Ohm and Fourier's law with Seebeck effect - derivation?

Question

In my work, I am dealing with a mathematical model which involves thermoelectric magnetohydrodynamics (TEMHD). Ohm's law becomes modified to become

$$ \mathbf{J}=\sigma\left(\mathbf{E}+\mathbf{u}\times\mathbf{B}-S\nabla T\right)$$

and Fourier's law becomes modified to become

$$ \mathbf{q}=-k\nabla T+ST\mathbf{J} $$

where $S$ is the Seebeck coefficient and all the other terms have their usual meanings. I am struggling to find a straightforward derivation or proof as to why they take this form when TEMHD effects are included, i.e. where do the $S\nabla T$ and $ST \mathbf{J}$ terms come from.

I do not have a background in thermodynamics and only a basic familiarity with Maxwell's equations. Could somebody pinpoint me to where I can find a straightforward first-principles derivation of where the above forms come from?

Answer 1

Check Landau & Lifshitz volumes 8 "Electrodynamics of Continuous Media" chapter III ("constant current"), paragraph named "Thermoelectric Phenomena" for the origin of the thermoelectric terms. Basically everything boils down to Onsager's reciprocity of kinetic coefficients described in vol.5 "Statistical Physics".

Also check vol. 10, chapter V ("Plasma in Magnetic Field") for kinetic coefficients in plasma (in particular the paragraph "hydrodynamics of magnetoactive plasma"). I am not sure if it has exactly what you need, but it might be a good starting point.