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?