From Boltzmann transport equation to fouriers law - infinite speed of propagation of information

The macroscopic Fourier's law can be derived from the Boltzmann transport equation using the relaxation time approximation for the collision integral.

This is demonstrated for an ideal gas in this lecture on pages 48-59.

Obviously, Fourier's law contradicts special relativity because the parabolic PDE implies infinite speed of propagation for the heat.

I am wondering, which assumption within the derivation leads to this contradiction.

Is it the assumption of the validity of BTE for microscale itself? (single-particle distribution functions)

Is it the relaxation-time approximation used to linearize the BTE?

Is it the method of moments, used to derive the energy balance equation?

Any ideas?

It comes from the Maxwellian distribution function (6.11) $$F \sim \exp\left({-\frac{\vec{v}^2}{2mT}}\right)$$ The relativistic version should be $$F \sim \exp\left({-\frac{\sqrt{m^2+\vec{p}^2}}{T}}\right).$$ And is now a distribution over $$\vec{p}$$. You could also subtract out the rest mass $$m$$ in the exponent because it cancels out when you normalize $$F$$. (I have used units where $$c=k_B=1$$ but hopefully it's obvious where they go.)