Exponents correlation length

The divergence m the correlation length is characterized by the critical exponent v defined by... 

The correlation length C = T -T diverges with the exponent v. Assuming that when T>T the site... 

Just as in phase transitions in statistical mechanical systems, observable quantities in PCA systems display singularities obeying simple power laws with universal critical exponents at the transition point. For example, letting ni be the number of sites with correlation length, and t be the correlation time, Kinzel [kinz85b] finds that for p ... 

The result is shown in Fig. 62 where the different Dc values, obtained from separate fits at each concentration, are also presented. The dynamic correlation length (c)/A = (3.4 + 0.7) c 067 005 is of the same order of magnitude as the value obtained from a static experiment on the PS/d-cyclohexane system [104]. The exponents for the concentration dependence of Dc and (c) are in agreement... 

As it was mentioned in Section 5.1, computer simulations demonstrate existence of the correlation length whose time development is, however, difficult to investigate in detail. At any rate, it corresponds approximately to the length scale o oc In t introduced earlier in the linear approximation. We can introduce the asymptotic (t —> oo) exponent a for the static tunnelling recombination similarly to (4.1.68) used for the diffusion-controlled problem ... 

The difference in the exponents (6.1.42) and (6.1.43) is large, which was indeed confirmed by computer simulations [13]. However, change of the correlation length in computer simulations is a more complicated problem. Indeed, assuming a = d/2, and = o, the exponent in the expression... 

In spite of the constant density of the gel, the friction of the poly(N-isopropylacrylamide) gel reversibly decreases by three orders of magnitude and appears to diminish as the gel approaches a certain temperature. This phenomenon should be universal and may be observed in any gel under optimal experimental conditions of the solvent composition and the temperature because the unique parameter describing the friction is the correlation length which tends to diverge in the vicinity of the volume phase transition point of gels. The exponent v for the correlation length obtained from the frictional experiment is far from the theoretical value. It will, therefore, be important to study a poly(N-isopropylacrylamide) gel prepared at the critical isochore where the frictional property of gel may be governed by the critical density fluctuations of the gel. 

Fig. 6.23 Logarithmic plots of the correlation length ( ) and zero-angle scattering intensity (/(0)) as a function of temperature reduced with respect to the Lilshitz temperature Tlp) for a blend of PE and PEP homopolymers with a PE-PEP diblock (details as Fig. 6.22) at a copolymer volume fraction

Measurements of static light or neutron scattering and of the turbidity of liquid mixtures provide information on the osmotic compressibility x and the correlation length of the critical fluctuations and, thus, on the exponents y and v. Owing to the exponent equality y = v(2 — ti) a 2v, data about y and v are essentially equivalent. In the classical case, y = 2v holds exactly. Dynamic light scattering yields the time correlation function of the concentration fluctuations which decays as exp(—Dk t), where k is the wave vector and D is the diffusion coefficient. Kawasaki s theory [103] then allows us to extract the correlation length, and hence the exponent v. 

At first, one would tend to reconsider conventional crossover due to mean-field criticality associated with long-range interactions in terms of the refined theories. Conventional crossover conforms to the first case mentioned—that is, small u with the correlation length of the critical fluctuations to be larger than 0. However, in the latter case one expects smooth crossover with slowly and monotonously varying critical exponents, as observed in nonionic fluids. Thus, the sharp and nonmonotonous behavior cannot be reconciled with one length scale only. 

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