Mean-field approximation various

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

In all of the discussion above, comparisons have been made between various types of approximations, with the nonlinear Poisson-Boltzmann equation providing the standard with which to judge their validity. However, as already noted, the nonlinear Poisson-Boltzmann equation itself entails numerous approximations. In the language of liquid state theory, the Poisson-Boltzmann equation is a mean-field approximation in which all correlation between point ions in solution is neglected, and indeed the Poisson-Boltzmann results for sphere-sphere [48] and plate-plate [8,49] interactions have been derived as limiting cases of more rigorous approaches. For many years, researchers have examined the accuracy of the Poisson-Boltzmann theory using statistical mechanical methods, and it is... 

An analytical approach for estimating the distributions PcTA(r/)(z) of Cta(T, t ) z for nonzero t was developed in Margolin and Barkai [26, 40], To treat the problem a nonergodic mean field approximation was used, in which various time averages were replaced by the time average intensity I qj, specific to a given realization. For short t -C T the result is... 

Obviously the enthalpy expression x a b in Eq. (3) neglects any correlation effects in the occupancy of lattice sites the probability that on neighboring lattice sites A-B-pairs occur is simply taken as the product < >A< >B of the respective volume fractions. This is a special case of a mean-field approximation (MFA), which is known to yield a critical behavior described by the Landau theory of phase transitions [100], which differs from the correct critical behavior expected [73,74] in the universal regime close to the critical point Xcrm 4>cri in Fig. 2. We shall discuss these various types of critical behavior in Sect. 2.2. 

Here, it is usual to make the Bom-Oppenheimer approximation that allows a classical treatment of the nuclei to be separated from a quantum mechanical description of the electrons. In this case, the wave function becomes just that of the electrons, and the nuclear-nuclear interaction is added to the energy as a sum over point particles. Consequently, the Hamiltonian operator H includes the kinetic energy of the electrons, the electron-electron interactions, and the electron-nuclei interactions. The wave function determined by solving this eigenproblem consists of a Slater determinant of the molecular orbitals for a molecule or, alternatively, the band structure of a solid. Unfortunately, direct solution of this equation is complicated by the electron-electron interactions. Often, it is necessary to introduce a mean-field approximation that neglects the individual dynamical electron-electron correlations but instead treats the electrons as moving in the average field created by the other electrons. Various corrections have been developed to improve upon this approximation [160, 167, 168]. 

Among adsorption isotherms of various types, those that can take into account the intermolecular interaction between adsorbed molecules are required to describe reaction (5). The Frumkin isotherm [91], which is based on the mean-field approximation, is the most frequently used in electrochemistry where electrochemical reactions involve adsorption and desorption processes. The Frumkin isotherm predicts the FWHM for the desorption peak to be 60 mV near the critical point, much broader than experimentally observed values of about 20 mV. To describe the phase transition of adsorbed molecules, Retter examined several isotherms and found that those based on the I sing model can reproduce the sharper phase transitions experimentally observed in the adsorption of heterocyclic compounds on mercury [92]. 

The use of low-order perturbation theory is probably the cheapest and conceptually simplest method for including correlation effects in a quantum-chemical calculation while maintaining a minimum of formal rigor. In particular, Mpller-Plesset perturbation expansions to various orders (commonly denoted MPn) have seen widespread use. For our purposes it is sufficient to discuss only the MP2 expansion, which is the lowest order that contributes beyond the mean-field approximation. 

Unfortunately, this approach does not give a deeper insight into a structure of surface films at the molecular level. The theory involves a concept of a certain averaging effects connected with heterogeneity of sohd surfaces. Moreover, molecular interactions are usually described in terms of a mean field approximation. As a consequence, the integral equation approach cannot elucidate many experimental findings. In particular, various phase transitions in adsorbed layers, such as the order-disorder transition, cannot be explained in the fiamework of this theory. 

Ni-Mo-Jf (X = V, Ta, Al, W, Cr) alloys by transmission electron microscopy (TEM) and the first-principles calculations. They discussed influences of the alloying elements on the ordering behavior in terms of the effective atomic interactions, electronic structures of the ordered compounds and so on. In these previous studies, however, the ordering behavior was interpreted based on the mean-field approximation even for the SRO and imperfectly ordered states. Thus, the following points about the atomistic ordering process are not clear (i) Do Ni-based 1 1/2 0 alloys form similar SRO structures in atomic level (ii) How do the various LRO structures develop from the 1 1/2 0 type SRO state (iii) What governs the SRO-LRO transition process in Ni-based 1 1/2 0 alloys ... 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

Figure A2.5.21. The heat eapaeity of an order-disorder alloy like p-brass ealeulated from various analytie treatments. Bragg-Williams (mean-field or zeroth approximation) Bethe-1 (first approximation also Guggenheim) Bethe-2 (seeond approximation) Kirkwood. Eaeh approximation makes the heat eapaeity sharper and higher, but still finite. Reprodueed from [6] Nix F C and Shoekley W 1938 Rev. Mod. Phy.s. 10 14, figure 13. Copyright (1938) by the Ameriean Physieal Soeiety.

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

The Maximum-Entropy Principle. The mean-field and the quasi-chemical approximations can be extended to larger clusters. Using the derivation of Section 3.1.3 to obtain a quasi-chemical approximation for a cluster with more sites is quite cumbersome. In this section we present an approach that is new and that unifies various approximations to deal with multi-site probabilities. It is based on the maximum-entropy principle. Suppose we have a cluster of n sites with occupations Xi, X2,. .., X . We define an entropy... 

---