The thermopower is a combination of the thermoelectric coefficients L12 and L11. They are not well-defined (or diverge) in the absence of scattering (i.e.\ in the clean limit). The thermopower, on the other hand, is well-defined and finite in the clean limit, and easily calculable by combining thermodynamic and hydrodynamic relations. The key observation is that the electron liquid must be in mechanical equilibrium when no current is allowed to flow.
(i) Show that this requires ∇P=ne∇Φ, where P is the pressure of the electron liquid, n is its local number density, and Φ is the local electric potential.
Since P is an intensive quantity, it can only depend on other intensive quantities, in this case, T and μ. We thus have
∇P=(∂μ∂P)T∇μ+(∂T∂P)μ∇T.(8.181)
(ii) Use appropriate Maxwell relations between thermodynamic quantities to show that (∂P/∂μ)T,V=(∂Ne/∂V)T,μ=Ne/V=n and (∂P/∂T)μ,V=(∂S/∂V)T,μ=S/V. The last steps follow from the extensitivity of S, Ne, and V, which are proportional to each other when the intensive quantities μ and T are fixed.
(iii) Now show that the thermopower in this clean limit is given by Eq.\ (8.180).
This exercise illustrates that the thermopower is not an ordinary transport coefficient, but more like a thermodynamic/hydrodynamic quantity. It is a measure of the entropy carried by mobile electrons, divided by their number. The derivation above does not assume non-interacting electrons, because electron–electron interactions do not alter the thermodynamic and hydrodynamic relations used. On the other hand, being in the clean limit is important, as impurity scattering can give rise to additional forces (effectively frictional forces) that alter the force equilibration relation. If the mean free path is a strong function of the carrier energy, this effect can also modify the thermopower.