# With what speed does a thermal front travel through a solid?

The heat equation provides theoretical temperature distributions as a function of space and time. Take for example an infinitely long rod that is suddenly heated up at $$t = 0$$ at its end $$x = 0$$. The heat equation predicts that the thermal front (defined as the surface where the temperature changes from its initial temperature to a different temperature) moves away from $$x = 0$$ at a very high (almost infinite or infinite) speed. Does this match experimental observations, or is there a finite limit of this speed and if so, what is it? I imagine doing experiments near zero Kelvin to minimize "thermal noise".

• Good question. Presumably it would move at about the speed of sound in the solid. Jan 8, 2022 at 7:58
• John Rennie is right. May be worth reading about "second sound" in superfluid liquid helium where great flows like a coherent wave, not diffusively. There the speed is the fastest in any terrestrial object and it is limited by the superfluid critical velocity. Most objects with conduct heat much more slowly, but this may give the right idea about the speed limits. Jan 8, 2022 at 11:05
• @KFGauss: Interesting. Second sounds also appears as a parameter in the "hyperbolic heat conduction" equation, which was proposed to upgrade the old parabolic heat equation, the latter being incompatible with relativity theory. Source: en.wikipedia.org/wiki/Relativistic_heat_conduction Jan 9, 2022 at 6:59

I'm not aware of experimental results that answer your question, but I'll take a shot from a theoretical perspective.

Considering pure conduction, thermal energy is transferred through a material via a series of collisions between adjacent molecules. The finite limit of the speed at which these collisions could propagate through any material would be the speed of sound in that material, as that is the limit by which pressure waves (in this case, vibrations of thermally excited molecules) can travel.

However, the speed of sound is the speed at which the individual collisions/vibrations travel. Since this occurs in random directions, unlike a directed sound wave, a discrete heat front will travel slower than the speed of sound. If you define the thermal front as any tiny deviation from initial temperature, then the question simplifies from thermal front velocity to velocity of a vibration. In that case, it should move at the speed of sound, since a random vibration will propagate in every direction at sound speed and cause some infinitesimal temperature change as it goes through. However, if you define the thermal front by a specific temperature change, then you can easily calculate the propagation speed with the heat equation.

Short version: the speed that the heat front moves will not be infinite, it will be equal to or less than the speed of light in the material. It will usually be much less than that. The temperature of the heat front or just behind the front itself will determine roughly what regime of speed it will move at. Pressure, density, composition of the material will also be important.

For your specific example of a rod, let's say it is at 20 K ("weird things" can happen at 0 K so we'll stay a bit above it), and the heat front is at 300 K. Also assuming it is "normal" matter, not exotic, and it is not a metal. In this temperature range, the thermal excitations in the heat front are the vibrations of atoms around their positions, and the speed of thermal energy transfer is the speed of sound on the solid.

For another example let's say the heat front is at 3000 K The thermal excitations in the heat front are now vibrations of the atoms and motion of the electrons. The energy of the latter will propagate through the material at the speed of light in the material. Then there will be disequilibrium between the election temperature and the ion temperature which will resolve/propagate through the material.

To learn more you can search for "solid state physics heat conduction"