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Quantum and Classical Statistical Mechanics of the Integrable Models in 1 + 1 Dimensions

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In a short but remarkable paper Yang and Yang [1] showed that the free energy of a model system consisting of N bosons on a line with repulsive δ-function interactions was given by a set of coupled integral equations. The Yangs’ chosen model is in fact the repulsive version of the quantum Nonlinear Schrodinger (NLS) model. We have shown that with appropriate extensions and different dispersion relations and phase shifts similar formulae apply to ‘all’ of the integrable models quantum or classical. These models include the sine-Gordon (s-G) and sinh-Gordon (sinh-G) models, the two NLS models (attractive and repulsive), the Landau-Lifshitz (L-L’) model which includes all four previous models, and so on. A significant feature is the bose-fermi equivalence of these models: the classical limit of the bose theories yields the classical statistical mechanics of the models in a simple way but in e.g. s-G fermion features remain even in classical limit. The models also include the integrable lattices and the Toda lattice is one of these. We comment on the statistical mechanics of the Toda lattice at the end of the paper. For simplicity the sinh-G model is used throughout this paper to exemplify the method of calculating the free energy, the method of ’generalised Bethe ansatz’.