FIG. 2: The thermopower vs. T and x for the t - J model (with J = 0) for t > 0 (a) and t < 0 (b). Note that x ∼ 0 . 7 corresponds to the Curie-Weiss metallic phase of NCO, cf. Fig. 1. The line-type and color-coding is the same as in Fig. 1. Note that for dopings below 0.5 it is not clear whether the t - J model adequately describes the physics of NCO. 5 of the quasiparticles for the FQHE at ν = 5 / 2 (provided they are non-Abelian). Thus, a non-zero entropy linear in N is obtained. From Eq. (8), we see that the ther- mopower is the derivative of the entropy with respect to the number of particles at constant T and V . When entropy is linear in particle number, as in non-Abelian FQHE states, ∂ S /∂N → S /N and the formulas are iden- tical. Our derivation provides a simple and straightfor- ward insight into the formula given previously $^{5}$. Before concluding, we point out an intriguing applica- tion of S$_{Kelvin}$ for high T$_{c}$ superconductors. For different families of high T$_{c}$ compounds, a universal curve of the thermopower, at T = 290 K , as a function of hole concen- tration p ∼ 1 − n has been observed $^{19}$. The thermopower, in all families, vanishes near optimal doping ( p ∼ 0 . 16) starting out positive at small p . Phillips et al . 20 appeal to the atomic limit of S$_{MH}$ as an explanation. Viewing this data 19 more generally, through the prism of S$_{Kelvin}$ (Eq. (8)) we conclude that the optimal filling, i.e., maxi- mum T$_{c}$ , additionally corresponds to a local maximum of the electronic entropy as a function of filling . This con- clusion is powerful, since we avoided the difficult issue of calculating either thermopower or entropy, merely using the link between them provided by S$_{Kelvin}$ . We now discuss how S$_{Kelvin}$ is applied to dissipation- less systems such as the FQHE where thermopower can be used as a possible non-Abelian quasiparticle detector $^{5}$. For a weakly disordered electron system (from Eq. (10) and (11)) S$_{Kelvin}$ essentially gives the dissipation-less thermopower where particle velocities are further approx- imated. If the system is dissipation-less and the parti- cle velocities are also energy independent, such as the FQHE, then we expect S$_{Kelvin}$ is exact. An expression for the thermopower in a 2D electron system in the pres- ence of a perpendicular magnetic field (the FQHE sys- tem) has been derived $^{5,12}$, assuming zero impurities, as S q e N (Eq. (6) in Ref. 5). Yang and Halperin show 5 that S ∼ k$_{B}$N log( d ) where d > 1 is the quantum dimension It is clear that S$_{MH}$ , which has served as a virtual workhorse for years, has a new competitor in S$_{Kelvin}$ . This simple minded approximation can be written in closed form and in many difficult regimes, where the ex- act Kubo-Onsager expressions are not useful, and S$_{Kelvin}$ provides an excellent guide to the physics of the system. We acknowledge support from DOE-BES Grant No. DE-FG02-06ER46319 at UCSC, and MRP acknowledges support from LPS-CMTC at University of Maryland and Microsoft Project Q. E. High Temperature Superconductors IV. CONCLUSION D. FQHE at ν = 5 / 2  0  0.2  0.4  0.6  0.8  1  0  2  4  6  8  10 -100 -50  0  50  100  150  200  250  300 (a)  t  > 0,  J =0 x T /| t | S * S Kelvin S$_{MH}$  0  0.2  0.4  0.6  0.8  1  0  2  4  6  8  10 -100 -50  0  50  100  150  200  250  300 (b)  t  < 0,  J =0 x T /| t | S * S Kelvin S$_{MH}$ 5 K. Yang and B. I Halperin, Phys. Rev. B. 79 , 115317 (2009). 1 I. Terasaki, Y. Sasago, and K. Uchinokura, Phys. Rev. B 56 , R12685 (1997); Y. Wang et al , Nature 423 , 425 (2003); M. L. Foo et al , Phys. Rev. Lett. 92 , 247001 (2004). 2 J. O. Haerter, M. R. Peterson, and B. S. Shastry, Phys. Rev. Lett. 97 , 226402 (2006); M. R. Peterson, B. S. Shas- try, and J. O Haerter, Phys. Rev. B. 76 , 165118 (2007); J. O. Haerter, M. R. Peterson, and B. S. Shastry, ibid. 74 , 245118 (2006). 3 A. P. Ramirez, in More is Different , edited by N. P. Ong and R. N. Bhatt (Princeton University Press, New Jersey, 2001). 4 B. S. Shastry, Phys. Rev. B. 73 , 085117 (2006). 12 Y. N. Obraztsov, Fiz. Tverd. Tela (Leningrad) 7 , 573 (1965) [Sov. Phys. Solid State 7, 455 (1965)]; N. R. Cooper, 6 G. Moore and N. Read, Nucl. Phys. B 360 , 362 (1991). 7 R. Willett et al ., Phys. Rev. Lett. 59 , 1776 (1987). 8 W. L. K Thomson, Proc. Roy. Soc. Edinburgh 123 Col- lected Papers I, pp. 237-41, (1854). 9 L. Onsager, Phys. Rev. 37 , 405 (1931); ibid . 38 , 2265 (1931). 10 R. Kubo, J. Phys. Soc. Jpn. 12 , 570 (1957). 11 B. S. Shastry, Rep. Prog. Phys. 72 , 016501 (2009).