called Mott-Heikes (MH) approximation introduced by Heikes (popularized by Mott $^{15}$) is written S$_{MH}$ = ( µ$_{0}$ − µ ( T )) /q$_{e}$T , where µ$_{0}$ ≡ µ ( T = 0). S$_{MH}$ is obtained by rather drastically replacing the first part of Eq. (4) by the zero temperature chemical potential µ$_{0}$ to make the theory sensibly behaved as T → 0. From thermodynam- ics, we know that − µ ( T ) T = ( ∂ S $_{∂N}$)$_{E,V}$ , and, hence, S$_{MH}$ relates thermopower to the partial derivative of entropy S with particle number N , at a fixed energy E and vol- ume V . We see below that S$_{Kelvin}$ is similar, but with more natural “held” variables, namely T and V . From Eq. (1), we can make a high frequency approxi- mation, where ω ≫ ω$_{c}$ ( ω$_{c}$ representing all finite charac- teristic energy scales), leading to the object S $^{∗}$. The for- mal expression and evaluation for S ∗ are discussed else- where 11 and we only quote the results. We have argued that S ∗ is the best possible approximation to the exact Kubo formula for strongly correlated systems 2 such as the t - J model, since the high frequency limit respects the single occupancy constraint and is closer to the DC limit than initially expected. It is not specifically suited for Hubbard type models, since the high frequency limit assumes ω ≫ U , and cannot capture the physics of cor- relations effectively $^{11}$. We will see that S$_{Kelvin}$ steps into this breach, and provides a very useful alternative for Hubbard type models $^{16}$. C. High frequency formula 2 is the susceptibility of any two operators ˆ A and ˆ B , where ρ , ˆ K = ˆ H − µ ˆ N , and ˆ J$_{x}$ are the charge density, the (grand) Hamiltonian, and the charge current operator, respectively, at finite wave vectors; ˆ H , µ , and ˆ N are the Hamiltonian, the chemical potential, and the total number operator, respectively. The susceptibility writ- ten in Eq. (3) is the Lehmann representation (where p$_{n}$ = exp( − βε$_{n}$ ) /Z is the probability of the quantum state | n 〉 with energy ε$_{n}$ and Z is the partition function and β = 1 /k$_{B}$T with k$_{B}$ the Boltzmann constant) which we find useful below. With Eq. (1), we can take different limits and obtain various interesting formulas. trical and electrical conductivities $^{11}$, where Taking the fast limit and using the continuity equa- tions to pass from densities to current operators, Eq. (1) gives the exact Kubo result 10 where q$_{e}$ is the charge of the carriers and ˆ J E the energy current. For narrow band systems, such as NCO, high T$_{C}$ superconductors, or heavy fermion systems, the so- To obtain an approximate thermodynamical expres- sion, we consider the slow limit of Eq. (1). S is among the few objects (along with Hall constant and Lorentz number) where this process gives finite and approximate results, unlike the electrical conductivity where the slow limit gives meaningless results $^{11}$. This limit is identified with Kelvin since he essentially took the equilibrium limit of an interacting gas of particles. The slow limit ( q$_{x}$ → 0, ω → 0) is easiest to compute starting from Eq. (1) To simplify we first consider the numerator of Eq. (5) which we rewrite by first using the Lehmann representation and then taking the q$_{x}$ → 0 limit. Note that ˆ ρ ( q$_{x}$ ) tends to a conserved quantity q$_{e}$N and cannot mix states of different S$_{Kelvin}$ = lim q $_{x}$→ 0 χ$_{ρ}$$_{(}$$_{q}$$_{x}$$_{)}$$_{,}$$_{ˆ}$ K ( − q x $_{)}$(0) T χ$_{ρ}$$_{(}$$_{q}$$_{x}$$_{)}$$_{,ρ}$$_{(}$ − q x $_{)}$(0) . (5) χ$_{ˆ}$ A, ˆ $_{B}$( ω ) = i ∫ ∞ 0 dte ( iω − 0 $^{+}$) $^{t}$〈 [ ˆ A ( t ) , ˆ B (0)] 〉 (2) = ∑ n,m p$_{n}$ − p$_{m}$ ε$_{m}$ − ε$_{n}$ + ω 〈 n | ˆ A | m 〉〈 m | ˆ B | n 〉 (3) S ( q$_{x}$, ω ) = χ$_{ρ}$$_{(}$$_{q}$$_{x}$$_{)}$$_{,}$$_{ˆ}$ K ( − q x $_{)}$( ω ) T χ$_{ρ}$$_{(}$$_{q}$$_{x}$$_{)}$$_{,ρ}$$_{(}$ − q x $_{)}$( ω ) , (1) D. The Kelvin formula II. THERMOPOWER FORMULAE A. The Kubo formula B. The Mott-Heikes formula S$_{Kubo}$ = 1 T ∫ ∞ 0 dt ∫ β 0 dτ 〈 ˆ J E x ( t − iτ ) ˆ J$_{x}$ (0) 〉 ∞ 0 dt β 0 dτ 〈 ˆ J$_{x}$ ( t − iτ ) ˆ J$_{x}$ (0) 〉 − µ ( T ) q$_{e}$T , (4)