Universal amplitude ratio

For a general discussion of universal amplitude ratios in the theory of critical phenomena, see Ref. 52. 

Probably our most important work is a high-precision study of the critical exponents u and 2A4 — 7 (and in particular the hyperscaling law dv = 2A4 — 7) and universal amplitude ratios for SAWs in both two and... 

B. Li, N. Madras, and A. D. Sokal, Critical exponents, hyperscaling and universal amplitude ratios for two- and three-dimensional self-avoiding walks, J. Stat. Phys. (to appear August 1995). 

At this point, we mention a further consequence of the universality principle alluded to above. For each universality class (such as that of the Ising model or that of the XY model, etc.) not just the critical exponents are universal, but also the scaling function F(H), apart from non-universal scale factors for the occurring variables (a factor for H we have expressed via the ratio C/B in eq. (84), for instance). A necessary implication then is the universality of certain critical amplitude ratios, where all scale factors for the variables of interest cancel out. In particular, ratios of critical amplitudes of corresponding quantities above and below Tc, A+j A [eq. (7)], C+jC [eq. (6)] and f+/ [eq. (38)] are universal (Privman et al., 1991). A further relation exists between the amplitude D and B and C 1 Writing M H -> oo) = XHl/ cf. eqs. (87) and (91), the universality of M(H) states that X is universal. But since 0 = B tfM H) = B t PXH = B] SC S H] X, a comparison with eq. (45) yields... 

Therefore, only two amplitudes are independent. It has been established theoretically [1, 5] and verified experimentally [6, 7] that all fluids and fluid mixtures, regardless of variety and complexity in their microscopic structure, belong to the same universality class, i.e. they have the same universal values of the critical exponents (Table 2) and of the critical-amplitude ratios (Table 1) as those of the 3-dimensional Ising model. The physical reason of the critical-point universality originates from the divergence of the order-parameter fluctuations near the critical point. 

In three dimensions, 9 = 2v l.26 and Ra = (To Q flk Tc=0. i9. Not only are the scaling relations (7.67) and (7.68) obtained from the simplified model in agreement with RG theory, the estimated value of the amplitude ratio R 0.39 is close to the universal theoretical value 7 o = 0-37. This value is confirmed by the most reliable experiments on fluids. Relationships between other thermodynamic properties and the diverging correlation length can be obtained in a similar fashion. 

The linear-model expressions for the critical amplitudes are presented in Table 10.5. The values of the universal critical-amplitude ratios implied by the restricted linear model are included in Table 10.3. For a corresponding set of expressions for the cubic model the reader is referred to the literature.More sophisticated parametric equations have also been considered in the literature, that are not discussed here. 

We note that the asymmetry coefficients a, b, and C3 affect the critical amplitudes but not the amplitude ratios which continue to have the same universal values listed in Table 10.3. Substitution of the parametric expressions for the scaling fields, sealing densities, and scaling susceptibilities from Table 10.4 into eqs 10.39 to 10.44 yields a linear-model equation of state consistent with complete scaling. 

Table 1. Predictions of universal quantities (critical exponents and one amplitude ratio)...

A warning According to Stoll and Domb , a supposedly ratio of universal amplitudes, determining the shape of the scaling function in Eq. (6d), depends on T even at temperatures above Tc. 

The photomultiplier, as shown in Fig. 6, is almost universally used as a photon counter, that is, the internal electron multiplication produces an output electrical pulse whose voltage is large compared with the output electric circuit noise. Each pulse in turn is the result of an individual photoexcited electron. The numbered electrodes, 1-8, called dynodes, are each successively biased about 100 V positive with respect to the preceding electrode, and an accelerated electron typically produces about 5 secondary electrons as it impacts the dynode. The final current pulse collected at the output electrode, the anode, would in this case contain 5 400,000 electrons. The secondary emission multiplication process is random, the value of the dynode multiplication factor is close to Poisson distributed from electron to electron. The output pulse amplitude thus fluctuates. For a secondary emission ratio of = 5, the rms fractional pulse height fluctuation is 1 /V<5 — 1 = 0.5. Since the mean pulse height can be well above the output circuit noise, the threshold for a pulse count may be... 

According to RG-calculations performed by Aharony and Halperin (1975), Brezin (1975), and Aharony and Hohenberg (1976), eqs. (221)-(226) are to be supplemented with the universal ratios among the critical amplitudes... 

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