Do electrons really diffuse when a temperature gradient is applied? In many websites and books, it is generally said that the charge carriers, be it electrons or holes, diffuse through the considered material when a temperature gradient is applied. However I have found exactly zero justification of such a claim, be it either by words or by a mathematical equation that would show that indeed, the dynamics of these charge carriers is driven by a diffusion-like equation.
Here's Wikipedia: reference, between a plethora of other sources.

At the atomic scale, an applied temperature gradient causes charge
  carriers in the material to diffuse from the hot side to the cold
  side.

I would like to know, and see, the mathematical derivation of such a claim. So far I have two ideas that could potentially lead to the answer, but I am unable to really proceed further.
The first one, is that the motion of particle means that the system is in a nonequilibrium state, where there is a non vanishing $\nabla \mu$ (chemical potential) at least in some region of said material. So the equation must contain that quantity, probably. Then, it must also contain $\nabla T$, because it is the driving force (basically the Seebeck effect). This really looks like this will involve Onsager reciprocal relations... but then I fall short in involving time, which looks like is required to justify the claim.
My other idea is to go to Solid State Physics, and make the assumptions/simplifications required for the Boltzmann transport equation to hold, for the charge carriers. But then, how would I involve the temperature, the chemical potential and how would I derive a diffusion-like equation from it?
 A: My current understanding shows no diffusion whatsoever. I carefully went through Callen's paper on the application of the Onsager relations to thermoelectricity, where the Kelvin relations are derived. This is non equilibrium thermodynamics, but it still assumes steady state. In other words, the transient time dependency of the problem is not looked upon. For this reason, I see no way to obtain any diffusion-like equation when dealing with Onsager's theory.
Now onto my other idea: the Boltzmann transport equation. By making reasonable assumptions (such as considering the charge carriers as quasiparticles with a well definite position, velocity, etc. as well as using the time relaxation approximation), and by following a particular textbook*, the equation reduces to $\frac{\partial f}{\partial t} + \vec{v} \cdot \frac{\partial f}{\partial \vec{r}}+e\vec E \cdot \frac{\partial f}{\partial \vec{p}}=-\frac{f-f_0}{\tau}$, where, if I understand well, $\vec E$ is the external applied field. In my case, which is an open circuit system, $\vec E=\vec 0$. Note that the electric field arising thanks to the Seebeck effect is still present in the equation.
Skipping many mathematical steps and a few reasonable assumptions, I reach that the equation reduces to $\frac{\partial f_0}{\partial t}+\frac{\partial f_0}{\partial \vec v} \cdot \frac{\xi}{\vec p T} \vec v \cdot \nabla T=\frac{f_1}{\tau}$. Where $f=f_0+f_1$ is a density of probability function, and $\tau$ is the scattering time. While I did not throw out the term with a time derivative, I do not get any term with a second derivative of spatial coordinates. Hence the possibility to reach a diffusion-like equation seems too remote. The equation governing the dynamics of the charge carriers looks like very complicated to me, and does not seem to reduce to a diffusion equation. If someone offers a different point of view (backed up with mathematical insights, not just gedanken experiments/justifications), that would be nice.
I'm really glad I investigated the so-called claim that charge carrier diffuse as if it was obvious, because to me at least, it is far from obvious and from what I could deduce, it shouldn't be the case.



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*Fundamentals of the theory of metals, by Abrikosov.

