Error phase transition

Figure 7. Vapor-Uquid-soUds (plastic crystal 2, plastic crystal 2, crystal 3) phase diagram of diamantane. This diagram is based on the data of Table II. The shaded area between vapor and plastic crystal 2 and crystal 3 phase transitions is indicative of the error range of the available data.

It is convenient to set Ap1 = 1, L = d - -dq = 1. Rounding errors are suppressed by replacing die intensity by 1/s2 (Porod s law) for big arguments (s > 8). A smooth phase transition zone (in all the example curves dz = 0.1) is considered by multiplication with exp (2nsdz/3)2 j. From this one-dimensional scattering intensity die correlation function is obtained by Fourier transformation. 

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. 

The value of %2 was deduced from the jump at the transition of and that of calculated %, because these two quantities should be proportional with each other. The value thus obtained was %2 = 0.6 + 0.1. In the actual calculation, the values of Ah, As, and x2 were adjusted within the error limits so that the calculated swelling curve fits the measured one as closely as possible. The results of calculation [21] are shown in Fig. 5a and b, where they should be compared with Fig. 3a and b, respectively. The parameter values used are given in Table 2. Of course the values of Ah, As, and %2 include relatively large arbitrariness, although the fact that we can fit the observed swelling curves using reasonable values of the parameters shows that this theory captures an essential point of the phase transition of neutral gels. 

Fig. 4.16 Phase diagram for aqueous solution of PEO iBio determined using SAXS and rheometry (Pople et al. 1997). The filled symbols mark the phase boundaries determined by SAXS, with the broken line as a guide to the eye, and the solid lines mark transitions determined using rheology (Deng et al. 1995 Pople et al. 1997). The error bars indicate uncertainties associated with the phase transitions determined from repeated heating and cooling ramps.

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

The accuracy and reproducibility of this method is such that, for example, the solid-solid phase transition of cholesterol acetate at 40 C has been detected by means of the temperature dependence of the contact angle of water 26], The discontinuity in the contact angle curve was about 0.3 of arc. For water on siliconized glass plates, the deviations of individual points in the plot of contact angle versus temperature were found to be about 0.1 128). in good agreement with the error limits estimated above. 

At lower temperatures, the spin concentration of PMQ4 increased as temperature decreased as illustrated in Figure 10. The reason for this is presently unknown. One possibility is that a phase transition from magnetic disorder to order could occur at a critical temperature, T<,. Three determinations of spin concentration versus temperature for PMQ4 were carried out with reproducible results within experimental error. The data was fitted using Equation 2. 

Applying superposition approximations to the Ising model, one finds an evidence for the phase transition existence but the critical parameter to is systematically underestimated (To is overestimated respectively). Errors in calculation of to are greater for low dimensions d. Therefore, the superposition approximation is effective, first of all, for the qualitative description of the phase transition in a spin system. In the vicinity of phase transition a number of critical exponents a, /3,7,..., could be introduced, which characterize the critical point, like oc f-for . M oc (i-io), or xt oc i—io for the magnetic permeability. Superposition approximations give only classical values of the critical exponents a = ao, 0 = f o, j — jo, ., obtained earlier in the classical molecular field theory [13, 14], say fio = 1/2, 7o = 1, whereas exact magnitudes of the critical exponents depend on the space dimension d. To describe the intermediate order in a spin system in terms of the superposition approximation, an additional correlation length is introduced, 0 = which does not coincide with the true In the phase... 

Numerical studies on quasi-species localization [42]—as they will be presented and discussed in Section IV—showed a sharpening of error thresholds with increasing chain lengths v. Sharpening means here that the transition zone from localized quasi-species to delocalized sequence distributions becomes narrower. This phenomenon is reminiscent of cooperative transitions observed with conformational equilibria in biopolymers. We shall investigate here whether or not error thresholds show analogy to phase transitions in the... 

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