predict exotic non-diffusive thermal transport phenomena, such as thermal backflow and vortices, in dielectric crystals.

Original authors: Enrico Di Lucente, Michele Simoncelli, Nicola Marzari

Published 2026-08-14

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CC BY 4.0

Original authors: Enrico Di Lucente, Michele Simoncelli, Nicola Marzari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Secret Life of Heat: When Warmth Flows Like a River

Imagine you are holding a hot cup of cocoa. You know the heat travels from the cup to your hands, but have you ever wondered how? In the world of solid materials, heat isn't a mysterious fog; it's actually made of tiny, invisible vibrations called phonons. Think of these phonons as a massive crowd of energetic dancers moving through a crystal lattice. Usually, these dancers are chaotic. They bump into each other, trip over impurities, and scatter in every direction, slowly diffusing heat from the hot side to the cold side. This is the "normal" way heat moves, described by a famous rule from the 1800s called Fourier's Law. It's like a crowded hallway where everyone is jostling; the flow is slow, messy, and predictable.

But what if the dancers suddenly decided to stop bumping into each other and started moving in perfect unison? What if, instead of a chaotic crowd, they formed a smooth, flowing river? This is the exciting world of phonon hydrodynamics. Just as water can flow smoothly in a pipe (a phenomenon called Poiseuille flow) or create swirling eddies, heat can sometimes behave like a fluid. This happens when the dancers are so coordinated that they conserve their momentum, gliding past one another without losing energy to friction. Scientists have been hunting for this "fluid-like" heat for decades because it could revolutionize how we cool down super-fast computer chips or harvest waste energy. But to understand it, we need to bridge the gap between the quantum mechanics of individual atoms and the big-picture flow of heat in a device.

The Paper's Journey: From Quantum Chaos to Fluid Rivers

This review article acts as a grand map, guiding us through the different levels of theory needed to understand how heat moves in crystals. The authors, Enrico Di Lucente, Michele Simoncelli, and Nicola Marzari, start at the very bottom: the quantum realm. Here, they use complex math involving "Green's functions" and "Dyson equations" to describe phonons as quantum particles interacting with each other. It's like watching a single dancer in slow motion, tracking every possible interaction. From this deep quantum view, they show how we can climb up to a semiclassical level, where phonons are treated more like a gas of particles bouncing around, described by the famous Boltzmann transport equation.

However, the real magic happens when they zoom out further to the mesoscopic level—the middle ground between individual atoms and a whole machine. Here, the authors introduce a new set of rules called the Viscous Heat Equations (VHE). These equations treat heat not as a diffusing gas, but as a viscous fluid. Just as honey has thickness (viscosity) that resists flow, the "phonon fluid" has a thermal viscosity. The paper demonstrates that when you solve these equations, you don't just get the standard, boring heat flow. You get something much wilder.

The authors show that in this fluid-like regime, heat can do things that seem impossible under normal rules. For instance, they predict thermal backflow, where heat actually flows backwards against the temperature gradient, creating swirling heat vortices (eddies) similar to water swirling in a bathtub drain. They also show how heat can travel as a wave, known as second sound, rather than just slowly spreading out. The paper explicitly argues against the idea that the old, simple rules (Fourier's Law) are enough for these scenarios; in fact, they show that Fourier's Law completely breaks down when heat starts behaving like a fluid.

Using advanced computer simulations based on the laws of quantum mechanics (first-principles calculations), the authors calculate exactly how "thick" or viscous this phonon fluid is in materials like graphite. They find that at specific temperatures (around 70 Kelvin for graphite) and in devices of certain sizes (around 10 micrometers), these exotic fluid effects become dominant. They even provide a "roadmap" for how to solve these new equations analytically, breaking the temperature field down into two parts: one related to how the fluid compresses and another related to how it swirls (vorticity).

The paper concludes by suggesting that this framework isn't just for heat; it could be a blueprint for understanding other "quasiparticles" in solids. While the authors don't claim to have built a new cooling device yet, they have provided the rigorous mathematical tools and the "fluid" perspective needed to design one. They show that by treating heat as a fluid with viscosity, we can predict and potentially control strange phenomena like negative thermal resistance, opening the door to a new era of thermal engineering where we don't just manage heat, we ride its waves.

Technical Summary: Thermal Transport in Crystals: From the Quantum Dyson Equation to Mesoscopic Phonon Hydrodynamics

Problem Statement
Thermal transport in dielectric, non-magnetic crystals is traditionally described by Fourier's law of heat conduction, which assumes diffusive behavior dominated by momentum-relaxing scattering. However, in specific materials and temperature regimes, heat transport exhibits non-diffusive, hydrodynamic phenomena such as Poiseuille-like flow, second sound, and thermal vortices. While the microscopic Peierls-Boltzmann transport equation (BTE) can theoretically describe these regimes, its computational complexity renders it unsuitable for simulating realistic devices with complex geometries. Conversely, macroscopic models like Fourier's law fail to capture hydrodynamic effects, and existing mesoscopic models (e.g., Guyer-Krumhansl, dual-phase-lag) often lack a rigorous derivation from first principles or fail to unify diffusive and hydrodynamic regimes. The central challenge is to establish a unified, rigorous theoretical framework that bridges the gap between fully quantum mechanical descriptions and mesoscopic continuum equations, enabling the prediction and control of non-diffusive heat transport in devices.

