Simplest Mean-Field Approach

The results of the simplest mean field approach are very impressing. However, some experimental observations, e.g., different temperature dependencies of the order parameter for different substances, a discontinuity of density at the N-Iso transition have not been explained. The main disadvantages of the simplest theory are (a) lack of the density (or volume) dependence of showing a jump at the transition (b) an oversimplified form of the potential well and (c) pure phenomenological nature of the depth of the potential well v. 

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]. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

The fraction / of charged monomers on an annealed polyelectrolyte is measured in a titration experiment. The simplest description of this experiment [90] is to assume that the polymer is homogeneous and to minimize its grand canonical free energy (its free energy at constant pH). In a mean field approach, the free energy of one chain is... 

The importance of the systematic hierarchy for solving the Schrodinger equation cannot be overemphasized, because it allows one, in principle, to systematically approach the exact result for a molecular property of interest. The simplest approach in this hierarchy is the Hartree-Fock (HF) method, which describes electron-electron interactions within a mean-field approach. " The electron-correlation effects neglected in this approach can be described by the so-called... 

There is a substantial body of theoretical work on micellization in block copolymers. The simplest approaches are the scaling theories, which account quite successfully for the scaling of block copolymer dimensions with length of the constituent blocks. Rather detailed mean field theories have also been developed, of which the most advanced at present is the self-consistent field theory, in its lattice and continuum guises. These theories are reviewed in depth in Chapter 3. A limited amount of work has been performed on the kinetics of micellization, although this is largely an unexplored field. Micelle formation at the liquid-air interface has been investigated experimentally, and a number of types of surface micelles have been identified. In addition, adsorption of block copolymers at liquid interfaces has attracted considerable attention. This work is also summarized in Chapter 3. 

The VSEPR approach is largely restricted to Main Group species (as is Lewis theory). It can be applied to compounds of the transition elements where the nd subshell is either empty or filled, but a partly-filled nd subshell exerts an influence on stereochemistry which can often be interpreted satisfactorily by means of crystal field theory. Even in Main Group chemistry, VSEPR is by no means infallible. It remains, however, the simplest means of rationalising molecular shapes. In the absence of experimental data, it makes a reasonably reliable prediction of molecular geometry, an essential preliminary to a detailed description of bonding within a more elaborate, quantum-mechanical model such as valence bond or molecular orbital theory. 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

Our purpose in these last two subsections has been to show how the simplest fundamental description of SEE for van der Waals solids can emerge from the hard-sphere model and mean field theory. Much of the remainder of the chapter deals with how we extend this kind of approach using simple molecular models to describe more complex solid-fiuid and solid-solid phase diagrams. In the next two sections, we discuss the numerical techniques that allow us to calculate SEE phase diagrams for molecular models via computer simulation and theoretical methods. In Section IV we then survey the results of these calculations for a range of molecular models. We offer some concluding remarks in Section V. 

A different approach was used by Razafimandimby et al. (1984), d Am-brumenil and Fidde (1985) and Fulde et al. (1988) for the Kondo lattice to calculate quasiparticle bands. They start from the observation, Nozieres (1974), that for r a Fermi liquid description can be used for the scattering by Kondo ions. Its phase shift is assumed to have a resonant behaviour around the Fermi energy, with T defining the energy scale. A periodic lattice of resonant scattering centers then leads to narrow quasipartiele bands, whieh have been caleulated within the KKR formalism. In the simplest approximation they are equivalent to those obtained from the mean-field approximation of the Anderson lattice. The method of Razafimandimby et al. has been used for a realistic... 

The simplest SAFT approach, usually referred to as SAFT-HS, describes assoeiating ehains of hard-sphere segments with the long-range attraetive interaetions deseribed at the van der Waals mean-field level. The Helmholtz... 

It is natural to expect that the simplest analytical approach to the iV-body cooperative system (5.2.21) would be provided by something similar to the mean field idea of thermodynamic phase transitions. In the present problem, the in-... 

Much of what has been done on the theory of the near-critical interface has been within the framework of the van der Waals theory of Chapter 3, so much of our present understanding of the properties of those interfaces comes from that theory or from some suitably modified or extended version of it. As we shall see, an interface thickens as Hs critical point is approached, and the gradients of denaty and composition in the interface are then small. Thus, the view that the interfadal region may be treated as matter in bulk, with a local free-energy density that is that of a hypothetically uniform fluid of composition equal to the local composition, with an additional term arising from the non-uniformity, and that the latter may be approximated by a gradient expansion, typically truncated in second order, is then most likely to be successful and perhaps even quantitatively accurate. In this section we shall see what the simplest theory of that kind— that which comes from treating simple models in mean-field approximation, as in Chapter 5— yields for the structure and tension of an interface near a critical point. 

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