Long-range fluctuations

Moreover, some uncertainty was expressed about the applicability to fluids of exponents obtained for tlie Ising lattice. Here there seemed to be a serious discrepancy between tlieory and experiment, only cleared up by later and better experiments. By hindsight one should have realized that long-range fluctuations should be independent of the presence or absence of a lattice. 

As a result of long-range fluctuations, the local density will vary with position in the classical Landan-Ginzburg theory of fluctuations this introdnces a gradient tenn. A Ginzburg number N is defined (for a... 

A long-range fluctuation contribution that describes the longer-term trends in the process. The new term process integration error (PIE2) covers what used to be called the time fluctuation error (TEE) in earlier literature. 

For solids in which IN([Xf) is very near to 1, often, although no magnetic order occurs, long-range fluctuations of coupled spins may take place, giving particular form to properties such as the (Stoner enhanced) magnetic susceptibility x, the electrical resistivity, and the specific heat of the solid. Spin fluctuations have been observed in actinides, and will be discussed in more detail in Chap. D. 

At the time of writing, the only evidence for critical fluctuations near the consolute point known to us comes from the work of Damay (1973). The thermopower of Na-NH3 plotted against T at the critical concentration is shown in Fig. 10.21. We conjecture that this behaviour is due to long-range fluctuations between two metallic concentrations, and that near the critical point, where the fluctuations are wide enough to allow the use of classical percolation theory, the... 

The point here is that the procedure of averaging over small volume Vo does not exclude the long-range fluctuations of a number of reactants in a system. 

It should be emphasized that the comparatively large change obtained in more recent work is mainly caused by the application of finite-size scaling. Under these circumstances, one certainly needs to reconsider how far the results of analytical theories, which are basically mean-field theories, should be compared with data that encompass long-range fluctuations. For the van der Waals fluid the mean-field and Ising critical temperatures differ markedly [249]. In fact, an overestimate of Tc is expected for theories that neglect nonclassical critical fluctuations. Because of the asymmetry of the coexistence curve this overestimate may be correlated with a substantial underestimate of the critical density. 

Second, predictions of p are substantially improved when account is made for ion pairs. The increase of the critical density is easily understood A certain free-ion density is needed for driving criticality. If pairs are formed, this free-ion density can only be achieved at a higher overall ion density. Nevertheless, all theories yield too low values if assessed by the more recent MC data. As mentioned, one reason for low critical densities may result from comparison with MC data that encompass long-range fluctuations. It will, however, be shown in the subsequent section that all available analytical theories seem to overestimate the degree of dissociation. Such an overestimate almost invariably leads to an underestimate of the critical density. 

The methods outlined in this section can be used even in states far from equilibrium when the contributions of the long-range fluctuations become important. 

Because of this long-range fluctuation error, samples taken at different times will give different results. Consequently, it is important to determine whether such trends exist and how they behave. Then we can be sure that process adjustments are effective by using appropriate sampling frequencies. 

A combination of SLS and DLS methods was used to investigate the behavior of nonionic micellar solutions in the vicinity of their cloud point. It had been known for many years that at a high temperature the micellar solutions of polyoxyethylene-alkyl ether surfactants (QEOm) separate into two isotropic phases. The solutions become opalescent with the approach of the cloud point, and several different explanations of this phenomenon were proposed. Corti and Degiorgio measured the temperature dependence of D pp and (Ig), and found that they can be described as a result of critical phase separation, connected with intermicellar attraction and long-range fluctuations in the local micellar concentration. Far from the cloud point, the micelles of nonionic surfactants with a large number of ethoxy-groups (m 30) may behave as hard spheres. ... 

The physical picture of such a phase transition is that the system cannot sustain bulk order if the field b is reduced to zero. However, below T there remain infinitly long range fluctuations and thus a tendency towards ordering. All temperatures below T are therefore critical temperatures. 

Physically this means that the limit to thermodynamic stability is reached when the system no longer resists long range fluctuations in concentration. 

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