Fluid properties, integral equations statistical mechanics

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

As in the case of the statistical mechanics of a fluid, these Boltzmann factors contain more information than is necessary in order to characterize the experimental properties of polymer systems. We therefore focus attention upon reduced distribution functions in order to make contact with the macroscopic observable properties of polymers. In the usual many-body problems encountered in statistical mechanics, the reduced distribution functions are the solutions to coupled sets of integro-differen-tial equations. - On the other hand, because a polymer is composed of several atoms (or groups of atoms) that are sequentially joined together by chemical bonds, these reduced distributions for polymers will obey difference equations. Therefore, by employing the limit in which a polymer molecule is characterized by a continuous chain, these reduced probability distributions can be made to obey differential, instead of difference, equations. This limit of a continuous chain then enables the use of mathematical analogies between polymers and other many-body systems. The use of this limit naturally leads to the use of the technique of functional integration. 

Many of the equilibrium properties of such systems can be obtained through the two-body reduced coordinate distribution function and the radial distribution function, defined in Eqs. (27.6-5) and (27.6-7). There are a number of theories that are used to calculate approximate radial distribution functions for liquids, using classical statistical mechanics. Some of the theories involve approximate integral equations. Others are perturbation theories similar to quantum mechanical perturbation theory (see Section 19.3). These theories take a hard-sphere fluid or other fluid with purely repulsive forces as a zero-order system and consider the attractive part of the forces to be a perturbation. ... 

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