Critical amplitude ratio

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. 

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. 

Equations 10.85 to 10.88 define what has been called a two-term crossover Landau model (CLM). In the classical limit (A/ic 1) the crossover function Y approaches unity and one recovers from eq 10.85 the classical expansion of eq 10.83. In the critical region (A/k 1) the crossover function approaches zero as Y x k/uA) and one recovers from eq 10.85 the power-law expansions specified in Table 10.5 with expressions for the critical amplitudes listed in Table 10.8. The values for the critical-amplitude ratios implied by the crossover Landau model are included in Table 10.3. The nonasymptotic critical behaviour is governed by u and A/cJ or, equivalently by u and by Nq, known... 

We recognize that many chromatographers, however, will Interpret a chromatogram by visual Inspection. This Interpretation Introduces difficulties because the resolution between two components Is Independent of the components amplitude whereas visual differentiation between peaks Is amplitude dependent. As an example, If one chooses to define each maximum In the chromatogram as a peak, the critical resolution between components of the same amplitude Is 0.5 (4). If the adjacent amplitude ratio Is 8 1, however, the critical resolution Is approximately 0.8 (4). 

Thus there exist relations between suitable ratios of critical amplitudes and derivatives of the scaling functions. We now consider the response function... 

Zinn, S.-Y. and Fisher, M.E. (1996) Univers2d surface-tension and critical-isotherm amplitude ratio in three dimensions, Physica A 226, 168-180. 

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... 

Draw resonance will occur when the resistance to extensional deformation decreases as the stress level increases. The total amount of mass between die and take-up may vary with time because the take-up velocity is constant but not necessarily the extrudate dimensions. If the extrudate dimensions reduce just before the take-up, the extrudate dimensions above it have to increase. As the larger extrudate section is taken up, a thin extrudate section can form above it this can go on and on. Thus, a cyclic variation of the extrudate dimensions can occur. Draw resonance does not occur when the extrudate is solidified at the point of take-up because the extrudate dimensions at the take-up are then fixed [171, 172]. Isothermal draw resonance is found to be independent of the flow rate. The critical draw ratio for almost-Newtonian fluids such as nylon, polyester, polysiloxane, etc., is approximately 20. The critical draw ratio for strongly non-Newtonian fluids such as polyethylene, polypropylene, polystyrene, etc., can be as low as 3 [173]. The amplitude of the dimensional variation increases with draw ratio and drawdown length. 

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. 

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). 

Fig. 7.27 Semi-log plot of the effective exponent 7efr of the coilective scattering function iScolK - 0) (upper part)j of the order parameter exponent (middle part) and the ratio of critical amplitudes C /Cl (lower part) versus the crossover scaling variable LmjN, which is proportional to the ratio of characteristic lengths m/fcross- L is the mean size of the lattices used in the finite size scaling analysis. Asymptotic limits are shown as dashed straight lines. Open circles refer to the bond fluctuation model which has (.aa = bb = 0, ab 5 0 for effective monomers two lattice spacings apart (Fig. 7.25), full dots to the version of the bond fluctuation model with ab = aa = bb = e and Zefr rj 14 (critical temperatures of this model are included in Fig. 7.4). Curves through the points are guides to the eye only. (From Deutsch and Binder." )...

Let ARc be the value of the open-loop amplitude ratio at the critical frequency (o. Gain margin GM is defined as ... 

Ising model. The ratios of the Ising and mean field critical amplitudes are given as... 

The steady-state solution for fiber spinning (Newtonian and isothermal case) was presented in Section 9.1.1, and it consists of Eqs. 9.26 and 9.28. Linearized (small disturbances) stability analysis involves (Fisher and Denn, 1976) the study of finite amplitude disturbances, and we do not present it. Rather, we present the results of such an analysis. The value of Dr = 20.21 is considered to be the critical draw ratio beyond which the flow becomes unstable. Figure 9.13 (Donnelly and Weinberger, 1975) shows experimental data that confirm the theory. More specifically, silicone oil (of viscosity equal to 100 Pa-s), which seems to be Newtonian, was extruded and the ratio of maximum to minimum filament diameters was plotted against the draw ratio. An instability appears at a draw ratio of about 17, or about 22 if we take into consideration about 14% die swell. The value of the critical draw ratio of 22 compares well with the theoretical value of 20.21. Pearson and Shah (1974) extended the analysis to a power-law fluid and included surface tension, gravitational. 

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