Uniqueness and the choice of the acceptable solutions of the MSA

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.

Rescaled mean spherical approximation for colloidal mixtures

“…For its practical use, however, a coupled set of algebraic equations, which possesses in general several solutions, has to be solved [5]. As it has been shown by Pastore [6], only one of such solutions is physical. Its identification, however, is still rather cumbersome [5,61.…”

Integral equation theory for nematic fluids

“…Finally, we obtain a system of algebraic equations for coefficients of factor correlation functions and Laplace transforms of a pair correlation function harmonics In (2.27) we should expect that multiple solutions occur, of which only one is acceptable [42]. To choose the physical solution one can utilize the condition…”

Solution of the Ornstein–Zernike Equation in the Critical Region

“…Ref. 35). This suggests that such solutions may be unstable with respect to the current iterative solution scheme.…”

“…(1). Imposing the additional constraint that the pair correlation functions remain bounded is sufficient to eliminate these unphysical solutions[35]. The assumptions underlying Eq.…”