Mean-field theoretical model

The mean-field theoretical model developed by Hao and Clem allows the anisotropic superconducting state thermodynamic parameters to be determined from measurements of M(H,T) outside of the linear Abrikosov regime (Hao et al. 1991) their model reduces to the Abrikosov linear region only in the vicinity of the transition. However, the determination of the mean-field upper critical field curve is fiirther complicated... 

In this section, theoretical foundations and application of the mean-field Zener model to III-V magnetic semiconductors are discussed in some detail. The capabilities of the model to describe various magnetic properties of (Ga,Mn)As are presented, too. In the final part, limitations of the model and its numerous refinements put recently forward are discussed. 

Therefore the model avoids two main difficulties the large amount of computer time which is normally needed for simulations and the loss of structural information which occurs in simple theoretical models (mean-field models) which do not take into account the structural aspects of the adsorbate layer. Mean-field-kind models fail in the prediction of phase transitions of the second order because at these points the long-range correlations appear. They also fail in describing the system s behaviour in the neighbourhood of the point of first-order kinetic phase transition. 

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. 

Feng, E.H., and Fredrickson, G.H. "Confinement of equilibrium polymers a field-theoretic model and mean-field solution". Macromolecules 39, 2364-2372 (2006). 

Feng EH, Lee WB, Fredrickson GH (2007) Supramolecular diblock copolymers a field-theoretic model and mean-field solution. Macromolecules 40(3) 693-702... 

Both the statics and the collective dynamics of composition fluctuations can be described by these methods, and one can expect these schemes to capture the essential features of fluctuation effects of the field theoretical model for dense polymer blends. The pronounced effects of composition fluctuations have been illustrated by studying the formation of a microemulsion [80]. Other situations where composition fluctuations are very important and where we expect that these methods can make straightforward contributions to our understanding are, e.g., critical points of the demixing in a polymer blend, where one observes a crossover from mean field to Ising critical behavior [51,52], or random copolymers, where a fluctuation-induced microemulsion is observed [65] instead of macrophase separation which is predicted by mean-field theory [64]. 

Figure 26.1 7 (a) Theoretical predictions for the heat capacity C near the critical temperature, for two-dimensional systems. The Bragg-Williams mean-field lattice model of Chapter 25 leads to a triangular function, while the exact solution of the two-dimensional Ising model shows a sharp peak. Source R Kubo, in cooperation with H Ichimura, T Usui and N Hashitsome, Statistical Mechanics, Elsevier Pub. Co., New York (1 965). (b) Experimental data for helium on graphite closely resembles the Ising model prediction. Source RE Ecke and JC Dash, Phys Rev B 28, 3738 (1983). 

Often, field-theoretic models are considered within the mean field approximation. Provided that the coarse-grained parameters have been identified and describe local... 

Rod- and disc-shaped mesogens are likewise capable of forming uniaxial nematic phases N . In the early 1970s, Alben [20] published a theoretical calculation based on a mean-field lattice model in which he predicted the appearance of a biaxial nematic phase Ng for mixtures containing certain ratios between discs and rods. The phase diagram corresponding to his prediction is schematically shown in Figure 5-16. 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

A number of theoretical models have been proposed to describe the phase behavior of polymer—supercritical fluid systems, eg, the SAET and LEHB equations of state, and mean-field lattice gas models (67—69). Many examples of polymer—supercritical fluid systems are discussed ia the Hterature (1,3). 

P Koehl, M Delame. A self consistent mean field approach to simultaneous gap closure and side-chain positioning m protein homology modelling. Nature Struct Biol 2 163-170, 1995. R Samudrala, J Moult. A graph-theoretic algorithm for comparative modeling of protein structure. J Mol Biol 279 287-302, 1998. 

Several methods have been employed to study chemical reactions theoretically. Mean-field modeling using ordinary differential equations (ODE) is a widely used method [8]. Further extensions of the ODE framework to include diffusional terms are very useful and, e.g., have allowed one to describe spatio-temporal patterns in diffusion-reaction systems [9]. However, these methods are essentially limited because they always consider average environments of reactants and adsorption sites, ignoring stochastic fluctuations and correlations that naturally emerge in actual systems e.g., very recently by means of in situ STM measurements it has been demon-... 

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]. 

In order to understand the thermodynamic issues associated with the nanocomposite formation, Vaia et al. have applied the mean-field statistical lattice model and found that conclusions based on the mean field theory agreed nicely with the experimental results [12,13]. The entropy loss associated with confinement of a polymer melt is not prohibited to nanocomposite formation because an entropy gain associated with the layer separation balances the entropy loss of polymer intercalation, resulting in a net entropy change near to zero. Thus, from the theoretical model, the outcome of nanocomposite formation via polymer melt intercalation depends on energetic factors, which may be determined from the surface energies of the polymer and OMLF. 

Figure 4 Overview of several theoretical predictions for the SE Brueckner-Hartree-Fock (continuous choice) with Reid93 potential (circles), self-consistent Green function theory with Reid93 potential (full line), variational calculation from [9] with Argonne Avl4 potential (dashed line), DBHF calculation from [16] (triangles), relativistic mean-field model from [22] (squares), effective field theory from [23] (dash-dotted fine).

Subsequently Furnstahl [20] in a more extensive study pointed out that within the framework of mean field models (both non-relativistic Skyrme as well as relativistic models) there exists an almost linear empirical correlation between theoretical predictions for both 04 and its density dependence, po, and the neutron skin, All lln — Rp, in heavy nuclei. This is illustrated for 208Pb in Fig. 5 (from ref.[20] a similar correlation is found between All and po). Note that whereas the Skyrme results cover a wide range of All values the RMF predictions in general lead to AR > 0.20 fm. 

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