Methodology
The authors construct a hierarchical theoretical framework that systematically connects three distinct levels of description:

Quantum Regime (NEGF Formalism): The review begins with the nonequilibrium Green's function (NEGF) formalism, specifically the Kadanoff-Baym equation (KBE) derived from the quantum Dyson equation. This provides a formally exact description of phonon kinetics, including driving terms, collision terms, and self-energy effects (anharmonic interactions). The authors utilize the Wigner mixed representation and gradient approximations to map the quantum kinetic equations onto phase-space variables.
Semiclassical Regime (Boltzmann Transport): Through the generalized Kadanoff-Baym (GKB) ansatz and the quasiparticle approximation, the KBE is rigorously reduced to the semiclassical Peierls-Boltzmann transport equation (BTE). The authors emphasize the role of the scattering matrix and the distinction between normal (momentum-conserving) and Umklapp (momentum-dissipating) scattering processes. They introduce the "relaxon" formalism, where the scattering operator is diagonalized to define collective excitations with well-defined lifetimes, distinguishing between odd relaxons (governing thermal conductivity) and even relaxons (governing thermal viscosity).
Mesoscopic Regime (Viscous Heat Equations): By projecting the linearized BTE (LBTE) onto the subspace of special relaxons (those associated with energy and momentum conservation), the authors derive the Viscous Heat Equations (VHE). These are coupled partial differential equations for the local temperature $T(\mathbf{r}, t)$ and the phonon drift velocity $\mathbf{u}(\mathbf{r}, t)$ . The VHE incorporate thermal viscosity, compressibility, and spatial nonlocality, effectively treating phonons as a fluid.

The authors further develop an analytical framework by recasting the steady-state VHE into decoupled modified Helmholtz and biharmonic equations for velocity potentials and stream functions. This allows for closed-form analytical solutions in specific geometries (e.g., 2D strips) and facilitates the decomposition of temperature profiles into compressibility and vorticity contributions.

Key Contributions

Rigorous Derivation of VHE: The paper provides a first-principles derivation of the viscous heat equations directly from the quantum Dyson and Kadanoff-Baym equations, bridging the gap between microscopic quantum mechanics and macroscopic hydrodynamics without empirical fitting.
Unification of Transport Models: The authors demonstrate that established models such as the Guyer-Krumhansl equations and the dual-phase-lag equation emerge as specific limits (steady-state, linear-isotropic, inviscid) of the more general VHE.
Relaxon Formalism for Transport Coefficients: The review clarifies the microscopic origin of thermal viscosity, linking it to even relaxons (collective excitations) and showing how thermal conductivity arises from odd relaxons. This allows for the calculation of all transport coefficients (conductivity, viscosity, specific heat) using first-principles simulations (DFT + LBTE).
Analytical Solutions and Novel Phenomena: By solving the VHE analytically, the authors derive closed-form expressions for temperature fields that reveal:
- Negative Thermal Resistance: A signature of hydrodynamic flow where heat flows against the temperature gradient in specific regions.
- Thermal Backflow and Vortices: The emergence of steady-state thermal backflow and heat vortices in device geometries, driven by the interplay of compressibility and vorticity.
- Compressibility Effects: The identification of phonon fluid compressibility as a defining feature distinguishing phonon hydrodynamics from electron hydrodynamics (which are often modeled as incompressible).

Results

Validation of Hydrodynamic Regimes: Numerical solutions of the VHE for graphite at intermediate temperatures (e.g., 70 K) reproduce Poiseuille-like heat flux profiles and second sound, matching spatially resolved LBTE results and recent experimental observations.
Boundary Condition Effects: The study analyzes the impact of boundary conditions (no-slip vs. frictionless/slip) on heat transport. It demonstrates that frictionless boundaries significantly flatten the heat flux profile, while no-slip conditions enhance the Poiseuille peak.
Fourier Deviation Number (FDN): The authors introduce the FDN as a dimensionless descriptor to quantify the departure from diffusive (Fourier) behavior. They show that hydrodynamic effects are maximized when the FDN is large, corresponding to weak Umklapp scattering and strong drift-temperature coupling.
Analytical Predictions: The analytical solutions predict that in the viscous limit, the temperature difference across a strip can become negative near contacts (thermal backflow), scaling as $1/x^2$ , a phenomenon absent in diffusive Fourier transport.
Comparison with Other Fluids: The paper highlights similarities and differences between phonon fluids and classical/electron fluids, noting that phonon hydrodynamics is inherently compressible due to the lack of a conserved phonon number, unlike electron fluids which are often treated as incompressible.

Significance and Claims
The paper claims to provide a unified and rigorous theoretical framework for thermal transport across all relevant length and time scales. Its primary significance lies in:

Bridging Scales: It successfully connects the fully quantum Dyson equation to mesoscopic continuum equations, clarifying the approximations and physical assumptions (e.g., quasiparticle validity, gradient expansion) required at each step.
Predictive Power: By combining the VHE with first-principles calculations, the framework enables the transparent connection of microscopic phonon physics to observable non-diffusive phenomena. This allows for the prediction of hydrodynamic effects in materials like graphite and diamond without adjustable parameters.
Guiding Experiments: The derived analytical solutions and identified signatures (e.g., negative thermal resistance, vortices) provide specific targets for experimentalists to detect, amplify, and control hydrodynamic heat transport.
Generalizability: The authors conclude by outlining a roadmap to generalize these tools to other quasiparticles, suggesting that the methodology can advance the study of collective quantum transport phenomena in solids beyond phonons.

The review does not propose new experimental setups but rather provides the theoretical tools and analytical solutions necessary to interpret existing and future experimental data in the context of phonon hydrodynamics. It emphasizes that while Fourier's law is sufficient for diffusive regimes, the VHE are essential for accurately modeling heat transport in devices where hydrodynamic effects are significant.

The Secret Life of Heat: When Warmth Flows Like a River

The Paper's Journey: From Quantum Chaos to Fluid Rivers

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