Mean-field approximations theory

The mean field approximate theories, ATA and CPA, are final simplifications of multiple scattering approaches to simulate a real disordered system by an equivalent effective system. 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

This profile of the phase boundary determined here looks very similar to that obtained by the mean-field approximation (19), but the result here only applies to the profile above the roughening temperature. Since this is a mean-field theory, fluctuations are also not considered correctly. 

Marcus theory, 31, 37, 38 Mean-field approximation, 61, 163 MetaUoporphyrin catalysts, 362-366, 637-686... 

It may seem that the prospeets are bleak for the GvdW approach to electrolytes but, in fact, the reverse is the ease. We need only follow Debye and Hiickel [18] into their analysis of the sereening meehanism, almost as successful as the van der Waals analysis of short-range fluids, to see that the mean-field approximation can be applied to the correlation mechanism with great advantage. In fact, we can then add finite ion size effects to the analysis and thereby unify these two most successful traditional theories. 

Certain difficulties remain, however, with this approach. First, such an important feature as a secondary structure did not find its place in this theory. Second, the techniques of sequence design ensuring exact reproduction of the given conformation are well developed only for lattice models of polymers. The existing techniques for continuum models are complex, intricate, and inefficient. Yet another aspect of the problem is the necessity of reaching in some cases beyond the mean field approximation. The first steps in this direction were made in paper [84], where an analog of the Ginzburg number for the theory of heteropolymers was established. 

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)... 

Figure 11. A schematic representation of the mean-field approximation, a central issue in the self-consistent-field theory. The arrows symbolically represent the lipid molecules. The head of the arrow is the hydrophilic part and the line is the hydrophobic tail. On the left a two-dimensional representation of a disordered bilayer is given. One of the lipids has been selected, as shown by the box around it. The same molecule is depicted on the right. The bilayer is depicted schematically by two horizontal lines. The centre of the bilayer is at z = 0. These lines are to guide the eye the membrane thickness is not preassumed, but is the result of the calculations. Both the potential energy felt by the head groups and that of the tail segments are indicated. We note that in the detailed models the self-consistent potential profiles are of course much more detailed than shown in this graph...

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

However, such a rigorous approximation need not be made in the FS theory as will now be shown with several examples. In fact, the mean-field approximation would lead in most cases to grossly incorrect results. 

The phase diagram for weakly segregated diblocks was first computed within the Landau mean field approximation by Leibler (1980). Because it has proved to be one of the most influential theories for microphase separation in block copolymers, an outline of its essential features is given here.The reader is referred to the original paper by Leibler (1980) for a complete account of the theory. 

A wide variety of theories have been developed for polymer solutions over the later half of the last century. Among them, lattice model is still a convenient starting point. The most widely used and best known is the Flory-Huggins lattice theory (Flory, 1941 Huggins, 1941) based on a mean-field approach. However, it is known that a mean-field approximation cannot correctly describe the coexistence curves near the critical point (Fisher, 1967 Heller, 1967 Sengers and Sengers, 1978). The lattice cluster theory (LCT) developed by Freed and coworkers (Freed, 1985 Pesci and Freed, 1989 Madden et al., 1990 Dudowicz and Freed, 1990 Dudowicz et al., 1990 Dudowicz and Freed, 1992) in 1990s was a landmark. 

PB equation is based on the mean-field approximation, where ions are treated as continuous fluid-like particles moving independendy in a mean electric potential. The theory ignores the discrete ion properties such as ion size, ion-ion correlation and ion fluctuations. Fail to consider these properties can cause inaccurate predictions for RNA folding, especially in the presence of multivalent ions which are prone to ion correlation due to the strong, long-range Coulomb interactions. For example, PB cannot predict the experimentally observed attractive force between DNA helices in multivalent ion solutions. 

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