Mean-field-theory approach

In this section we briefly outline a general mean-field theory approach to arbitrary PCA and then apply the formalism to a particular class of one-parameter rules. We then compare the theoretical predictions to numerical simulations on lattices of dimension 1 < d < 4. 

Helfand (25,26,27,28,29) has formulated a statistical thermodynamic model of the microphases similar to that of Meier. This treatment, however, requires no adjustable parameters. Using the so-called mean-field-theory approach, the necessary statistics of the molecules are embodied in the solutions of modified diffusion equations. The constraint at the boundary was achieved by a narrow interface approximation which is accomplished mathematically by applying reflection boundary conditions. 

This implies that the critical exponent y = 1, whether the critical temperature is approached from above or below, but the amplitudes are different by a factor of 2, as seen in our earlier discussion of mean-field theory. The critical exponents are the classical values a = 0, p = 1/2, 5 = 3 and y = 1. 

For general rules, a first-order statistical approximation for limiting densities Pi t —> oo) can be obtained by a method akin to the mean-field theory in statistical mechanics (more sophisticated approaches will be introduced in chapter 4). 

Analytical approaches to understanding the effect of molecular flexibility on orientational order have concentrated on both the isotropic-nematic and the nematic-smectic transition [61, 62] and mean field theory has shown that cholesteric pitch appears not to depend on the flexibility of the molecule [63]. 

Prausnitz and coworkers [91,92] developed a model which accounts for nonideal entropic effects by deriving a partition function based on a lattice model with three categories of interaction sites hydrogen bond donors, hydrogen bond acceptors, and dispersion force contact sites. A different approach was taken by Marchetti et al. [93,94] and others [95-98], who developed a mean field theory... 

The most essential step in a mean-field theory is the reduction of the many-body problem to a scheme that treats just a small number of molecules in an external field. The external field is chosen such that it mimics the effect of the other molecules in the system as accurately as possible. In this review we will discuss the Bragg Williams approach. Here the problem is reduced to behaviour of a single chain (molecule) in an external field. Higher order models (e.g. Quasi-chemical or Bethe approximations) are possible but we do not know applications of this for bilayer membranes. 

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. 

Semianalytical mean-field theories of block copolymer micellization were formulated by Noolandi et al. [ 197] and by Leibler et al. [198]. In the approach of Noolandi et al., the micellar characteristics were obtained through a minimization of the Gibbs free energy for an isolated micelle. This was applied to PS-PB micelles, and the obtained theoretical values were in good agreement with the experimental ones. 

If we understand FM or magnetic properties of quark matter more deeply, we must proceeds to a self-consistent approach, like Hartree-Fock theory, beyond the previous perturbative argument. In ref. [11] we have described how the axial-vector mean field (AV) and the tensor one appear as a consequence of the Fierz transformation within the relativistic mean-field theory for nuclear matter, which is one of the nonperturbative frameworks in many-body theories and corresponds to the Hatree-Fock approximation. We also demonstrated... 

The weakness of this boundary condition is being able to justify a large enough distance to be comparable with an infinite distance, or perhaps 1000R. In practice, this would require B to be in excess over A by about 109 times A more reasonable approach to this outer boundary condition would be to require that there be no loss or gain of matter over this boundary, as there is an approximately equal tendency for the B reactants to migrate towards either A reactant upon each side of the boundary. The proper incorporation of this type of boundary condition into the Smoluchowski model leads to the mean field theory of Felderhof and Deutch [25] and is discussed further in Chap. 8 Sect. 2.3 and Chap. 9 Sect 5. 

Experiments and modem physics [3] has shown that the way KT and other thermodynamic properties diverge when approaching the critical point is described in a fundamentally wrong way by all classical, analytical equations of state like the cubic equations of state and is path dependent. The reason for this is that these equations of state are based on mean field theory,... 

There is a substantial body of theoretical work on micellization in block copolymers. The simplest approaches are the scaling theories, which account quite successfully for the scaling of block copolymer dimensions with length of the constituent blocks. Rather detailed mean field theories have also been developed, of which the most advanced at present is the self-consistent field theory, in its lattice and continuum guises. These theories are reviewed in depth in Chapter 3. A limited amount of work has been performed on the kinetics of micellization, although this is largely an unexplored field. Micelle formation at the liquid-air interface has been investigated experimentally, and a number of types of surface micelles have been identified. In addition, adsorption of block copolymers at liquid interfaces has attracted considerable attention. This work is also summarized in Chapter 3. 

