Download A Guide To Entropy And The Second Law Of Thermodynamics by Jakob Yngvason, Elliott H.Lieb PDF

By Jakob Yngvason, Elliott H.Lieb

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Extra info for A Guide To Entropy And The Second Law Of Thermodynamics (Princeton)

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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.

As A , B , C, D can be parametrized in terms of g, X, u, we can set: A==A(u) B==B(u) C==C(t*) D==D(u) Then: C=A=A(u) D = B = A ( X ~~u) K * } Infact the bottom right and the top left corner transfer matrices are the same and so are the bottom left and the top right; besides we know that replacing u by (X — u) merely rotates everything of 90 degrees (it's the property (ii) of symmetry). We can use these symmetries of the weight function because the boundary conditions we gave (all spins up) don't break them; however, at least in the antiferromagnetic case and in the hard hexagon case these symmetries may be broken.

14f) = ^ , ^ . ^ 1 , ^ 1 ) • • to( 2. e. two values for the set which contributes; infact for all a, b, c and d.

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