# Jakob Yngvason, Elliott H.Lieb's A Guide To Entropy And The Second Law Of Thermodynamics PDF

By Jakob Yngvason, Elliott H.Lieb

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Take or2 = • • • = am = 0 as the ground state: it gives a contribution one both t o F(0) and to F(l). 58). Then allow two of the {ar} to be one, and so on. 3) + W - l + 1. 66) simplifies andfinallywe get the local density in terms of elliptic function products: This result holds only for densities p < pCf when g lies between 0 and 1. There are many other phases of the hard hexagon model, in which however one uses identities like these (there are about 40 identities listed by Ramanujan). 53 DUALITY It's possible to convert the Ising spin model on a square lattice to a vertex model if the Boltzmann weight is an even function of the four spin round a face: < o G :0 In that case instead of specifying w by the values of a, b, c, d, one could specify it by the values of their products along the four edges (fig.

22 a) we have to introduce three other corner transfer matrices which are similarly defined (fig. 22 b, c, d). <*>' M 14f) B(*' | *") 4> « 1 fig. 22 (b) J. I # fig. 22 (a) C (^ i r) \$ W fig. 22 (c) fig. rl ^ _M fig. 22 e Then the partition function of the full lattice is (fig. 44) is oyer all spin sets <£•••<£'" subject to the restriction that <7i = a\ = a" = cr'/f. 45) 46 and the same for B, C and D. 45): < * >== TrSABCD TrABCD \0 , 0 Ma. <*"> V Note that the position of S in the numerator product is irrilevant because it commutes with each of A, B, C and D.

T r" ) - P « X . '? 28) Let's simplify the notation. 29) 36 These equations have to be true for all values of the six external spins, so there are 2 6 ( = 64) equations to be satisfied. For the Ising model these equations reproduce the star triangle relations and the Onsager's commutation relations. 29) reads: f eK'ca+L'*c+M cc cogh ^ + Lf bd + MQ = = « * / * * ' « + • » ' * / cosh (Kl + Lc + Ma) and these 64 equations factor into two sets of 8 equations: 2 cosh (Kl + Lc + Ma) = A e * W z / a * + M , e c 2 cosh (Kb + Lf + Md) = AeK'fd+L'bd+M'bf Graphically: M These are just the star triangle relations.