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, provides a unified way to describe that memory.

"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 length scales 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 material and experimental scale, heat transport may then appear diffusive, quasi-ballistic, spatially 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."