Ising criticality theories

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. 

Critical theories take into account concentration fluctuations and describe phase transitions near the critical point. The critical theory for this phase transition is the Ising model which also describes liquid-vapour transitions near their critical points. The prediction of the three-dimensional Ising model for the critical exponent /3 = 0.3 is in good agreement with experiments as shown in Fig. 5.3(b). Mean-field and cri-... 

A field-theory based on this simple expansion will already yield three-dimensional Ising critical behavior. Note that the non-trivial critical behavior is related to fluctuations vaW or a- b- Fluctuations in the incompressibility field U or the total density a + are not important. 

Both the statics and the collective dynamics of composition fluctuations can be described by these methods, and one can expect these schemes to capture the essential features of fluctuation effects of the field theoretical model for dense polymer blends. The pronounced effects of composition fluctuations have been illustrated by studying the formation of a microemulsion [80]. Other situations where composition fluctuations are very important and where we expect that these methods can make straightforward contributions to our understanding are, e.g., critical points of the demixing in a polymer blend, where one observes a crossover from mean field to Ising critical behavior [51,52], or random copolymers, where a fluctuation-induced microemulsion is observed [65] instead of macrophase separation which is predicted by mean-field theory [64]. 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

The calculations that have been carried out [56] indicate that the approximations discussed above lead to very good thermodynamic functions overall and a remarkably accurate critical point and coexistence curve. The critical density and temperature predicted by the theory agree with the simulation results to about 0.6%. Of course, dealing with the Yukawa potential allows certain analytical simplifications in implementing this approach. However, a similar approach can be applied to other similar potentials that consist of a hard core with an attractive tail. It should also be pointed out that the idea of using the requirement of self-consistency to yield a closed theory is pertinent not only to the realm of simple fluids, but also has proved to be a powerful tool in the study of a system of spins with continuous symmetry [57,58] and of a site-diluted or random-field Ising model [59,60]. 

The theoretical foundation for describing critical phenomena in confined systems is the finite-size scaling approach [64], by which the dependence of physical quantities on system size is investigated. On the basis of the Ising Hamiltonian and finite-size scaling theory, Fisher and Nakanishi computed the critical temperature of a fluid confined between parallel plates of distance D [66]. The critical temperature refers to, e.g., a liquid/vapor phase transition. Alternatively, the demixing phase transition of an initially miscible Kquid/Kquid mixture could be considered. Fisher and Nakashini foimd that compared with free space, the critical temperature is shifted by an amoimt... 

The approximations of the superposition-type like equation (2.3.54), are used in those problems of theoreticals physics when other-kind expansions (e.g., in powers of a small parameter) cannot be employed. First of all, we mean physics of phase transitions and critical phenomena [4, 13-15] where there are no small parameters at all. Neglect of the higher correlation forms a(ml in (2.3.54) introduces into solution errors which cannot be, in fact, estimated within the framework of the method used. That is, accuracy of the superposition-like approximations could be obtained by a comparison with either simplest explicitly solvable models (like the Ising model in the theory of phase transitions) or with results of direct computer simulations. Note, first of all, several distinctive features of the superposition approximations. 

According to RG theory [11, 19, 20], universality rests on the spatial dimensionality D of the systems, the dimensionality n of the order parameter (here n = 1), and the short-range nature of the interaction potential 0(r). In D = 3, short-range means that 0(r) decays as r p with p>D + 2 — tj = 4.97 [21], where rj = 0.033 is the exponent of the correlation function g(r) of the critical fluctuations [22] (cf. Table I). Then, the critical exponents map onto those of the Ising spin-1/2 model, which are known from RG calculations [23], series expansions [11, 12, 24] and simulations [25, 26]. For insulating fluids with a leading term of liquid metals [27-29] the experimental verification of Ising-like criticality is unquestionable. 

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