Uniqueness and the choice of the acceptable solutions of the MSA

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Abstract
We prove rigorously, in the general case of arbitrary potential and number of components, that if a solution of the mean spherical approximation for simple fluids exists it is unique within the class of correlation functions corresponding to positive and bounded structure factors. The longstanding problem of the choice of the acceptable solutions of the system of nonlinear algebraic equations which arises with some methods used to derive analytical solutions is carefully examined with a particular emphasis on the factorization method for Yukawa mixtures. It is shown that a local criterion, related to the boundedness of the pair correlation functions, already suggested by Baxter but apparently ignored in the recent literature, is sufficient to select the unique (acceptable) solution of the model. A very simple and practical implementation of the method is given. The dependence of the asymptotic behaviour of the pair correlation functions on the parameters is briefly discussed.