Generalized mean field approximation

For a given Hamiltonian the calculation of the partition function can be done exactly in only few cases (some of them will be presented below). In general the calculation requires a scheme of approximations. Mean-field approximation (MFA) is a very popular approximation based on the steepest descent method [17,22]. In this case it is assumed that the main contribution to Z is due to fields which are localized in a small region of the functional space. More crudely, for each kind of particle only one field is... 

In addition to the nearest-neighbor interaction, each ion experiences the electrostatic potential generated by the other ions. In the literature this has generally been equated with the macroscopic potential 0 calculated from the Poisson-Boltzmann equation. This corresponds to a mean-field approximation (vide infra), in which correlations between the ions are neglected. This approximation should be the better the low the concentrations of the ions. 

From this equation and Eqs. (2) and (4) the energy of the system can be obtained. The entropy is more difficult to derive, and we refer to the literature [4,6]. Generally, the quasichemical gives better results than the mean-field approximation, since it allows for local order. We note that for the three-dimensional lattice gas no exact analytical solution exists. 

The general expression for the BWG (mean field) approximation then gives... 

The Quasi-Chemical Approximation. The mean-field approximation ignores all correlation in the occupation of neighboring sites. This is incorrect when there is a strong interaction between adsorbates at such sites. The simplest way to include some correlation is to work with probabilities of occupations of two sites (XY) instead of one site (X). Approximations that do this are generally called pair approximations (not to be confused with pair interactions). There are more possibilities to reduce multi-site probabilities as in eqn. (8) to 2-site probabilities than to 1-site probabilities. This leads to different types of pair approximations. The best-known approximation that is used for Ising models is the Kirkwood approximation, which uses for example ... 

In principle, an analytic matrix inversion needed in Eqs. (C.42), (C.45) and (C.46) is feasible but tedious, but in general it does not give very lucid equations, while the numerical matrix inversion is readily carried out by computers. In two cases, however, the analytic inversion is worth carrying out since the resultant equations show clearly the contrast with a mean-field approximation. 

A mean-field estimate of this probability can be made for the general case of an ideal chain in cf-dimensional space by replacing a chain with an ideal gas of N monomers in the pervaded volume of a coil R. The probability of a given monomer to contact any other monomer within this mean-field approximation is simply the overlap volume fraction (j>, of a chain inside its pervaded volume, determined as the product of the monomer volume and the number density of monomers in the pervaded volume of the coil NjR ... 

In general, a mean-field approximation is defined by any set of occupation numbers rt/t by means of a corresponding Fock operator matrix element, and the dependence of the results on the specific set of occupation numbers turned out to be very weak in practical calculations. This approximation has also been developed independently by Beming etal (2000). 

Doi first proposed the generalized dynamic equations for the concentrated solution of rod-like polymers. Such constitutive equations can be derived from the molecular theory developed by Doi and Edwards (1986). The basis for the molecular theory is the Smoluchowski equation or Fokker-Planck equation in thermodynamics with the mean field approximation of molecular interaction. 

In general, the systems that have a tree structure, i.e. without cycle, can easily be studied, and in particular, all mean field approximations are obtained by replacing the initial system by a system with such a structure. 

Generalizations [71] of the bare anisotropic-planar-rotor model (2.5) include other multipolar interactions such as dipolar and octopolar terms with and without in-plane crystal-field modulations. Several such combinations were analyzed in the mean-field approximation, Landau theory, and spin-" wave expansion [71]. The quadrapole-quadrupole model written in the form... 

A polymer chain at an interface is subject to a potential t/(r), which in general is composed of two components. There is an external component that arises directly from the effects of the wall. To this must be added an internal potential which reflects the mean field experienced by one segment due to the physical presence of all other segments. In the mean field approximation, this latter potential is simply proportional to the segment concentration c(r)... 

Cotter has examined the postulates underlying the mean field approximation in the light of Widom s analysis of this general problem and has concluded that thermodynamic consistency requires that u should be proportional to V regardless of the nature of the intermolecular pair potential. However, in what follows we have assumed a dependence as in the original formulation of the theory by Maier and Saupe. 

---