Heat, it turns out, has a memory.
A cooling cup of coffee may not seem particularly thoughtful. At the scale of a kitchen, heat appears to follow a straightforward rule: it moves from warmer places to cooler ones. Leave the cup unattended long enough, and the disappointing result offers convincing evidence that this rule works.
But shrink the system to the dimensions of a modern computer chip—or observe it over just trillionths of a second—and this simple description can become incomplete. Heat flowing at one place and time may still carry the influence of a temperature disturbance that occurred earlier or somewhere else in the material.
In other words, heat can retain a kind of physical memory.
A new theoretical framework developed by Jianjun "JJ" Dong, the Thomas and Jean Walter Professor in the Department of Physics at Auburn University, and Yi Zeng, an Auburn alumnus who earned his Ph.D. in mechanical engineering in 2019 and is now a research scientist at the Department of Energy's National Laboratory of the Rockies provides a unified way to describe that memory. Appropriately enough, their study, 'Unified Statistical Theory of Heat Conduction in Nonuniform Media,' was published in Physical Review B this summer — just in time to put heat conduction to the test.
"Heat does not remember in the way that we remember a person or an event," Dong said. "Its memory is stored in the microscopic motion of the material. The heat flowing at this moment can still carry information about a temperature disturbance that occurred earlier."
When the textbook rule begins to bend
For roughly two centuries, scientists and engineers have relied on Fourier's law to describe heat conduction. It assumes that heat flow at a particular location responds immediately to the temperature gradient at that same location.
This local and instantaneous description works remarkably well for familiar objects and at ordinary scales. It helps engineers predict how buildings retain heat, how engines cool and how warmth spreads through cookware. In simple terms, Fourier's law describes heat flow as responding to conditions "here and now."
At very small lengths and short times, however, "here and now" may no longer tell the complete story.
In crystalline solids, heat is often carried by collective atomic vibrations called phonons. At large scales, enormous numbers of phonons scatter and interact, producing the smooth diffusion described by Fourier's law. At very short distances and times, some of those phonons can travel significant distances before scattering. The resulting heat flow can also retain the influence of an earlier disturbance.
Depending on the materials and the experimental scale, heat transport may then appear diffusive, quasi-ballistic, spatial-nonlocal or even wave-like. Scientists have developed equations for each of these regimes, but those equations are usually introduced as separate models suited to different conditions.
The new framework places those behaviors inside one mathematical description.
"Instead of choosing a different theory every time heat begins to behave differently, we wanted to identify the deeper microscopic structure connecting those behaviors," Zeng said. "In our framework, familiar diffusion, nonlocal transport and wave-like or hydrodynamic behavior emerge as different limits of the same underlying theory."
A thermal response map
At the heart of the framework is what physicists call a spatiotemporal kernel. In less technical language, it is a thermal response map.
The map specifies how strongly a temperature gradient at one location and an earlier time contributes to the heat flowing at another location now. Its temporal dimension describes memory: how long the influence of an earlier disturbance persists. Its spatial dimension describes reach, or nonlocality: how far that influence extends through the material.
A traffic slowdown offers a familiar analogy. Traffic at one location may still be affected by congestion that developed earlier and elsewhere along the road. Similarly, heat flow can retain the influence of an earlier temperature disturbance some distance away. Fourier's law appears when both dimensions collapse.
When the response has appreciable duration or spatial extent, a single conductivity value no longer contains all the relevant information.
"Thermal conductivity is an extremely useful number, but a single number cannot always tell the full story," Dong said. "At very small length scales and very short times, we need to know how far the thermal response extends and how long its influence persists."
The kernel is not merely an empirical fitting function. Statistical mechanics connects it to equilibrium fluctuations of heat flow inside the material. In this sense, the work generalizes the Green–Kubo relation, a foundational result that connects transport coefficients to microscopic fluctuations.
Ordinary thermal conductivity becomes the large-scale, long-time summary of a richer response.
Silicon remembers—but distance matters more
To demonstrate the framework, the researchers constructed the kernel for crystalline silicon at room temperature. They then used it to model transient thermal grating experiments, in which laser pulses create a temporary temperature pattern and researchers observe how that pattern fades.
The calculations covered spatial periods from tens of micrometers down to nanometers, spanning conventional diffusion and deep quasi-ballistic transport..
The results revealed a particularly interesting distinction. Under the conditions studied, the dominant departure from ordinary Fourier behavior came from spatial nonlocality – how far the heat flow response extends – rather than from temporal memory alone.
Silicon contains phonons with a broad range of mean free paths—the average distances they travel before scattering. At small scales, phonons with longer mean free paths cannot be treated as though they transfer energy only to their immediate surroundings.
Temporal memory still mattered, but its strongest influence appeared during the earliest stages of the thermal response.
Put simply: silicon remembers, but in this case how far its heat carriers travel matters more than how long their influence lasts.
The full framework reproduced the transition away from conventional diffusion across multiple length scales. When the researchers also accounted for scattering at the boundaries of a thin silicon membrane, the calculations agreed quantitatively with published experimental measurements.
Not every heat wave tells the same story
The theory also helps clarify the interpretation of wave-like temperature response.
Under certain conditions, temperature disturbances can propagate in a way that resembles a wave rather than simply spreading through diffusion. This phenomenon is sometimes called "second sound" and is often associated with hydrodynamic behavior, in which phonons move collectively.
The new analysis shows that an oscillating or wave-like temperature signal does not, by itself, prove that phonons have entered a hydrodynamic regime.
Wave-like features can also arise through an interplay between temporal memory and spatial nonlocality. Because phonons span a broad range of characteristic distances, their contributions can fall out of step, producing spatial dephasing that that suppresses the observable thermal wave.
For room-temperature silicon, the researchers found that this spatial dephasing strongly suppresses coherent, second-sound-like oscillations.
"Seeing a wave-like temperature signal is exciting, but it is not enough to identify the microscopic mechanism," Zeng said. "Our theory provides a way to examine how temporal memory and spatial nonlocality contribute to wave-like temperature response and to distinguish different physical origins of non-Fourier behavior."
From interfaces to future electronics
The work is fundamentally theoretical, but the questions it addresses are increasingly practical.
As computer chips become smaller and operate faster, their dimensions and operating times approach the natural distances and relaxation times of phonons. Heat may no longer dissipate in the smooth, predictable manner assumed by conventional device models.
A more complete description could eventually help researchers evaluate materials, interfaces and device geometries when ordinary thermal conductivity no longer provides enough information. The framework also establishes a way to think about interfaces between materials. Interface heat flow is commonly summarized as Kapitza resistance. Within the new framework, conventional Kapitza resistance can be understood as a limiting description of a richer spatially nonlocal response occurring near the interface.
The study uses an illustrative interface model rather than a material-specific calculation. Nevertheless, it shows how an apparent temperature jump can arise from a finite boundary region in which microscopic heat correlations are redistributed. Future first-principles calculations will be needed to construct and validate such kernels for real interfaces.
Because the framework is based on equilibrium heat-flux fluctuations, it is not restricted to perfectly ordered crystals or to a picture of phonons travelling independently like particles. The theory also formally connect with established descriptions of heat transport in disordered harmonic solids.
Future work will focus on calculating the kernels directly from atomistic simulations and applying the theory to disordered, composite and interfacial systems. Efficient mathematical representations will also be needed before the full framework can be incorporated into simulations of large devices.
For now, the study offers a new way of thinking about an old and seemingly simple phenomenon.
Heat may not remember a face, hold a grudge or recall where it left its keys. But at the smallest scales, it carries traces of where it has been and what happened before—and those traces may become increasingly important to the technologies of the future.
