# Application of Integrable Systems to Phase Transitions by C.B. Wang

By C.B. Wang

The eigenvalue densities in a number of matrix versions in quantum chromodynamics (QCD) are finally unified during this ebook through a unified version derived from the integrable platforms. Many new density types and loose power services are accordingly solved and offered. The part transition types together with severe phenomena with fractional power-law for the discontinuities of the unfastened energies within the matrix types are systematically categorized by way of a transparent and rigorous mathematical demonstration. The tools right here will stimulate new learn instructions akin to the $64000 Seiberg-Witten differential in Seiberg-Witten thought for fixing the mass hole challenge in quantum Yang-Mills conception. The formulations and effects will gain researchers and scholars within the fields of part transitions, integrable platforms, matrix types and Seiberg-Witten theory.

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87) , where l ≥ 1. According to the Cayley-Hamilton theorem for J (l) , choose α (l) = 2 2 (Λ + Λ2 − 4b(l) )/2, where Λ = Λ(η) = tr J (l) , b(l) = det J (l) and b(l) > 0. We can transform gj (j = 1, . . , 2m) into a new set of parameters gj (j = 1, . . , 2m) ms s q by a linear transformation so that W (η) = l−1 s=0 η q=0 glq+s+1 Λ , where each ms (s = 0, . . , l − 1) is the largest integer such that s + lms ≤ 2m − 1. 88) for η in the outside of the cuts to be discussed in the following. Then there is ωl (η) = 1 2 W (η) + y(η), where −y(η) is equal to ms l−1 ηs s=0 glq+s+1 q=0 q + r=[q/2]+1 q r μq 2 2[q/2] (l) q−2[q/2] q b(l) α [q/2] 2r b(l) α (l) q−2r .

24) which agrees with the result E(g) = E(0) + 1 2 1 b − 1 9 − b2 − ln b2 . 25) obtained in [3] for W (η) = 12 η2 + gη4 . If we think E as a function of 2g2 b2 , it can be seen that E has an extreme minimum point at 2g2 b2 = 2, or at g4 = g4c , where g2 g4c = − 122 , which is always not positive. For the non-symmetric density discussed above, g4 is positive at such point. If W (η) = g3 η3 + g4 η4 is degenerated to W (η) = g4 η4 by taking a → 0, the free energy becomes E = 3/8 − ln b. It is the same result as W (η) = g2 η2 + g4 η4 is degenerated to W (η) = g4 η4 .

We can transform gj (j = 1, . . , 2m) into a new set of parameters gj (j = 1, . . , 2m) ms s q by a linear transformation so that W (η) = l−1 s=0 η q=0 glq+s+1 Λ , where each ms (s = 0, . . , l − 1) is the largest integer such that s + lms ≤ 2m − 1. 88) for η in the outside of the cuts to be discussed in the following. Then there is ωl (η) = 1 2 W (η) + y(η), where −y(η) is equal to ms l−1 ηs s=0 glq+s+1 q=0 q + r=[q/2]+1 q r μq 2 2[q/2] (l) q−2[q/2] q b(l) α [q/2] 2r b(l) α (l) q−2r . 90) 40 2 Densities in Hermitian Matrix Models Fig.