Self-consistent integral equations thermodynamic consistency

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

In simple terms, eqn [52] are diffusion equations in the component densities, which take into account the noise in the system. Dynamics of the molecular ensemble is based on the assumption that for each type of site a the local flux is proportional to the local site concentration 4> . At equilibrium, = constant, which results in the familiar self-consistent field equations for an inhomogeneous polymer system. When the system is not in equilibrium, the negative gradient -V/i,j(r) represents an effective thermodynamic force that drives collective relaxation processes. The time integration of eqn [52] generates an ensemble of density fields with the Boltzmann distribution. 

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