Order parameter field theory

The basic idea of a Ginzburg-Landau theory is to describe the system by a set of spatially varying order parameter fields, typically combinations of densities. One famous example is the one-order-parameter model of Gompper and Schick [173], which uses as the only variable 0, the density difference between oil and water, distributed according to the free energy functional... 

Early attempts to develop theories that accounted for the power-law behavior and the actual magnitudes of the various critical exponents include those by van der Waals for the (liquid + gas) transition, and Weiss for the (ferromagnetic + paramagnetic) transition. These and a later effort called the Landau theory have come to be known as mean field theories because they were developed using the average or mean value of the order parameter. These theories invariably led to values of the exponents that differed significantly from the experimentally obtained values. For example, both van der Waals and Weiss obtained a value of 0.50 for (3, while the observed value was closer to 0.35. 

Using the potential V(h)°ch2 in Eq. (110), one recognizes that Eq. (110) is formally identical to a Ginzburg-Landau theory of a second-order transition for T>TC(D), with h(x,y) the order parameter field [186,216]. Therefore, it is straightforward to read off the correlation length , associated with this transition at TC(D), namely... 

We return here to the simple mean field description of second-order phase transitions in terms of Landau s theory, assuming a scalar order parameter cj)(x) and consider the situation T < Tc for H = 0. Then domains with = + / r/u can coexist in thermal equilibrium with domains with —domain with exists in the halfspace with z < 0 and a domain with 4>(x) = +

0 (fig. 35a), the plane z = 0 hence being the interface between the coexisting phases. While this interface is sharp on an atomic scale at T = 0 for an (sing model, with = -1 for sites with z < 0, cpi = +1 for sites with z > 0 (assuming the plane z = 0 in between two lattice planes), we expect near Tc a smooth variation of the (coarse-grained) order parameter field (z), as sketched in fig. 35a. Within Landau s theory (remember 10(jc) 1, v 00 01 < 1) the interfacial profile is described by... 

These three theories predict the global phase behavior of microemulsion systems in terms of bending elastic properties (first approach), molecular interaction parameters (second approach), and expansion coefficients of order parameter fields (third approach). In this review, we try to fill the gap between the early years approach and item 1, above. 

In Fig. 10, we compaxe the structure of the three-ring defect obtained from simulation and theory for s = 0.3ii with R = Soq. In both cases, the director field is shown superimposed to a contour plot of 5, the scalar order parameter. Both theory and simulation exhibit the third ring in addition to the usual two Saturn rings. In these plots we can observe how the strength of S decreases continuously from its bulk value to a minimum at the defects. Once more, in contrast to the field theory results, the Monte Carlo data show layers of low and high values of S close to the spheres surfaces that are correlated with modulations in density see Fig. 10). 

The first ten years of this period saw several important developments which escalated interest and research in liquid crystals. Among these, there was the publication by Maier and Saupe [34] of their papers on a mean field theory of the nematic state, focusing attention on London dispersion forces as the attractive interaction amongst molecules and upon the order parameter. This theory must be regarded as the essential starting point for the advances in theoretical treatments of the liquid crystal state which followed over the years. 

Thus T is the predicted transition temperature from mean-field theory. This mean-field picture of the phase transition is only valid when c is so large that spatial modulations in the local order parameter field are suppressed however, we see from (7.246) that this is unlikely, especially when the dimension d is small. 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

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. 

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 large size of now is responsible for mean-field theory being reliable for large N Invoking the Ginzburg criterion one says mean field theory is self-consistent if the order parameter fluctuation in a correlation volume is much smaller than the order parameter itself. 

Recently the density dependence of the symmetry energy has been computed in chiral perturbation effective field theory, described by pions plus one cutoff parameter, A, to simulate the short distance behavior [23]. The nuclear matter calculations have been performed up to three-loop order the density dependence comes from the replacement of the free nucleon propagator by the in-medium one, specified by the Fermi momentum ItF... 

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

Crystal field theory enables us to define certain parameters in order to characterize and distinguish the different europium site configurations. Crystal field parameter calculation involves several steps ... 

Most of the experimental results on CJTE can be explained on the basis of molecular field theory. This is because the interaction between the electron strain and elastic strain is fairly long-range. Employing simple molecular field theory, expressions have been derived for the order parameter, transverse susceptibility, vibronic states, specific heat, and elastic constants. A detailed discussion of the theory and its applications may be found in the excellent review by Gehring Gehring (1975). In Fig. 4.23 various possible situations of different degrees of complexity that can arise in JT systems are presented. 

A weakly bound state is necessarily nonrelativistic, v Za (see discussion of the electron in the field of a Coulomb center above). Hence, there are two small parameters in a weakly bound state, namely, the fine structure constant a. and nonrelativistic velocity v Za. In the leading approximation weakly bound states are essentially quantum mechanical systems, and do not require quantum field theory for their description. But a nonrelativistic quantum mechanical description does not provide an unambiguous way for calculation of higher order corrections, when recoil and many particle effects become important. On the other hand the Bethe-Salpeter equation provides an explicit quantum field theory framework for discussion of bound states, both weakly and strongly bound. Just due to generality of the Bethe-Salpeter formalism separation of the basic nonrelativistic dynamics for weakly bound states becomes difficult, and systematic extraction of high order corrections over a and V Za becomes prohibitively complicated. 

This result is inconsistent with the fact that the differential equation developed by Heaviside from Maxwell s original equations describe circular polarization. The root of the inconsistency is that U(l) gauge field theory is made to correspond with Maxwell-Heaviside theory by discarding the commutator Am x A(2). The neglect of the latter results in a reduction to absurdity, because if S3 vanishes, so does the zero order Stokes parameter ... 

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