Exponent renormalization group

The scaling argument provides only the exponent but not the absolute numerical value for the constant. Therefore, for quantitative results, it should be completed by some more refined technique like the afore-mentioned renormalization group method [49],... 

The key aspects of the modern understanding of phase transitions and the development of renormalization group theory can be summarized as follows. First was the observation of power-law behavior and the realization that critical exponents were, to some extent, universal for all kinds of phase transitions. Then it became clear that theories that only treated the average value of the order parameter failed to account for the observed exponents. The recognition that power-law behavior could arise from functions that were homogeneous in the thermodynamic variables and the scale-invariant behavior of such functions... 

The values of the exponents quoted in Table XII have been estimated numericcilly by renormalization group techniques. Intuitively, there should be a close relationship between conductivity and percolation probability, and one would guess that their critical exponents should be identical. This is not true. Dead ends contribute to the mass of the infinite network described by the percolation probability, but not to the electric current it carries. Figure 39 shows the different growth of the percolation probability and the conductivity. It is convenient to set the conductivity equal to unity at = 1, as in Fig. 39. We note, in passing, that diffusivity is proportional to conductivity, in agreement with Einstein s result in statistical physics that diffusivity is proportional to mobility. 

In summary, by using a multistage real-space renormalization group method, we show that the finite-size scaling can be applied in Mott MIT. And the dynamic and correlation length critical exponents are found to be z = 0.91 and v = 1, respectively. At the transition point, the charge gap scales with size as Ag 1/L0-91. 

The universal exponent (i >/(3) depends on the type of theory. Renormalization group theory predicts /3 = 4 while mean-field theory predicts /g =3. 

This relation is the basic equation governing a renormalization group in real space. According to Eq. (159) we shall obtain the critical exponent v for correlation length ... 

After these caveats, fig. 17 shows qualitatively the dimensionality dependence of the order parameter exponent /5, the response function exponent y, and correlation length exponent v. Although only integer dimensionalities d = 1,2, 3 are of physical interest (lattices with dimensionalities d = 4,5, 6 etc. can be studied by computer simulation, see e.g. Binder, 1981a, 1985), in the renormalization group framework it has turned out useful to continue d from integer values to the real axis, in order to derive expansions for critical exponents in terms of variables = du — d or e1 = d — dg, respectively (Fisher, 1974 Domb and Green, 1976 Amit, 1984). As an example, we quote the results for r) and v (Wilson and Fisher, 1972)... 

Another feature arising from field-density considerations concerns the coexistence curves. For one-component fluids, they are usually shown as temperature T versus density p, and for two-component systems, as temperature versus composition (e.g. the mole fraction x) in both cases one field is plotted against one density. However in three-component systems, the usual phase diagram is a triangular one at constant temperature this involves two densities as independent variables. In such situations exponents may be renormalized to higher values thus the coexistence curve exponent may rise to p/(l - a). (This renormalization has nothing to do with the renormalization group to be discussed in the next section.)... 

R N. The exponent v = 0.588 has been calculated using renormalization group techniques [9, 10], enumeration techniques for short chain lengths and Monte Carlo simulations [13]. 

Over the last 10 years or so, a great deal of work has been devoted to the study of critical phenomena in binary micellar solutions and multicomponent microemulsions systems [19]. The aim of these investigations in surfactant solutions was to point out differences if they existed between these critical points and the liquid-gas critical points of a pure compound. The main questions to be considered were (1) Why did the observed critical exponents not always follow the universal behavior predicted by the renormalization group theory of critical phenomena and (2) Was the order of magnitude of the critical amplitudes comparable to that found in mixtures of small molecules The systems presented in this chapter exhibit several lines of critical points. Among them, one involves inverse microemulsions and another, sponge phases. The origin of these phase separations and their critical behavior are discussed next. 

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