Mean-field solutions

Figure 14. Various results for the mean-field solution, Eqs. (160)-(162), of a dipole in a field that is ramped from Hi = 0 to Hf = (a) Fields mx s) and 1x s) at the ramping speed r = 1. Curves...

In this Section we introduce a stochastic alternative model for surface reactions. As an application we will focus on the formation of NH3 which is described below, equations (9.1.72) to (9.1.76). It is expected that these stochastic systems are well-suited for the description via master equations using the Markovian behaviour of the systems under study. In such a representation an infinite set of master equations for the distribution functions describing the state of the surface and of pairs of surface sites (and so on) arises. As it was told earlier, this set cannot be solved analytically and must be truncated at a certain level. The resulting equations can be solved exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. This procedure is well-suited for the description of surface reaction systems which includes such elementary steps as adsorption, diffusion, reaction and desorption.The numerical part needs only a very small amount of computer time compared to MC or CA simulations. 

Ellis et al. [30] discussed the quantum fluctuations about the mean-field solution that would correspond in field theory to quantum fluctuations in the lightcone and could be induced by higher-genus effects in the string approach. Such effects would result in stochastic fluctuations in the velocity of light as of the order of... 

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

FIG. 17. Comparison between Monte Carlo simulation (symbols) with decamers and the mean-field solution (lines) of the simplified model with the harmonic grafting potential. The net osmotic pressure is given as a function of separation for various value of surface o and y. solid line and circles y —1.0, one charge per 50.77 A2. Dotted line and squares y = 1.0, one charge per 101.54 A2. Dashed line and diamonds y = 0.875, one charge per 50.77 A2 [61]. 

P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell Unlv. Press, Ithaca, NY (1979). (Scaling description for polymers In good solvents, emphasizing the "blob" structure as compared to homogeneous "mean-field" solutions.)... 

Before we discuss the stability condition 0 on the mean field solution, we first describe the RPA formalism. 

Just as there exist the so-called Thouless stability conditions on the Hartree-Fock solutions in nuclear physics (Thouless, 1960, 1961 Rowe, 1970) and in quantum chemistry (CiZek and Paldus, 1971), one has stability conditions on the mean field solutions in lattice dynamics problems (Fredkin and Werthamer, 1965). The mean field solutions are obtained from the condition AA . = 0 (see Section IV,A). They are stable i.e., they correspond with a local minimum in the free energy if > 0. Substituting the mean field solution (109) into the equation (107) for AA ., the term with Apf vanishes and we can express the stability condition as... 

The problem with this localized, stable, mean field solution is that it has a much lower symmetry than the experimentally observed hexagonal symmetry of /3-nitrogen. We have conjectured that the higher symmetry is... 

Fig. 6. Free energy (at zero pressure) for a-nitrogen and /3-nitrogen, in different mean field models (closed lines). The dashed free rotor curve has been calculated from the isotropic (/, 12, ly) - (0,0, 0) term of the ab initio potential by adding the free rotor expression for the free energy. The dashed jump model curve has been obtained from the localized mean field solution (with the full anisotropic potential) by adding an entropy term -kBT In 6 (see the text).

Newman M. E. J., Moore C. and Watts D. J. (2000). Mean-field solution of the small-world network model. Physical Review Letters. 84, pp 3201-3204. 

From the rigorous treatment of the double-layer problem on the molecular level, it becomes clear that the Gouy-Chapman theory of the interface is equivalent to a mean field solution of a simple primitive model (PM) of electrolytes at the interface (6). To consider the correlation between ions, integral equations that describe the PM are devised and solved in different approximations. An exact solution of the PM of the electrolyte can be obtained from the computer simulations. This solution can be compared with the solutions obtained from different integral equations. For detailed discussion of this topic, refer to the review by Camie and Torrie (6). In many cases, the molecular description of the solvent must be introduced into the theory to explain the complexity of the observed phenomena. The analytical treatment in such cases is very involved, but initial success has already been achieved. Some of the theoretical developments along these lines were reviewed by Blum (7). 

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

If the noise term is turned off, the system is driven towards the nearest saddle point. Therefore, the same set of equations can be used to find and test mean-field solutions. The complex Langevin method was first applied to dense melts of copolymers [74], and later to mixtures of homopolymers and copolymers [80] and to diluted polymers confined in a slit under good solvent conditions [77]. Figure 2 shows examples of average density configurations (p ) for a ternary block copolymer/homopolymer system above and below the order/disorder transition. 

The minimum value in EKs p(r)] corresponds to the exact ground-state dectron density. To determine the actual energy, variations in E p(r)] must be optimized with respect to variations in p, subject to the orthonormality constraints. In KS-DFT, an arriiidal reference system of noninteracting dectrons is constracted, which has exactly the same electron density as the real molecular system. Therefore, from a computational viewpoint, the KS version of DFT leads to a mean-field solution, which is only an approximation. 

The scattered X-ray intensity I(q) can be estimated in the nematic phase from Gaussian fluctuations about the mean field solution =0 ... 

Within the mean-field approximation, the free energy per chain of the system is obtained by inserting the mean-field solution into the free-energy expression. 

The equilibrium structures predicted by SCMFT correspond to the solutions obtained at the extrema of the free-energy functional of the system. These solutions do not necessarily ensure the minimization of the free-energy functional. The mean-field solution may, for example, correspond to a saddle point. In order to investigate the stability of the ordered phases, we have to consider the effect of the higher-order contributions to the free-energy functional. In particular, the Gaussian fluctuation contributions derived above can be used to predict the stability of any ordered structure. In what follows we formulate the theory of Gaussian fluctuations in ordered phases [15,27,31,32]. 

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