Mean-field level theory

Theoretically, several aspects of the Thommes-Findenegg experiment can be analyzed at the mean-field level [157]. A key quantity of a mean-field theory of confined fluids is the (Helmholtz) free energy, given by... 

In practice, the scheme as explained above is not implemented. The consecutive generation of all possible chain conformations is a very expensive step. The reason for this is that there are of the order of ZN number of conformations, where Z is the lattice coordination number. A clever trick is to generate a subset of all possible conformations and to use this set in the SCF scheme. This approach is known in the literature as the single-chain mean-field theory, and has found many applications in surfactant and polymeric systems [96]. The important property of these calculations is that intramolecular excluded-volume correlations are rather accurately accounted for. The intermolecular excluded-volume correlations are of course treated on the mean-field level. The CPU time scales with the size of the set of conformations used. One of the obvious problems of this method is that one should make sure that the relevant conformations are included in the set. Typically, the set of conformations is very large, and, as a consequence, the method remains extremely CPU intensive. 

An important prerequisite for any theory of potential use for describing critical phenomena is the proper description of fluctuations. In their original form, both DH and MSA theories are not suited for this purpose. One of the major achievements is the development of various approaches, and in particular the GDH theory, which enable the description of fluctuations at the mean-field level. In fact, GDH theory provides the only description of ionic... 

A basic approach for the description of polymer chains in the continuum is the Gaussian thread model [26, 31]. Treating interactions among monomers in a mean-field-like fashion, one obtains the self-consistent field theory (SCFT) [11, 32-36] which can also be viewed as an extension of the Hory-Huggins theory to spatially inhomogeneous systems (like polymer interfaces in blends, nucrophase separation in block copolymer systems [11, 13], polymer bmshes [37, 38], etc.). However, with respect to the description of the equation of state of polymer solutions and blends in the bulk, it is stiU on a simple mean-field level, and going beyond mean field to include fluctuations is very difficult [11, 39-42] and outside the scope of this article. 

In eleetronie strueture theory, the single-eonfiguration pieture (e.g., the Is22s22p4 deseription of the Oxygen atom) forms the mean-field starting point the eonfiguration interaetion (Cl) or perturbation theory teehniques are then used to systematieally improve this level of deseription. 

Even at the level of mean field approximation (MFA), the GL theory gives significant information such as penetration depth (l) and coherence length ( ) of the superconducting samples. Many unconventional properties of superconductivity connected with the break down of the simple MFA has been studied both analytically (Tesanovich, 1999)... 

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