Order parameter consistency relation

Let us consider now behaviour of the gas-liquid system near the critical point. It reveals rather interesting effect called the critical opalescence, that is strong increase of the light scattering. Its analogs are known also in other physical systems in the vicinity of phase transitions. In the beginning of our century Einstein and Smoluchowski expressed an idea, that the opalescence phenomenon is related to the density (order parameter) fluctuations in the system. More consistent theory was presented later by Omstein and Zemike [23], who for the first time introduced a concept of the intermediate order as the spatial correlation in the density fluctuations. Later Zemike [24] has applied this idea to the lattice systems. 

For values of the parameters consistent with convenient device fabrication, this relation gives a value for x of the order of 50 ms. 

However for a specific inflationary model, the four observable quantities As, At, ns and nj can be expressed at lowest order in term of the physical quantities 14 and the two slow-roll parameters < /,. and 5k, so that there exist some consistency relations (between the tensor spectral index and the scalar-to-tensor ratio), which in principle allow to test inflation, and to reconstruct the potential on a small region (as we can have access to V as well as V and V" with the slow roll parameters). Note however that this consistency relation crucially relies on the detection of the tensor modes (and hence, on the H-polarization of CMB as it is probably the most efficient way to detect the tensor modes), which may very well be an extraordinarily difficult task if /, happens to be very small. 

From the potential Vnem(vt). one can obtain the rod orientation distribution function 1/ (u), and hence the order parameter S, by a self-consistent calculation. At equilibrium, the distribution function V (u) is related to the potential Fnemfu) by Boltzmann s equation ... 

A similar analysis applies to the triclinic-monoclinic (/T -72/c) transition that occurs at 298 K across the compositional join CaAl2Si208-SrAl2Si208 near 85 mol % Sr, except that the order parameter varies with composition with a form similar to Equation 7 (Phillips et al. 1997). With increasing Sr-content, the Si MAS-NMR spectra (Fig. 17) clearly show a decrease in the number of peaks that corresponds to a change in the number of crystallographically distinct Si sites from four (/1) to two (72/c). The order-parameter could be related to the difference in chemical shift between the Tlo site of the 72/c phase (-85.4 ppm) and the peak for the Tlmz site of the 71 samples, which is well-resolved and moves from -89.5 to -86.7 ppm with increasing Sr-content. These results yielded a critical exponent P = 0.49+0.2, consistent with the second-order character of the transition. 

The values of the dielectric constant obtained from the relations (11) and (14) should be consistent, and this is indeed the case in our calculations. To end this section, we should mention that Eq. (14) is similar to the expression derived by Klapp and Patey [25] for positionally frozen dipolar fluids once the local freezing order parameters are set to zero. 

The assignment of the order of the herringbone transition was discussed in the literature over many years. The LEED results [92, 93] (see Fig. 26) were consistent with an interpretation in terms of a first-order transition with pronounced rounding elfects. However, these data could not mle out a continuous transition so that this study was not accurate enough to decide the order of the transition. In addition, there is the problem to relate the LEED superlattice spot intensities to the proper long-range order parameter of the system [93, 108] (see the presentation in Section III.D.l). 

Phase transitions in which the square of the soft-mode frequency or its related microscopic order parameter goes to zero continuously with temperature can be defined as second order within the framework of the Ginzburg-Landau model [110]. The behavior is obviously classical and consistent with mean field... 

Fig. 2.3.2. Variation of the free energy with the order parameter calculated from the Maier-Saupe theory for different values of A/k TV. The minima in the curves occur at values of s which fulfil the consistency relation (2.3.13). ...

On a more phenomenological level, the relation between Ps and 0 is described by Landau theories. Since the predictions of Landau models will be used in several of the following sections, a short introduction is given here. The basic part gg of the Landau free energy g consists, for chiral and nonchiral compoimds, of a series of powers of the tilt order parameter... 

To relate macroscopic properties, especially the results of chirality measurements, to mesoscopic and further to pseudoscalar molecular properties, experimental data should be available in order to develop and check structure-property relations, mechanisms, and models. A set of usable data for the chiral nematic phase consists of the composition of the phase, p, HTP, V.2, Ki, K2, and K3. On the microscopic scale the (HTP)i as well as the microscopic order paramet S, D, A, B, the helicity tensor Qy or the chirality interaction tensor W y, and a chirality tensor Cy or an equivalent quantity should be known for every component. At the moment, the coordinates W y can only be estimated because a variation of the order of the chiral dopant for a constant host order needs to be known for their measurement. Some data are collected in TABLE 1 in order to give a feeling for the size and sign of available quantities [25-33] but, as can be seen, no complete sets of data for a system have yet beoi given in the literature. 

The basic idea of the system orientational state is to determine the share of the sampling chain links nj rj) and riy i]), oriented along direetions x andy, depending on the orientational state of the environment (on the value of the order parameter rf), within the model of orientationally self-correlated wanderings on the regular lattice. If the order parameter (in accordance with definition (2.35)) is expressed through the values of nj, if) and riyf tj), a self-consistent equation is found which relates the order parameter ij to the functions of 4(7)-... 

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