The intermediate order parameter

How could we take into account the fluctuations of the order parameter Let us return to the well-studied example of the gas-liquid system. A general equation of the state of gases and liquids proved in statistical physics [9] has a form p = nk T - n2G(x) where G(x) is some integral containing the interaction potential of particles and the joint correlation function x(r). Therefore, the equation for the long-range order parameter n contains in itself the functional of the intermediate-order parameter x r)-... 

A new principal element of the Waite-Leibfried theory compared to the Smoluchowski approach is the relation between the effective reaction rate K(t) and the intermediate order parameter x = Xab (r,i). In its turn, the Smoluchowski approach is just an heuristic attempt to describe the simplest irreversible bimolecular reactions A + B- B,A + B- B and A + B -> 0 and cannot be extended for more complicated reactions. The Waite-Leibfried approach is not limited by these simple reactions only it could be applied to the reversible reactions and reaction chains. However, in the latter case the particular linearity in the joint correlation function x = Xab (r, ) does not always mean linearity of equations since additional non-linearity caused by particle densities can arise. 

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. 

Equation (10) shows that we can always accomplish our objective if we can measure the full canonical distribution of an appropriate order parameter. By full we mean that the contributions of both phases must be established and calibrated on the same scale. Of course it is the last bit that is the problem. (It is always straightforward to determine the two separately normalized distributions associated with the two phases, by conventional sampling in each phase in turn.) The reason that it is a problem is that the full canonical distribution of the (an) order parameter is typically vanishingly small at values intermediate between those characteristic of the two individual phases. The vanishingly small values provide a real, even quantitative, measure of the ergodic barrier between the phases. If the full -order parameter distribution is to be determined by a direct approach (as distinct from the circuitous approach of Section IV.B, or the off the map approach to be discussed in Section IV.D), these low-probability macrostates must be visited. 

Applying superposition approximations to the Ising model, one finds an evidence for the phase transition existence but the critical parameter to is systematically underestimated (To is overestimated respectively). Errors in calculation of to are greater for low dimensions d. Therefore, the superposition approximation is effective, first of all, for the qualitative description of the phase transition in a spin system. In the vicinity of phase transition a number of critical exponents a, /3,7,..., could be introduced, which characterize the critical point, like oc f-for . M oc (i-io), or xt oc i—io for the magnetic permeability. Superposition approximations give only classical values of the critical exponents a = ao, 0 = f o, j — jo, ., obtained earlier in the classical molecular field theory [13, 14], say fio = 1/2, 7o = 1, whereas exact magnitudes of the critical exponents depend on the space dimension d. To describe the intermediate order in a spin system in terms of the superposition approximation, an additional correlation length is introduced, 0 = which does not coincide with the true In the phase... 

The parameters C, D, E and F involve components of polarizability derivative tensors. Qjs is the displacive order parameter in the low-temperature phase and Qjq corresponds to which is present in the intermediate pseudo-phase. The damping constant T qjh) includes the effect of coupling between the hard mode and the reorientational relaxation of displaced lead atoms. 

The two polarizations Pp and Pap may be taken as secondary order parameters coupled with the genuine order parameters. As a result, depending of the model, the theory predicts transitions from the smectic A phase into either the synclinic ferroelectric phase at temperature Tp or into an anticlinic antiferroelectric phase at Tap- One intermediate ferrielectric phase is also predicted that has a tilt plane in the i + 1 layer turned through some angle

tilt plane in the i layer. The models based on the two order parameters are of continuous nature (9 may take any values) and, although conceptually are very important, caimot explain a variety of intermediate phases and their basic properties. 

The main contribution of X-ray magnetic scattering to the study of Er has been to shed further light on the phases in the intermediate temperature interval between and Tc, and more recently to characterize the temperature dependence of the magnetic order parameter near 7. X-ray magnetic scattering has revealed a great wealth of new information on these intermediate phases that fits most easily with the cycloidal model. 

The molecules in a nematic liquid crystal tend to be parallel to a unique direction known as the director which is identical with the optic axis of the phase. The molecular orientation fluctuates with respect to the director and the extent of these fluctuations is reflected by the orientational order parameters they are defined in Section 6. As we shall see the order parameters for cylindrically symmetric particles are chosen to be unity in a perfectly ordered phase or crystal while for the disordered phase or liquid the order parameters vanish. In a nematic phase the order parameters are intermediate between these two extremes. The magnitude of the orientational fluctuations are controlled by the energy of the molecule as it changes its orientation with respect to the director. For a cylindrically symmetric molecule this energy is determined entirely by the angle between the director and the molecular symmetry axis. 

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