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A new method for the thermodynamics of the BCS model. (English) Zbl 0658.60141
The authors analyze a simplified version of the B.C.S. - model to understand phase transitions. Although Bogolyubov and his school have developed perturbative methods to analyze the full B.C.S.-model, these are rigorously valid only in the limiting case of zero temperature. The authors deal with a quasi-spin model, with variable energy and coupling terms.
The method used is based on the recent applications of Varadhan’s large deviation principle to quasi-spin systems by W. Cegla, J. T. Lewis and G. A. Raggio [Commun. Math. Phys. 118, No.2, 337-354 (1988)] and the Berzin-Lieb estimates for the free energy. The technique consists in introducing a large number of subsystems and approximating the subsystem hamiltonian by a total spin hamiltonian. This approximation is shown to converge when the number of subsystems goes to infinity. This limit and the thermodynamic limit are shown to commute.
The variational principle arising from the application of the large deviation principle is shown to yield the gap equation. The existence of bifurcation at a critical parameter value is demonstrated. It is shown that the order parameter does not vanish below this critical temperature.
Reviewer: Y.Prahalad

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
82B26 Phase transitions (general) in equilibrium statistical mechanics
82B10 Quantum equilibrium statistical mechanics (general)
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