A perfect harmonic crystal (PHC) has infinite thermal conductivity (see here for example, or also (1)).
Does this mean that in a PHC there is instantaneous heat transfer?
Fourier's law would seem to suggest that this is the case, since when the thermal conductivity is infinite the time derivative of heat blows up. However, I don't know how faithfully this equation describes heat propagation, even in an idealized system like a PHC.
If infinite thermal conductivity really implies instantaneous heat propagation, this bothers me a little, even if I know that the PHC is an idealization. What bothers me is that in my mind even in a PHC you need some time for the vibration to propagate, and therefore it seems absurd that heat propagation is infinite.
To be clearer, what I am imagining is this: I take a still PHC (no vibrations...we are considering the classical model here), and I impart momentum to a finite group of atoms, making them vibrate. Will this vibration propagate in the whole system instantaneously?
Notice that I avoided talking about temperature, since a temperature cannot be defined in a PHC, because the system can never thermalize (1).
Since the work of Born and Debye, it has been known that the idealised model, with harmonic forces between atoms, is useless for an investigation of thermal conductivity, since it would lead to an infinitely large conductivity.
For in the model with harmonic forces [...] one can think of the motions of the atoms as built up from mutually independent sound waves. Once an arbitrary energy distribution of these vibrations exists, it will exist forever. A thermal equilibrium will then not be established, and in general one can therefore not speak of a temperature at all.
-R. Peierls, “Zur kinetischen Theorie der Wärmeleitung in Kristallen” Ann. Phys. 395, 1055–1101 (1929)