Heat has a ‘memory’: physicists have developed a new theory of heat transfer

On the usual scales, heat behaves predictably: it flows from hotter regions to colder ones, and engineers usually find Fourier’s law sufficient. But at the nanoscale and over time scales of trillionths of a second, this description may no longer be adequate. The current flow of heat can depend not only on the ‘here and now’ temperature, but also on what happened earlier and at another point in the material.

Physicists Yi Zeng and Jianjun Dong have proposed a unified theoretical framework that accounts for this physical ‘memory’ and the spatial non-locality of heat transfer. The paper has been published in the journal *Physical Review B*.

Details

The classical Fourier law assumes that the heat flux at every point is instantaneously determined by the local temperature gradient. For typical scales and timescales, this works very well. However, in modern nanostructures, energy transfer becomes more complex.

In crystalline solids, heat is largely carried by phonons — collective vibrations of atoms. On large scales, they scatter many times, and the result resembles ordinary diffusion. But over very short distances, some phonons manage to travel a considerable distance before colliding. Consequently, the heat flow can be influenced by a temperature perturbation that occurred earlier or is not immediately nearby.

The authors have described this effect using a space-time response kernel. This kernel shows the extent to which a temperature difference at a specific point and at a specific moment in the past influences the heat flux at another location in the present. The temporal component of the kernel accounts for the ‘memory’, whilst the spatial component accounts for the range of this influence.

In this framework, ordinary diffusion, quasi-ballistic transport and hydrodynamic regimes are not separate theories, but rather different limiting cases of a single description. The approach also generalises the Green–Kubo relation, which links a material’s transport properties to its microscopic fluctuations.

To test how the model works, the researchers calculated heat transfer in crystalline silicon at room temperature and compared it with experiments using a transient temperature gradient. Under the conditions considered, spatial non-locality proved to be the main source of deviation from Fourier’s law: what mattered more was how far the phonons managed to carry energy, rather than the duration of the ‘memory’ itself. The temporal effect was most pronounced in the very earliest stages of the thermal response.

When phonon scattering at the boundaries of a thin silicon membrane was included in the calculation, the model was quantitatively consistent with previously published experimental measurements.

The study also refines the interpretation of the so-called ‘second sound’—the wave-like propagation of a temperature perturbation. According to the authors’ calculations, the thermal wave itself does not yet prove that the phonons have transitioned to a hydrodynamic regime: a similar signal may arise from a combination of temporal memory and spatial non-locality.

Why this is important

As computer chips become smaller, their dimensions approach the characteristic distances that phonons travel before scattering, whilst the duration of electronic processes approaches their relaxation time. Under such conditions, thermal conductivity alone may not be sufficient to accurately predict where and how quickly heat will dissipate.

A unified model could potentially allow for more accurate comparisons of materials, the interfaces between them, and the geometry of miniature devices. This could be useful in the design of electronics, where heat dissipation is becoming one of the main limitations on performance.

However, this is currently just a theoretical tool, not a new cooling technology. For real-world interfaces, the authors used an illustrative model. The next step is to calculate such cores from atomistic simulations and test them for specific materials, composites and interphase boundaries.

Source

Phys.org / Auburn University, 3 September 2026: Heat has a memory—and a new theoretical framework can track it.

Original research paper: Yi Zeng, Jianjun Dong, ‘Unified statistical theory of heat conduction in non-uniform media’, Physical Review B 113, 235204. Published 8 June 2026. DOI: 10.1103/hx28-4ksc. The article is published under an open access licence (CC BY 4.0).