From the theoretical viewpoint, much of the phase behaviour of blends containing block copolymers has been anticipated or accounted for. The primary approaches consist of theories based on polymer brushes (in this case block copolymer chains segregated to an interface), Flory-Huggins or random phase approximation mean field theories and the self-consistent mean field theory. The latter has an unsurpassed predictive capability but requires intensive numerical computations, and does not lead itself to intuitive relationships such as scaling laws. 

There have been few attempts to generalize mean-field theories to the unrestricted case. Netz and Orland [227] applied their field-theoretical model to the UPM. Because such lattice theories yield quite different critical properties from those of continuum theories, comparison of their results with other data is difficult. Outhwaite and coworkers [204-206] considered a modification of their PB approach to treat the UPM. Their theory was applied to a few conditions of moderate charge and size asymmetry. 

The TDLDA approach used here for describing electronic dynamics is basically an effective mean-field theory. In particular, the possibly crucial dynamical correlations are missing. While the latters are expected to play a minor role at low excitations, they may become dominant in far from... 

Recently, the stiff-chain polyelectrolytes termed PPP-1 (Schemel) and PPP-2 (Scheme2) have been the subject of a number of investigations that are reviewed in this chapter. The central question to be discussed here is the correlation of the counterions with the highly charged macroion. These correlations can be detected directly by experiments that probe the activity of the counterions and their spatial distribution around the macroion. Due to the cylindrical symmetry and the well-defined conformation these polyelectrolytes present the most simple system for which the correlation of the counterions to the macroion can be treated by analytical approaches. As a consequence, a comparison of theoretical predictions with experimental results obtained in solution will provide a stringent test of our current model of polyelectrolytes. Moreover, the results obtained on PPP-1 and PPP-2 allow a refined discussion of the concept of counterion condensation introduced more than thirty years ago by Manning and Oosawa [22, 23]. In particular, we can compare the predictions of the Poisson-Boltzmann mean-field theory applied to the cylindrical cell model and the results of Molecular dynamics (MD) simulations of the cell model obtained within the restricted primitive model (RPM) of electrolytes very accurately with experimental data. This allows an estimate when and in which frame this simple theory is applicable, and in which directions the theory needs to be improved. 

Abstract In this chapter we review recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions. We will discuss an improved density functional approach to go beyond mean-field theory for the cell model and an integral equation approach to describe stiff and flexible polyelectrolytes in good solvents and compare some of the results to computer simulations. Then we review some recent theoretical and numerical advances in the theory of poor solvent polyelectrolytes. At the end we show how to describe annealed polyelectrolytes in the bulk and discuss their adsorption properties. 

Mean-field theory can be used to predict the effects of mutation rate and parent fitness on the moments of the mutant fitness distribution (Voigt et al, 2000a). In this analysis, only the portion of the mutant distribution that is not dead (zero fitness) or parent (unmutated) is considered. The mutant effects are averaged over the transition probabilities without the cases of mutations to stop codons or when no mutations are made on a sequence. In order to obtain the fitness distribution, two probabilities are required (1) the probability pi(a) that a particular amino-acid identity a exists at a residue i, and (2) the transition probability that one amino acid will mutate into another Q = 1 — (1 — pm)3. The probability vectors p a) can be determined through a mean-held approach (Lee, 1994 Koehl and Delarue, 1996 Saven and Wolynes, 1997). The amino acid transition probabilities Q are calculated based on the special connectivity of the genetic code and the per-nucleotide mutation rate. Removing transitions to stop codons and unmutated sequences only requires the proper normalization of the probabilities pi and the moments. For example, the first moment of the fitness improvement w of the uncoupled fitness function is written as... 

This approach leads to values of y that depend on the solvent quality on the interface. For the case where the interface serves as a good solvent, the mean-field theory predicts v = 0.75 [31] while the numerical calculation by Le Guillou and Zinn-Justin yields v = 0.77 [32], leading to predictions of y = 3 and 2.86, respectively. For cases where the interface gives rise to the theta sol-... 

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