energy so ε$_{m}$ → ε$_{n}$ . Thus, The derivative with respect to µ in Eq. (6) is within the grand canonical ensemble and performed with a fixed V and T . The denominator of Eq. (5) is treated similarly yielding q 2 $_{e}$β [ 〈 ˆ N $^{2}$〉 − 〈 ˆ N 〉 $^{2}$] = q 2 $_{e}$d 〈 ˆ N 〉 /dµ . Combining it with Eq. (6), yields 3 lim q $_{x}$→ $_{0}$χ$_{ρ}$$_{(}$$_{q}$$_{x}$$_{)}$$_{,}$$_{ˆ}$ K ( − q x $_{)}$(0) = lim q $_{x}$→ 0 ∑ n,m p$_{n}$ − p$_{m}$ ε$_{m}$ − ε$_{n}$ 〈 n | ˆ ρ ( q$_{x}$ ) | m 〉〈 m | ˆ K ( − q$_{x}$ ) | n 〉 = lim ε $_{m}$→ ε n ∑ n,m p$_{n}$ − p$_{m}$ ε$_{m}$ − ε$_{n}$ 〈 n | ˆ ρ ( q$_{x}$ ) | m 〉〈 m | ˆ K ( − q$_{x}$ ) | n 〉 = lim ε $_{m}$→ ε n ∑ n,m p$_{n}$ 1 − e β ( ε $_{n}$− ε m ) ε$_{m}$ − ε$_{n}$ 〈 n | ˆ ρ ( q$_{x}$ ) | m 〉〈 m | ˆ K ( − q$_{x}$ ) | n 〉 = q$_{e}$ ∑ n,m βp$_{n}$δ$_{ε}$ n ,ε m 〈 n | ˆ N | m 〉〈 m | ˆ K | n 〉 = q$_{e}$β [ 〈 ˆ N ˆ K 〉 − 〈 ˆ N 〉〈 ˆ K 〉 ] = q$_{e}$ [ d dµ 〈 ˆ H 〉 − µ d dµ 〈 ˆ N 〉 ] . (6) S$_{Kelvin}$ = 1 q$_{e}$T d $_{dµ}$〈 ˆ H 〉 − µ d $_{dµ}$〈 ˆ N 〉 d $_{dµ}$〈 ˆ N 〉 . (7) S$_{Kelvin}$ = 1 q$_{e}$ ( ∂ S $_{∂µ}$)$_{T,V}$ ( ∂N ∂µ )$_{T,V}$ = 1 q$_{e}$ ( ∂ S ∂N )$_{T,V}$ (8) = − 1 q$_{e}$ ( ∂µ ∂T )$_{N,V}$ (9) S$_{Mott}$ = T π $^{2}$k2 B 3 q$_{e}$ d dµ ln [ ρ$_{0}$ ( µ ) 〈 ( v x $_{p}$) $^{2}$τ ( p, µ ) 〉$_{µ}$ ] |$_{µ}$ → µ$_{0}$ , (10) S$_{Kelvin}$ = T π $^{2}$k2 B 3 q$_{e}$ d dµ ln [ ρ$_{0}$ ( µ )] |$_{µ}$ → µ$_{0}$ , (11) S$_{MH}$ = T π $^{2}$k2 B 6 q$_{e}$ d dµ ln [ ρ$_{0}$ ( µ )] |$_{µ}$ → µ$_{0}$ (12) A. Free electrons III. APPLICATIONS OF THERMOPOWER FORMULAE To further simplify Eq. (7) we note a relation found in textbooks on thermodynamics in the grand canonical ensemble: 〈 ˆ H 〉 ≡ E = Ω + T S + µN (Ω the grand po- tential), so that ( ∂E $_{∂µ}$)$_{T,V}$ = µ ( ∂N ∂µ )$_{T,V}$ + T ( ∂ S $_{∂µ}$)$_{T,V}$ and hence where we used, to go from the second equality to the last equality, a Maxwell relation obtained with dF = −S dT − pdV + µdN , and equating ∂ $^{2}$F ∂T ∂N = ∂ $^{2}$F ∂N∂T . We refer to the last two equivalent equations (Eqs. (8) and (9)) as the Kelvin formula for the thermopower. This formula is unknown in the literature as far as we are aware. which differs from the exact answer (Eq. (10)) in the neglect of the relaxation time τ and particle velocity v x p in the logarithm. S$_{MH}$ , to the same order, gives a formula often ascribed to Mott and ρ$_{0}$ ( µ ) is the sin- gle particle density of states per unit volume per spin. In this non-interacting electron context, S$_{Kelvin}$ gives (to O ( T $^{3}$)), To gain insight into the strengths and weaknesses of the various thermopower formulations discussed above we consider non-interacting degenerate electrons treated within the limit of elastic scattering at the Born level with an energy momentum dependent relaxation time τ ( p, ω ). This is a modestly dissipative system, but at such a simple level that the Boltzmann-Bloch equation is an adequate description. The solution for S is available in textbooks and a useful benchmark for various approx- imations. In the low temperature limit $^{11}$, to O ( T $^{3}$), Note that S$_{MH}$ is similar to S$_{Kelvin}$ . The distinction is that in S$_{Kelvin}$ , the number derivative of the entropy is taken at constant T rather than at constant E . Thus, in the low T limit of a metal, where µ ( T ) ∝ T $^{2}$, they differ in the linear T coefficient by a significant factor of two. We show below that for non-interacting electrons, scat- tered by impurities, S$_{Kelvin}$ is closer to the exact result than S$_{MH}$ . Further, we see that the approximation of ex- changing the slow and fast limits has some justification in dissipation-less systems, such as in the FQHE where S$_{Kelvin}$ is identical to that found by several workers (see below).