Mean-field approach

There has been considerable interest in the simulation of lipid bilayers due to their biological importance. Early calculations on amphiphilic assemblies were limited by the computing power available, and so relatively simple models were employed. One of the most important of these is the mean field approach of Marcelja [Marcelja 1973, 1974], in which the interaction of a single hydrocarbon chain with its neighbours is represented by two additional contributions to the energy function. The energy of a chain in the mean field is given by ... 

A recent survey analyzed the accuracy of tliree different side chain prediction methods [134]. These methods were tested by predicting side chain conformations on nearnative protein backbones with <4 A RMSD to the native structures. The tliree methods included the packing of backbone-dependent rotamers [129], the self-consistent mean-field approach to positioning rotamers based on their van der Waals interactions [145],... 

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. 

In this part we consider a mean-field approach restricted to the case h = 0 and b = 0. Then Eq. (10) leads to [27]... 

As explained above, despite the fact that the interactions do not depend on the concentration, the diffuse intensity maps of Ala V and AtgC are very different. This behavior is in total contradiction with a mean field approach According to the Krivoglaz-Clapp-Moss formula", a minimum of V (c/) corresponds of a maximum of topology changes observed in the diffuse intensity maps. 

Although differences between mean-field estimates and exact limiting densities are to be expected because of the presence of correlations in actual evolutions, the mean-field approach nonetheless typically comes to within about 10 — 20% of the exact values for all elementary complex rules. 

The simplest approximation to make is simply that the initial distribution of live" sites is completely random and that any site-site correlations are negligible i.e. we first take a conventional Mean-Field approach (see section 7.4). In this case, the equilibrium density can be written down almost by inspection. The probability of a site having value 1 (= p) is equal to the probability that it had value 1 on the previous time step multiplied by the probability that it stays equal to 1 (i.e. the probability that a site has either 2 or 3 live neighboring sites) plus the probability that the site was previously equal to 0 multiplied by the probability that it become 1 (i.e. that it is surrounded by exactly 3 live sites). Letting p and p represent the density at times t and t + 1, respectively, simple counting yields ... 

Equations 22.3-22.14 represent the simplest formulation of filled phantom polymer networks. Clearly, specific features of the fractal filler structures of carbon black, etc., are totally neglected. However, the model uses chain variables R(i) directly. It assumes the chains are Gaussian the cross-links and filler particles are placed in position randomly and instantaneously and are thereafter permanent. Additionally, constraints arising from entanglements and packing effects can be introduced using the mean field approach of harmonic tube constraints [15]. 

The bundle model of polymer crystallization will be discussed first. This mean-field approach describes the metastable configurational equilibrium of the undercooled polymer chain in solution or in the melt, and we will summarize and update previous work of ours [7-9]. The concept of the bundle, i.e. an aggregate of a few parallel polymer segments or stems connected by folds (see Fig. 1 see also [10]) and stabilized by attractive crystal-like interac-... 

Green function method, that can be considered as a generalization of the BHF approach. The results are compared with those from various many-body approaches, such as variational and relativistic mean field approaches. In view of the large spread in the theoretical predictions we also examine possible constraints on the nuclear SE that may be obtained from information from finite nuclei (such the neutron skin). 

In the following the present status of calculations of the SE is reviewed first the ones in which a microscopic nucleon-nucleon (NN) interaction is used, followed by the phenomenological mean field approaches. In section 4 we discuss possible constraints that can be obtained from empirical information. 

Shao, B., Walet, N., and Amado, R. D. (1992), Mean Field Approach to the Algebraic Treatment of Molecules Linear Molecules, Phys. Rev. A 46, 4037. 

Figure 9 shows the result of a model calculation (using a mean-field approach) for a delocalized complex with 0 = 0.60, in which there is equal coupling to each of four modes Uh (0 =... 

There exists, however, one special group of branched structures that, in spite of an incisive constraint imposed onto the reaction, can be well described by a mean field approach. This case occurs when a monomer bears two types of functional groups [1,16,51], say A and B, where only group A can react with one of the (f-1) B groups of another monomer unit. Figure 5 shows as examples two cases of an AB2 polycondensation. 

The percolation simulations clearly allow ring formation it also represents self-avoiding statistics, i.e., includes the excluded volume effects in good solvents. Finally, the probability of placing units on lattice sites becomes more and more dependent on whether a site in the neighborhood is already occupied. In other words the percolation experiment becomes a non-mean field approach when the occupation reaches the critical percolation threshold. Therefore, strong deviations were expected between the more accurate percolation and the Flory-Stockmayer mean field approaches. Physicists were of the opinion that the mean field results must be basically wrong. 

Furthermore, a related and common criticism of the MFT method is that a mean-field approach cannot correctly describe the branching of wave packets at crossings of electronic states [67, 70, 82]. This is true for a single mean-field trajectory, but is not true for an ensemble of trajectories. In this context it may be stressed that an individual trajectory of an ensemble does not even possess a physical meaning—only the average does. 

It is probable that numerous interfacial parameters are involved (surface tension, spontaneous curvature, Gibbs elasticity, surface forces) and differ from one system to the other, according the nature of the surfactants and of the dispersed phase. Only systematic measurements of > will allow going beyond empirics. Besides the numerous fundamental questions, it is also necessary to measure practical reason, which is predicting the emulsion lifetime. This remains a serious challenge for anyone working in the field of emulsions because of the polydisperse and complex evolution of the droplet size distribution. Finally, it is clear that the mean-field approaches adopted to measure > are acceptable as long as the droplet polydispersity remains quite low (P < 50%) and that more elaborate models are required for very polydisperse systems to account for the spatial fiuctuations in the droplet distribution. 

There are many sources of this paradoxical situation, in which a theoretical understanding lags far behind experiment in such a practically relevant area as electro-diffusion. There was a period of intense qualitative development in this area in the 1920s until the early 1950s when the modern classics of chemical physics developed the theory of electrolytic conductance and related phenomena [11]—[13]. These works were mainly concerned with the mean field approach to microscopic mechanisms determining such properties of electrolyte solutions as ion diffusivity, dielectric susceptibility, etc. in particular, they were concerned with the effects of an externally applied stationary and alternating electric field upon the above properties... 

Figure 27 presents values of W(T) calculated by Dietl et al. (2001a) in comparison to the experimental data of Shono et al. (2000). Furthermore, in order to establish the sensitivity of the theoretical results to the parameter values, the results calculated for a value of Xc that is 1.8 times larger are included as well. The computed value for low temperatures, W = 1.1 /xm, compares favorably with the experimental finding, W = 1.5 fim. However, the model predicts a much weaker temperature dependence of IV than observed experimentally, which Dietl et al. (2001a) link to critical fluctuations, disregarded in the mean-field approach. 

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