Approximation mean-field

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  

Ising model harbors a critical point. It can be shown (see [bax82]) that the correlation length = [ln(Ai/A2)] h If H = 0, however, then it can also be shown that limy g+(Ai/Aj) = 1 and, thus, that oo at // = 7 = 0. Since one commonly associates a divergent correlation length with criticality, it is in this sen.se that 7/ = T = 0 may be thought of as a critical point. 

Suppose the external field H = 0. We see from the above equation that when the temperature lies below the Curie temperature = Jq, there will exist a nonzero spontaneous magnetization, Mq, given implicitly by Mq = ta,nh JqMo/kBT). As for the behavior near Tc, we first set t = T — so that t 0 as T T.  

Using this new variable, the mean-field equation can be written as 

Since for T just less than Tc, Mq is small but nonzero, we can approximate tanh (Mq) by Mo + Mq/3. Solving this approximate equation for Mo, we find that 

Equation (4.76) states that the grand potential calculated by thermodynamic perturbation theory is an upper bound for its true value. Hence, one wishes to minimize the deviation between Q (T, p 1) and (T, /x 0) + (// — Ho)x=o-In this case, we take as an ansatz 

Clearly, Eq. (4.84) ignores correlations in the occupation-number patterns (i.e., the configurations of the lattice fluid). This (independent) assumption is, however, required to be consistent with the mean-field ansatein Eq. (4.77). 

Finally, the best estimate of fl is found by minimizing the fimctional fl [/A] with respect to for fixed values of T and p that is, we require the functional derivative [26, 30] to satisfy 

We split the saddle-point integration into two steps [27, 63] First, we approximate the functional integral over W+ by the most probable value of the integrand. 

The real field, W+, gives rise to an imaginary contribution to Wa and Wb that, in turn, corresponds to a strongly oscillating behavior of the integrand. To evaluate those oscillating contributions, the standard procedure is to extend the auxiliary field W+ 

Consequently, the integrand has a stationary phase at w+ along the shifted path of integration, and this region yields the dominant contribution to the integral. From the condition of stationary phase, we obtain 

Inserting this expression into the saddle-point equation for, we obtain 

Substituting backthe saddle-pointvalue, iv+,into the partition function, Eq. (5.15), we find 

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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... 

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

In mean field approximation we obtain for the imaginary-time correlation functions [296]... 

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]. 

As pointed out by Flory [16], the principle of equal reactivity, according to which the opportunity for reaction (fusion or scission) is independent of the size of the participating polymers, implies an exponential decay of the number of polymers of size / as a function of /. Indeed, at the level of mean-field approximation in the absence of closed rings, one can write the free energy for a system of linear chains [11] as... 

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... 

For a given Hamiltonian the calculation of the partition function can be done exactly in only few cases (some of them will be presented below). In general the calculation requires a scheme of approximations. Mean-field approximation (MFA) is a very popular approximation based on the steepest descent method [17,22]. In this case it is assumed that the main contribution to Z is due to fields which are localized in a small region of the functional space. More crudely, for each kind of particle only one field is... 

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... 

The result is valid below the roughening temperature, but above the roughening temperature this mean-field approximation is not sufficient... 

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... 

In order to perform the calculation., of the conductivity shown here we first performed a calculation of the electronic structure of the material using first-principles techniques. The problem of many electrons interacting with each other was treated in a mean field approximation using the Local Spin Density Approximation (LSDA) which has been shown to be quite accurate for determining electronic densities and interatomic distances and forces. It is also known to reliably describe the magnetic structure of transition metal systems. 

Mean Field Approximation as a first order approximation, we will ignore all correlations between values at different sites and parameterize configurations purely in terms of the average density at time t p. The time evolution of p under an arbitrary rule [Pg.73]

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... 

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