Mean field analysis

As for theoretical research on the chiral symmetry breaking, Frank was the first to show that a linear autocatalysis with an antagonistic nonlinear chemical reaction can lead to homochirality [16]. His formulation with rate equations corresponds to the mean-field analysis of the phase transition in a nonequilibrium situation [17], and other variants have been proposed [6, 18-23]. All these analyses have been carried out only for open systems where... 

On the basis of what has been discussed, we are in the position to provide a unified understanding and approach to the composite elastic modulus, yield stress, and stress-strain curve of polymers dispersed with particles in uniaxial compression. The interaction between filler particles is treated by a mean field analysis, and the system as a whole is macroscopically homogeneous. Effective Young s modulus (JE0) of the composite is given by [44]... 

Dyson obtained a condition for the sustainment of catalytic activities in these collections of molecules, by taking an abstract model. For simplicity he classified molecules into two states depending on whether they have catalytic activity or not. Furthermore, he assumed that the ratio of the synthesis of catalytic molecules is amplified as the fraction of catalytic molecules is larger that is, a positive feedback process is assumed. This model is mapped to a kind of Ising model. With the aid of mean-field analysis in statistical physics, he showed that the catalytic activities can be sustained depending on the number of molecules and their species. Although his model is abstract, the result he obtained probably can be applied to any system with a set of catalytic molecules, be it protein, lipids, or other polymers. 

In the mean-field analysis the free energy of the Nem phase is estimated as... 

Parameters derived by means of mean field analysis of experimental data 5 3j represents the 3d spin while the exchange constants J 2 Jj2 refer to the... 

The above mentioned collapse on the scaling plot can be used to evaluate the validity of the mean-field analysis. In this sense the method is self-consistent only if has been properly... 

For an open system, described by (13), any field-induced density depletion, dN/dEo) yx < 0, could only be expected in case of dramatic rise in orientational polarizability of the molecules upon dilution. A mean-field analysis [69, 82] of a water-mimicking Ising model in electric field explored the assumption that molecular dipole alignment perturbs hydrogen bonding. Over an interval of intermediate field strengths, the model-system featured a density drop akin to the... 

The EA model contains the two essential ingredients for a spin glass - disorder and competition as discussed above - but it is a crude approximation to real spin glasses. Therefore at the beginning we have to discuss how well this model reproduces spin-glass properties. Up to now there is no analytic solution available for the short-range EA model. Even modern methods like real space renormalization yield inconclusive results (Kinzel and Fischer 1978, Tatsumi 1978). Hence numerical methods like the Monte Carlo simulation have been applied to answer this question the results of which will be briefly summarized now. Then the mean-field analysis of the EA model will be discussed, and finally the question... 

The infinite-range model of Sherrington and Kirkpatrick (SK) (1975) was originally introduced as a model for which the mean field analysis of EA would be exact. Here, each spin couples equally with every other spin in the system. One takes in eq. 10 the mean [7 ] , = Jg/N and the variance AJ = J/VN in order to get a non-trivial thermodynamic limit (N is the number of spins in the system). 

A configuration snapshot for penetrable ions in Fig. 11 gives impression that counterions are stacked. The impression is further supported by the presence of a peak around r = 0 in the correlation function. The presence of stacked formations is, furthermore, linked to the presence of instability within the Kirkwood (mean field) analysis [11, 60-63], Using Fourier transformed Eq. (15) of the mean-field analysis we find... 

The mean field analysis of the Frost s model of frustrated smectics describes most of the reported experimental observations on polar smectics. Situations where fluctuations are important are however not correctly described the understanding of multiply re-entrant behavior, the appearance of nematic bubbles and the scaling properties of the SmA2-SmAcritical point for instance require more elaborate analysis. Some of these points will be discussed in the following sections. 

T. McPherson, A. Kidane, I. Szleifer, K. Park, Fhevention of protein adsorption by tethered poly (ethylene oxide) layers Experiments and single-chain mean-field analysis, Langmuir, 14 (1998) 176-186. 

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