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?

• Are you referring to the thermoelectric effect? I generally think of this electron drift/diffusion phenomenon at a temperature difference as electrons in the hot end gaining more thermal energy and thus "taking up more space" as their random motions become more violent and rapid. They will then eventually drift towards the colder end, because there is more "space". In the same way that a gas expands to an area of lower pressure. – Steeven Sep 20 '18 at 11:50
• Turn it around - what do you need for a mobile entity not to diffuse? – Jon Custer Sep 20 '18 at 12:31
• @Steeven yes, I am referring to the thermoelectric effect (as can be seen from the tag and the mention of the Seebeck effect). I get that part, but this thinking says nothing about the way the electrons "drift" towards the colder end (in reality it depends on the Seebeck coefficient sign, they can drift toward the hotter part). The mention of the gas might be a good and relevant one, because it's clear that a diffusion-like equation (or Fokker-Planck) is taking place at a microscopic level, for individual molecules. – thermomagnetic condensed boson Sep 20 '18 at 13:54
• @JonCuster Very good point. Indeed, all seems to point out at diffusion, so intuition tells us this should be the case. Now the hardest part is to show that this is indeed the case, under certain assumptions. Because of course, it should be possible to tweak the system so much that in the end the charge carriers motion is more ballistic than diffusion-like, even though in the general case it should be diffusion-like. – thermomagnetic condensed boson Sep 20 '18 at 13:56
• Regarding the sign of the Seebeck coefficient, it is opposite for an opposite semiconductor (p-type), e.g. when the majority charge carrier is holes. Generally, the "thing" being "pushed" from hot to cold end is charge carriers, whichever they might be, rather than electrons as I used in my example. – Steeven Sep 20 '18 at 15:03

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.