Heat Can Flow Backward in Pure Crystals, EPFL Theory Confirms

Heat is supposed to flow downhill - from hot to cold, always. Every engineer knows it, every coffee drinker feels it. Yet in a narrow class of highly ordered materials, something stranger happens: heat can swirl into vortices and even drift back toward warmer regions. This counterintuitive behavior, called phonon hydrodynamics, has been observed experimentally since the 1960s, but nobody had a clean theoretical explanation for when it is maximized and why.

A team at EPFL's School of Engineering has now built that explanation. Led by Nicola Marzari of the Theory and Simulation of Materials group, the researchers developed an analytical framework - not just a numerical simulation - that reveals the precise conditions under which heat backflow peaks in a two-dimensional strip of crystalline graphite. Their paper appeared in Physical Review Letters.

Phonons: The Particles That Carry Heat

When a solid gets hot, its atoms vibrate. Quantum mechanics treats those vibrations as quasiparticles called phonons, and phonons are how heat moves through a material. In most solids, phonons collide chaotically and scatter heat in all directions - a diffusive, messy process. In highly pure, defect-free crystals held at low temperatures, however, phonon-phonon collisions become cooperative rather than scattering. The phonons start behaving less like particles in a gas and more like molecules in a flowing fluid. That is the hydrodynamic regime.

Once phonon flow becomes fluid-like, all the mathematics of fluid mechanics becomes available. Vorticity - the tendency of a flow to curl and swirl - appears. So does compressibility, which describes whether the flow can be squeezed or bunched. And when those two properties interact in certain ways, heat can reverse course entirely, moving from a cooler region back toward a warmer one. The second law of thermodynamics is not violated: the net heat flow still runs from hot to cold; only a local portion reverses within the device.

Incompressibility Is the Key

The EPFL team's central finding is that heat backflow reaches its maximum when the phonon flow is nearly incompressible - that is, when the fluid of phonons cannot be easily squeezed or compressed as it encounters resistance. In that limit, the flow cannot pile up at a barrier; instead, it is redirected. Some of it turns back toward its source.

"Thanks to our analytical framework, we have shown that heat backflow is maximized when the flow is nearly incompressible," said first author Enrico Di Lucente, now a postdoctoral researcher at Columbia University. "Previous work relied on numerical modelling, which describes temperature patterns but doesn't fully explain how physical quantities influence each other."

The distinction between numerical modelling and an analytical framework is important. A simulation can tell you what happens in a specific material under specific conditions. An analytical model tells you why it happens and lets you vary parameters freely - making it far more useful for designing experiments or devices.

The temperature profile of any hydrodynamic heat-flow system, the team showed, can be decomposed into two independent components: vorticity and compressibility. When compressibility is low, vorticity dominates, and the flow curls back on itself. That localized reversal reduces heat buildup at hotspots and allows a more coordinated, efficient distribution of thermal energy across the device.

From Crystalline Graphite to Smartphone Batteries

The researchers demonstrated their framework using first-principles simulations of a 2D strip of crystalline graphite - a model system chosen for its exceptionally high purity and well-understood phonon behavior. Graphite is not a hypothetical material; it is cheap, abundant, and already used in electronics. That makes the choice practical as well as illustrative.

The implications extend well beyond graphite. Di Lucente offered a concrete example: a smartphone with a hydrodynamic component designed to redirect thermal energy away from the battery. Batteries degrade faster and pose safety risks when they overheat. A material or device layer that exploits heat backflow to actively steer thermal energy away from sensitive components could extend battery lifespan and improve safety - without requiring active cooling fans or liquid-cooling loops.

Marzari emphasized that the mathematical formulations apply not just to phonons but to any microscopic carrier - electrons, magnons, or more exotic quantum particles. The approach allows the ease with which any carrier travels through a material to be calculated directly from quantum mechanics' fundamental equations, known as first-principles methods.

"In addition to this theoretical development, our first-principles simulations provide a realistic description of physical systems quickly and inexpensively compared to the cost of building new experimental setups," Marzari said. "At the same time, they can indicate where experimental efforts should be focused to develop more heat-efficient electronics."

Sectors That Stand to Benefit

Consumer electronics represent just one application. The team noted potential relevance across industrial heat management, energy storage systems, data centers, and cloud computing infrastructure - all sectors where waste heat is both a significant operating cost and an engineering constraint. Data centers, for instance, currently spend roughly 30 to 40 percent of their total energy budget on cooling alone. Any material-level advance that reduces that burden has immediate economic value.

The current work is entirely theoretical and computational. No device has been built, and no experimental measurement of enhanced backflow has yet been reported using the EPFL framework as a guide. The next step, according to the authors, is to work with experimentalists to design specific material geometries and temperature conditions that maximize the effect in the laboratory. That transition from theory to experiment will test whether the predictions hold in real, imperfect materials rather than idealized simulations.

The scale of the backflow effect is also worth noting: Di Lucente described it as "very small" in absolute terms. The practical question is whether it can be amplified enough - through careful material selection, device geometry, and operating conditions - to make a measurable difference in real electronics. That remains an open engineering challenge.

Still, the value of the analytical framework is not contingent on any particular application succeeding. By establishing the physical conditions that maximize phonon hydrodynamics, the EPFL team has given experimentalists a precise roadmap for where to look - and that is a contribution that stands regardless of how quickly the technology matures.