Order parameter fluctuations

Equation (A3.3.57) must be supplied with appropriate initial conditions describing the system prior to the onset of phase separation. The initial post-quench state is characterized by the order parameter fluctuations characteristic of the pre-quench initial temperature T.. The role of these fluctuations has been described in detail m [23]. Flowever, again using the renomialization group arguments, any initial short-range correlations should be irrelevant, and one can take the initial conditions to represent a completely disordered state at J = xj. For example, one can choose the white noise fomi (i /(,t,0)v (,t, 0)) = q8(.t -. ), where ( ) represents an... 

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. 

Order parameter fluctuations allow for extra neutrino processes having no exponential suppression in a broad region of temperatures near Tc. 

The coupling to the soft mode introduces here oscillatory terms that can become important close to Tc. The fact that the amplitude, as well as the coherence length of the order parameter fluctuations, increase on approaching Tc from either side also probably brings higher order coupling terms into... 

They can serve therefore as a test for Ti dispersion. In Fig. 12 the relaxation results are shown for D-RADP-15. The solid lines are a fit of the theory [19] to the data. Above Tc the lit is excellent, whereas below Tc it probably suffers from the fact that the phase transition is already diffuse and only nearly of second order. This proves that a soft mode component is needed to explain the data. Furthermore, the fact that the ratio ti/t2 remains unchanged above and below Tc proves that the order parameter fluctuations are in the fast motion regime on both sides of the transition. 

Order-parameter fluctuations can be generalized by introducing the wave-vector f3 in a Fourier representation,... 

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. 

The order-parameter fluctuations are temperature- and system-dependent and their decay rate is related to the transport coefficients (5) Usually the magnitude of the fluctuations are characterized by a correlation length . Along a critical isochore or isopleth, the correlation length diverges as... 

The dec8y rate of the order-parameter fluctuations is proportional to the thermal diffusivity in case of pure gases near the vapor-liquid critical point and is proportional to the binary diffusion coefficient in case of liquid mixtures near the critical mixing point (6). Recently, we reported (7) single-exponential decay rate of the order-parameter fluctuations in dilute sugercritical solutions of liquid hydrocarbons in CO for T - T 10 C. This implied that the time scales associated with thermal diffusion and mass diffusion are similar in these systems. 

The above equation provides a basis for correlating the temperature dependence of a transport coefficient such as mass diffusivity in the supercritical region. The effects of composition, solute, and solvent characteristics can also be introduced into the correlations via and A which are system-dependent amplitudes. However, a rigorous ftest of the applicability of equation 5 requires independent measurements of the decay rate of the order-parameter fluctuations, the correlation length, and the viscosity. 

In this study, we employed PCS to measure the decay rate of the order-parameter fluctuations in dilute supercritical solutions of heptane, benzene, and decane in CC - The refractive index increment with concentration is much larger than the refractive index increment with temperature in these systems. Therefore the order-parameter fluctuations detected by light scattering are mainly concentration fluctuations and their decay rate T is proportional to the binary diffusion coefficient, D = V/q. The... 

CO -benzene, and CO -n-decane. The critical densities and the corresponding compositions are plotted in Figure 1. The three hydrocarbons in order of higher to lower solubility in C0 were heptane, benzene, and decane. The measured binary diffusion coefficients or the decay rates of the order-parameter fluctuations at various temperatures and pressures are listed in Tables I, II, and III for CO -heptane, CO -benzene, and CO -decane systems respectively. In Figure 2, the critical lines of the three binary systems in the dilute hydrocarbon range are shown in the pressure-temperature space. dP/dT along the critical lines of CO.-heptane and CO -benzene systems are similar and lower than dP/dT along the critical line of CO -decane system, which indicates that C02 and decane form more asymmetric mixtures relative to CO with heptane or benzene. 

In Eqs. (53) and (54) , denotes the correlation length of order parameter fluctuations in the disordered phase,... 

It must be remembered, however, that the Leibler-type mean field theory [197] is believed to be accurate for the limit of infinite chain length, N—for finite N effects of order parameter fluctuations are important and change the character of the transition from second order to first order even for symmetric composition [185,186,192,210,211]. With a self-consistent Hartree approximation that one believes to be valid for large N, Eq. (41) gets replaced by [234]... 

A final general consideration is that Landau theory is expected to give an accurate representation of changes in the physical and thermodynamic properties over wide PT intervals when a transition is accompanied by significant spontaneous strains. This is because the relatively long ranging influence of strain fields acts to suppress order parameter fluctuations. 

This estimate, however, would not characterize the incompatibility between the polymer species too well, but rather quantify the inability of the mean field theory to cope with Ising-like order parameter fluctuations. 

Certainly, restricting the window size limits order parameter fluctuations to far less than those explored in a grandcanonical simnlation and each subsimulations resembles more closely a simulation in the canonical ensemble than in the grandcanonical ensemble. We emphasize, however, that local density (order parameter) fluctuations are not restricted and that, ideally, configurations... 

A. Sandvik (1998) Critical temperature and the transition from quantum to classical order parameter fluctuations in the three-dimensional Heisenberg an-tiferromagnet. Phys. Rev. Lett. 80, p. 5196... 

Because of the broken translational and rotational symmery, order-parameter fluctuations in ordered phases are anisotropic. The importance of such anisotropic fluctuations was mentioned in the work of Bates and co-workers, who conjectured that anisotropic fluctuations might play a role in stablizing nonclassical ordered structures (Hamley et al., 1993 Bates et al., 1994a,b). The concept was also invoked by this group in understanding the effect of shear on the HEX-to-DlS transitions in diblock copolymer melts (Ahndal etal., 1996). 

Fig. 11. Schematic variation with temperature 7 plotted for several quantities near a critical point Tic specific heat Ch (top), ordering susceptibility xr (middle part), and correlation length of order parameter fluctuations (bottom). The power laws which hold asymptotically in the close vicinity of Tc are indicated.

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