Mean-field nature

About 30 years after Buback and Franck s pioneering study on the critical point of molten NH4CI and 10 years after Singh and Pitzer s report on a mean-field nature of the liquid-liquid critical point in an electrolyte solution,... 

We also note that the stiffness of our chains is lower by a factor of 15 compared to the theoretically predicted region. Some numerical discrepancy was expected due to the mean-field nature of the theory. May be, however, the theory also underestimates the stiffening the chains undergo upon ordering. When we measure the persistence length in the simulation defined by the decay of orientational correlation between... 

What are the limits of the approximated expression Eq. 4 Mainly those due to the mean-field nature of PB. For, say, 99 % of the studied systems, the ions are monovalent, ion-ion correlations in water can be safely ignored, and the standard expression is valid. This is no more the case in presence of multivalent counterions (or monovalent ions in solvent of low e). That opens to the fascinating concept of electrostatic attraction between hke-charged colloids, subject of numerous false analyses, debates, and controversies in the literature for 30 years. Figure 1 presents Monte Carlo (MC) simulations data for the force vs. separation law within the primitive model (two latex colloids and ions in continuous solvent) in presence of counterions of increasing valence. While the PB/DLVO prediction remains everywhere repulsive, the exact MC behavior deviates at intermediate separation and develops an attractive well deeper and deeper as the valence increases above 3. This non mean-field effect is due to the repulsions and correlations among the counterions localized in the intersticial region (discreteness of the condensed layer). The same type of colloidal attraction is responsible for a liquid-gas (concentrated solution-dilute solution) phase separation, observed... 

The mean-field nature of our entanglement model is perhaps its weakest point. It has long been recognized that mean-field theories are rather rough approximations near the critical points for systems with d <4 dimensions. For example, the experimentally measured exponent for the liquid-gas critical point is (J= 0.33 and not S = 0.5. Mean-field theories generally predict more abrupt transitions than are actually observed. The discrepency between mean-field theories and experiments stems from the absence of fluctuation phenomena which are often more important near the transition. 

By expressing the mean-field interaction of an electron at r with the N- 1 other electrons in temis of a probability density pyy r ) that is independent of the fact that another electron resides at r, the mean-field models ignore spatial correlations among the electrons. In reality, as shown in figure B3.T5 the conditional probability density for finding one ofA - 1 electrons at r, given that one electron is at r depends on r. The absence of a spatial correlation is a direct consequence of the spin-orbital product nature of the mean-field wavefiinctions... 

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]. 

We have seen that Lagrangian PDF methods allow us to express our closures in terms of SDEs for notional particles. Nevertheless, as discussed in detail in Chapter 7, these SDEs must be simulated numerically and are non-linear and coupled to the mean fields through the model coefficients. The numerical methods used to simulate the SDEs are statistical in nature (i.e., Monte-Carlo simulations). The results will thus be subject to statistical error, the magnitude of which depends on the sample size, and deterministic error or bias (Xu and Pope 1999). The purpose of this section is to present a brief introduction to the problem of particle-field estimation. A more detailed description of the statistical error and bias associated with particular simulation codes is presented in Chapter 7. 

It should be noted that the results of VCS and BHF calculations using the same NN interaction disagree in several aspects. For instance for SNM the VCS and BHF calculations saturate at different values of density [5]. As for the SE using only two-body interactions the VCS approach yields a smaller value for 04 than BHF (see Table 1), and as a function of density in VCS the SE levels off at p 0.6 fm-3, whereas the BHF result continues to increase. Therefore it seems natural to ask whether the inclusion of more correlations by extending the BHF method (which is basically a mean field approximation) will lead to results closer to those of VCS. 

The above observation suggests an intriguing relationship between a bulk property of infinite nuclear matter and a surface property of finite systems. Here we want to point out that this correlation can be understood naturally in terms of the Landau-Migdal approach. To this end we consider a simple mean-field model (see, e.g., ref.[16]) with the Hamiltonian consisting of the single-particle mean field part Hq and the residual particle-hole interaction Hph-... 

To summarize, it has been found that the SH method is able to at least qualitatively describe the complex photoinduced electronic and vibrational relaxation dynamics exhibited by the model problems under consideration. The overall quality of SH calculations is typically somewhat better than the quality of the mean-field trajectory results. In particular, this holds in the case of several curve crossings (see Fig. 2) as well as when the dynamics and the observables of interest are essentially of adiabatic nature— for example, for the calculation of the adiabatic population dynamics associated with a conical intersection (see Figs. 3 and 12). Furthermore, we have briefly discussed various consistency problems of a simple quasi-classical SH description. It has been shown that binned electronic population probabilities and no momentum adjustment for classically forbidden transitions help us to improve this matter. There have been numerous suggestions to further improve the hopping algorithm [70-74] however, the performance of all these variants seems to depend largely on the problem under consideration. 

The mapping approach outlined above has been designed to furnish a well-defined classical limit of nonadiabatic quantum dynamics. The formalism applies in the same way at the quantum-mechanical, semiclassical (see Section VIII), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasi-classical approximations are needed to get from one level to another. Most of the established mixed quantum-classical methods such as the mean-field-trajectory method or the surface-hopping approach do invoke additional assumptions. The comparison of the mapping approach to these formulations may therefore (i) provide insight into the nature of these additional approximation and (ii) indicate whether the conceptual virtues of the mapping approach may be expected to result in practical advantages. 

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. 

The second question concerns one particular aspect of general applicability of the simple mean field equations outlined above as opposed to more sophisticated statistical mechanical descriptions. In particular, the equilibrium Poisson-Boltzmann equation (1.24) is often used in treatments of some very short-scale phenomena, e.g., in the theory of polyelectrolytes, with a typical length scale below a few tens of angstroms (1A = 10-8 cm). On the other hand, the Poisson-Boltzmann equation implicitly relies on the assumption of a pointlike ion. Thus a natural question to ask is whether (1.24) could be generalized in a simple manner so as to account for a finite ionic size. The answer to this question is positive, with several mean field modifications of the Poisson-Boltzmann equation to be found in [5], [6] and references therein. Another ultimately simple naive recipe is outlined below. 

The only problem necessary for developing the condensation theory is to add to the above-mentioned equation of the state the equation defining the function x(r)- Unfortunately, it turns out that the exact equation for the joint correlation function, derived by means of basic equations of statistical physics, contains f/iree-particle correlation function x 3), which relates the correlations of the density fluctuations in three points of the reaction volume. The equation for this three-particle correlations contains four-particle correlation functions and so on, and so on [9], This situation is quite understandable, since the use of the joint correlation functions only for description of the fluctuation spectrum of a system is obviously not complete. At the same time, it is quite natural to take into account the density fluctuations in some approximate way, e.g., treating correlation functions in a spirit of the mean-field theory (i.e., assuming, in particular, that three-particle correlations could be expanded in two-particle ones). 

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