Localized mean-field theory

They state that there is scope for improvement to this method by using a more sophisticated model such as the non-local mean field theory [W2] or molecular simulation [103]. 

FIG. 9 Changes of the monolayer film critical temperature with the concentration of impurities obtained from the Monte Carlo simulations (open circles) and resulting from the mean field theory (solid line). (Reprinted from A. Patrykiejew. Monte Carlo studies of adsorption. II Localized monolayers on randomly heterogeneous surfaces. Thin Solid Films, 205 189-196, with permision from Elsevier Science.)... 

To conclude, we can draw an analogy between our transition and Anderson s transition to localization the role of extended states is played here by our coherent radiant states. A major difference of our model is that we have long-range interactions (retarded interactions), which make a mean-field theory well suited for the study of coherent radiant states, while for short-range 2D Coulombic interactions mean-field theory has many drawbacks, as will be discussed in Section IV.B. Another point concerns the geometry of our model. The very same analysis applies to ID systems however, the radiative width (A/a)y0 of a ID lattice is too small to be observed in practical experiments. In a 3D lattice no emission can take place, since the photon is always reabsorbed. The 3D polariton picture has then to be used to calculate the dielectric permittivity of the disordered crystal see Section IV.B. 

Because of its diatomic nature and permanent quadrupole moment, the physisorp-tion of nitrogen at 77 K presents special problems. The application of DFT is facilitated if the molecules are assumed to be spherical, which was the approach originally adopted by Seaton et al. (1989) and also by Lastoskie et al. (1993). The analytical procedures already outlined in Chapter 7 (Section 7.6) do not depend on the meniscus curvature and are in principle applicable to both capillary condensation and micropore filling. The non-local version of the mean field theory (NLDFT), which was used by Lastoskie, gave excellent agreement with computer simulation when applied to the carbon slit pore model. However, as pointed out earlier, these computational procedures are not entirely independent since they involve the same model parameters. 

The variation of 5(7) near the N-I phase transition will be measured in this experiment and will be compared with the behavior predicted by Landau theory, " " which is a variant of the mean-field theory first introduced for magnetic order-disorder systems. In this theory, local variations in the environment of each molecule are ignored and interactions with neighbors are represented by an average. This type of theory for order-disorder phase transitions is a very useful approximate treatment that retains the essential features of the transition behavior. Its simplicity arises from the suppression of many complex details that make the statistical mechanical solution of 3-D order-disorder problems impossible to solve exactly. 

It is believed that electron correlation plays an important role with the anomalously high resistivity exhibited in marginal metals. Unfortunately, although the Mott-Hubbard model adequately explains behavior on the insulating side of the M-NM transition, on the metallic side, it does so only if the system is far from the transition. Electron dynamics of systems in which U is only slightly less than W (i.e. metallic systems close to the M-NM transition), are not well described by a simple itinerant or localized picture. The study of systems with almost localized electrons is still an area under intense investigation within the condensed matter physics community. A dynamical mean field theory (DMFT) has been developed for the Hubbard model, which enables one to describe both the insulating state and the metallic state, at least for weak correlation. 

It has been pointed out that any relationship between the exchange integral and the Weiss field is only valid at 0 K, since the former considers magnetic coupling in a pair-wise manner and the latter results from a mean-field theory (Goodenough, 1966). Finally, it is also essential to understand that Eq. 8.43 is strictly valid only for localized moments (in the context of the Heitler-London model). One might wonder then whether the Weiss model is applicable to the ferromagnetic metals, in which the electrons are in delocalized Bloch states, for example, Fe, Co, and Ni. This will be taken up later. 

With realizations of the Vycor glass as shown in Fig. 1, mean field theory applied to the lattice model (1) provides a simple and sufficiently realistic method to examine fluid adsorption behavior on a coarse-grained level. In particular, the local density on each site pi = (riitj) is self-consistently determined by... 

The LDA-fU theory may be regarded as an approximate GW method [37]. The screened Coulomb and exchange parameters U and J are usually estimated in a supercell approximation [39]. However, there is some arbitrariness in the choice of the localized orbitals when performing the partitioning of the Hamiltonian. A further step in the improvement of LDA-I-U consists in adding dynamical effects — frequency dependence in H r,r u). This may be performed using a DMFT-type approach (DMFT= Dynamical Mean Field Theory) [40] as part of the so-called LDA-b-1- approaches [41]. 

Finally a method which shows promise for the future is d5mamic mean field theory. Dynamical mean field theory uses an approximation to the local spectral density functional (rather than energy density functional) and a set of correlated local orbitals. For solids this local description is combined with a periodic description such as DFT using EDA to provide a method of dealing with both localised and delocalised electrons." Anisimov et applied this method to the photo-... 

Van der Waals consciously omitted contributions of the profile shape to the inter-facial excess entropy. In other words, at each position in the transition layer the local entropy is only determined by the local density, p z) On the other hand, the total Helmholtz energy is considered to depend both on P lz) and on the profile p (z) over the entire transition range, see later in this subsection. All of this is in line with the assumptions made in mean field theories for low-molecular mass molecules. ... 

It is important to note that the treatment of different localized mean fields around different parts of a molecule does not break the molecule into pieces. That is, the connectivity of the molecule is preserved in this IMF theory thus, this theory can treat molecules of any length (including high polymers). 

The mean field Cahn-Hilliard approach (Eq. 7) describes the intrinsic profile ( >(z) about the internal interface between two coexisting phases. It involves only one dimension, i.e., depth z, as a lateral homogeneity is assumed [7]. Capillary wave excitations may however cause lateral fluctuations of the depth Ie(x,y) at which the internal interface is locally positioned. As a result the effective interfacial width may be broadened beyond its intrinsic value (Eqs. 10 and 12). The mean field theory predicts the temperature dependence of the intrinsic width in a good agreement with experimental data presented here and reported by others (e.g., [76,89] reanalyzed by [88] or [96,129]). Some other experimental results [95,97,98] indicate the width larger than its intrinsic value... 

This argiimcntation is further supported by the predictions of a simple mean-field theory of the ferroelectric transition, which was originally presented in Ref. 257. Within this theory, we neglect any stratification (i.e., inhomogeneities of the local density) as well as any oscillations in the order parameter (which are indeed observed in the computer simulations). We also neglect nontrivial interparticle correlations. Our system can then be viewed as a system composed of N uncorrelated dipolar particles individually interacting with the mean field... 

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