Phonons on their own lead to the common heat equation. One sees that for example in insulators or non doped semiconductors. However in metals (or conductors), the electrons are the ones that are mostly responsible for the heat transfer, which extremely surprisingly to me, is also of the form of the common heat equation. In heavily doped semiconductors, both electrons and phonons play a major role on the thermal transfer, and again the heat equation applies. So... knowing that the electrons responsible for the heat transfer in a metal move at very high speeds (Fermi velocity), while phonons have a speed about 3 order of magnitude lower than electron's, and also that they are quite different from each other, how come they both produce the same heat equation?