The mean field approximation

We have already made use of the so-called mean-field approximation by assuming that (1) all adsorbed species are distributed randomly over the surface and (2) there is no interaction betv een the adsorbed species. This is an approximation that is seldom fulfilled. Usually there vill be either an attractive or repulsive interaction 

At high coverages, adsorbate interactions will always be present, implying that preexponential factors and activation energies are dependent on coverage. In the following we shall assume that the mean-field approximation is valid, but one should be aware that it may be a source of error. The alternative to this approximation is to perform Monte Carlo simulations (see Chapter 7). 

We assume that each molecule is subject to an average internal field which is independent of any local variations or short-range ordering. Consistent with the symmetry of the structure, viz, the cylindrical distribution about the preferred axis and the absence of polarity, we may postulate that the orientational energy of a molecule 

The exact nature of the intermolecular forces need not be specified for the development of the theory. However, in their original presentation Maier and Saupe assumed that the stability of the nematic phase arises from the dipole-dipole part of the anisotropic dispersion forces. The second order perturbed energy of the Coulomb interaction between a pair of molecules 1 and 2 is given by 

We assume that the rotational motions of the molecules are independent of their translational motions. Let (2) (2) (2) 

Cotter has examined the postulates underlying the mean field approximation in the light of Widom s analysis of this general problem and has concluded that thermodynamic consistency requires that u should be proportional to V regardless of the nature of the intermolecular pair potential. However, in what follows we have assumed a dependence as in the original formulation of the theory by Maier and Saupe. 

A rigorous molecular theory of a fluid system based on a pairwise interaction potential as complicated as Eq. [2] is impossibly difficult. A simple but adequate approach is to derive a theory in the mean field approximation. That is, we derive a single molecule potential that serves to orient the molecule along the symmetry axis (the director) of the nematic phase. The single-molecule potential represents (approximately) the mean field of intermolecular forces acting on a given molecule. Mean field theories have been found capable of describing the qualitative behavior of many different cooperative phenomena. The 

In order to derive a mean field approximation to the potential, we first have to express V12 in terms of a polar coordinate system based on the director, n, as the polar axis. The coordinate axes for the molecules 1 and 2 must be rotated from that shown in Fig. 1(a) to that shown in Fig. 1(b). The primed angles now describe the orientations of the molecules with respect to the new rotated coordinate system. Mathematically, the rotation of the coordinate axes transforms the spherical harmonics into the form 

The averaging of V12 over all orientations of the intermolecular vector r has an influence only on the Wigner rotation matrices  

Here rigCr) is the molecular distribution function for the separation of pairs of molecules and n is the number density of molecules in the fluid. Use of Eq. [11] in taking the average of Fi2 gives 

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To improve upon die mean-field picture of electronic structure, one must move beyond the singleconfiguration approximation. It is essential to do so to achieve higher accuracy, but it is also important to do so to achieve a conceptually correct view of the chemical electronic structure. Although the picture of configurations in which A electrons occupy A spin orbitals may be familiar and usefiil for systematizing the electronic states of atoms and molecules, these constructs are approximations to the true states of the system. They were introduced when the mean-field approximation was made, and neither orbitals nor configurations can be claimed to describe the proper eigenstates T, . It is thus inconsistent to insist that the carbon atom... 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

Within the mean-field approximation one can minimize the free energy (64) with respect to a fixed profile C q and this leads to the minimized free energy F 1). Next, one can define the free energy difference... 

R. D. Vigil, F. T. Willmore. Oscillatory dynamics in a heterogeneous surface reaction Breakdown of the mean-field approximation. Phys Rev E 54 1225-1231, 1996. 

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]. 

The thermodynamic quantities and correlation functions can be obtained from Eq. (1) by functional integration. However, the functional integration cannot usually be performed exactly. One has to use approximate methods to evaluate the functional integral. The one most often used is the mean-field approximation, in which the integral is replaced with the maximum of the integrand, i.e., one has to find the minimum of. F[(/)(r)], which satisfies the mean-field equation... 

