Does heat transfer happen instantanoeusly in a perfect harmonic crystal?

Question

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).

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(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)

Answer 1

Even in a perfectly harmonic crystal, an initially localized excitation of the atoms from their equilibrium position will not propagate through the system instantaneously! If you look at the acoustic branches of the crystal oscillation dispersion relationship, you will see that the fastest velocity of propagation of any disturbance will be the sound velocity (longitudinal or transverse).

Answer 2

Let's assume you have a 1D lattice of songs with spacing $a$ and one atom per unit cell. Let's further assume the harmonic approximation.

If you simulate heat by displacing a single atom, the resulting wave from this displacement will excite phonons of all wavevectors $k$. This is because you are starting with essentially a delta function in space, or equivalently something that is uniform in wavevector space.

The dispersion relation for phonons in a 1D chain is $\omega(k)=2\omega_0|\mathrm{sin}(ka/2)|$ which implies $\frac{d \omega}{d k}= a \omega_0 \mathrm{cos}(ka/2)$. This means the largest phonon speed is $a\omega_0$, not infinity.

Answer 3

My take on this is that I do not really see a way to assign a speed to the heat propagation because temperature is not even defined, as stated by Peierls. So this would be my answer, and I post it as a complement to the already two magnificient answers posted by @freecharly and @user157879.

Nevertheless, here are some comments:

• I think you may have a misunderstanding with the thermal conductivity $\kappa$ and the speed at which heat is propagated. In the regular conductive heat equation $\frac{\partial T}{\partial t} = \nabla \cdot \kappa\nabla T$, the speed at which heat is propagated is infinite. There are several ways to see this. One way is to solve the equation for a particular setup (i.e. boundary conditions and initial condition). You'll realize that the solution has a non retarded time dependence. In other words, the temperature will vary instantly due to a source, regardless of the distance between the source and the point considered. This is valid regardless of the value of $\kappa$, so clearly $\kappa$ is not the responsible for the speed of heat propagation.

• The greater the thermal conductivity, the greater the amount of heat is propagated. Not the fastest, where speed is thought of the distance at which the initial perturbation is propagated, divided by time. Now sure, one can define a diffusivity constant and imagine the diffusion of heat through time and a greater kappa would make a material seem to thermalize faster than a material with a lesser kappa. So if one define a speed of heat propagation based on the time a material thermalize (where the criteria to define thermalization is subjective), indeed, an infinite $\kappa$ would be related to an infinite speed due to the instantaneous thermalization.

• For some reason, you seem focused on phonons as mode of heat transfer. In metals, the electronic conduction plays a major role. So it seems you are implicitely dealing with a non metallic crystal with your question as stated.

• Non relativistic QM, which is used in describing the harmonic crystal, is not Lorentz invariant. Many times there is no reason to expect things to have finite speed of propagation in non Lorentz invariant theories. Take the free electron, intially localized in some region. One instant later (or say dt), no matter how far from the original region one is, one will find a non vanishing wavefunction. This means the electron can be found at any arbitrary large distance soon after being measured to be at any particular position.
  Taking the analogy of heat conduction with electricity conduction, one finds a finite speed (and lesser than c) at which current and EM waves travel in a material, regardless of the value of the conductivity $\sigma$ (which is analogous to $\kappa$ for heat conduction). Note that Maxwell equations describing EM waves propagation is Lorentz invariant, unlike the Schrödinger equation used to describe the Hamiltonian of a harmonic crystal, and the conduction heat equation where Fourier law applies.