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. 

For an analytical treatment of Eq. (18) we make a mean-field approximation in layers, where the index i is now decomposed into the layer index k and lattice position j within the layer Si s /.. The mean-field approximation in the layer leads to the layer order parameter = (T. Its evolution is obtained from (18) as... 

With a finite value of A(i 0, the interface starts to move. In the mean-field approximation of a similar model, one can obtain the growth rate u as a function of the driving force Afi [49]. For Afi smaller than the critical value Afi the growth rate remains zero the system is metastable. Only above the critical threshold, the velocity increases a.s v and finally... 

This profile of the phase boundary determined here looks very similar to that obtained by the mean-field approximation (19), but the result here only applies to the profile above the roughening temperature. Since this is a mean-field theory, fluctuations are also not considered correctly. 

In order to find approximate solutions of the equations for Ci t) and gi,..j t) one can use regular approximate methods of statistical physics, such as the mean-field approximation (MFA) and the cluster variation method (CVM), as well as its simplified version, the cluster field method (CFM) . In both MFA and CFM, the equations for c (<) are separated from those for gi..g t) and take the form... 

The mean-field approximation consists of replacing Hi with its mean-field value H, obtained by replacing the spins Sj with an average value < Si > ... 

Another consequence of linearity is that the mean-field approximation becomes exact. Prom equations 7.82 and 7.88, we can immediately see that the mean-field iterative equation... 

In order to improve upon the mean-field approximation given in equation 7.112, we must somehow account for possible site-site correlations. Let us go back to the deterministic version of the basic Life rule (equation 7.110). We could take a formal expectation of this equation but we first need a way to compute expectation values of Kronecker delta functions. Schulman and Seiden [schul78] provide a simple means to do precisely that. We state their result without proof... 

Note that the rates of product formation and reactant conversion indeed have the dimensions of mol per unit of time, and that these rates are proportional to the number of sites, or, in fact, the amount of catalyst present in the reactor. Also, in the case of a second order reaction, e.g. betv een adsorbed species A and B, we write the rate in the form r = Nk0j 0 by applying the mean-field approximation. Here the rate is proportional to both the total number of sites on the surface and the probability of finding a species A adjacent to a species B on the surface, the latter being proportional to the coverages of A and B. In the mean-field approximation A and B are distributed randomly over the N available sites this only tends to be valid when the adsorbents repel each other. Thus the rate is not r= k(N0/ )(N02,) since the reactants need to be on adjacent sites. Another important consideration is that we want the rate to be linearly proportional to the amount of catalyst in the reactor, in accordance with r = Nk0A0B for a second order surface reaction. 

What is the mean-field approximation in the kinetics of catalytic reactions, and when does it break down In such cases, is the rate larger or smaller than expected on basis of the mean-field approximation ... 

It was assumed that the volume V was reduced by the presence of other particles to the free volume V — Nvf) where N is the number of particles. In arriving at the binding energy effect the mean field approximation was used, which says that the soft (negative) part of the pair potential was sampled in an uncorrelated manner as if the system was an ideal gas. The corresponding free energy per particle in the bulk fluid is... 

Here is the soft and normally attractive part of the pair potential. This simple bilinear form of functional lacks correlation effects except that which is introduced by the truncation of the integral at the onset of the inner hard part of the potential. We are then using an extended form of the mean field approximation as did van der Waals in his original... 

It may seem that the prospeets are bleak for the GvdW approach to electrolytes but, in fact, the reverse is the ease. We need only follow Debye and Hiickel [18] into their analysis of the sereening meehanism, almost as successful as the van der Waals analysis of short-range fluids, to see that the mean-field approximation can be applied to the correlation mechanism with great advantage. In fact, we can then add finite ion size effects to the analysis and thereby unify these two most successful traditional theories. 

III. METHODS FOR TREATING THE LATTICE GAS A. The Mean-Field Approximation... 

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