Correlation length and critical exponent

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 cx In 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 intermediate asymptotic exponent could be also defined as dlnn(f) 

This intermediate asymptotic exponent is useful to demonstrate the formation in time of a new asymptotic law shown in Fig. 6.3. In a given temporal interval a t) approaches its asymptotic value close to a = J/2. 

This estimate should be made more precise. To do it, let us use some results of the numerical solution of a set of the kinetic equations derived in the superposition approximation. The definition of the correlation length 0 in the linear approximation was based on an analysis of the time development of the correlation function Y r,t) as it is noted in Section 5.1. Its solution is obtained neglecting the indirect mechanism of spatial correlation formation in a system of interacting particles, i.e., omitting integral terms in equations (5.1.14) to (5.1.16). Taking now into account such indirect interaction mechanism, the dissimilar correlation function, obtained as a solution of the complete set of equations in the superposition approximation 

Assuming that the reaction kinetics in the region where (6.1.5) holds is defined entirely by the single scale ( o, o). we arrive at the asymptotic solution for the correlation functions (for equal initial concentrations) in a form Xu r,t) = X r]), Y r,t) = Y r]), z r,t) = z r]), where the automodel variable r] = r/ o is introduced. Analysis of the numerical solution of equations at long times, o. shows that Y(r, t) z r, t) for all r. 

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Percolation in the two-dimensional lattice gas is very instructive in connection with the universality concept If percolation occurs at finite lattice-gas correlation length, the critical exponents are the same as for random percolation. This can be easily understood if one takes into account that near the percolation threshold the lattice-gas correlation length I is much smaller than the typical duster radius thus, the large clusters average over the effects of correlation. This argument breaks down only at the critical point of the two-dimensional lattice-gas where both and vary simultaneously in fact, we have seen that at this point some exponents do change. 

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 ... 

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. 

In view of the anomalous critical behavior of the correlation length and the osmotic compressibility, it appeared of interest to characterize the behavior of other properties. Bell-ocq and Gazeau investigated how the interfacial tension between the coexisting phases on the one hand and the difference of density of these phases on the other hand vanished at various points of the critical line P (Fig. 25) [152]. The aim of the experiments was to determine the associated critical exponents and and check whether the scaling laws that relate v,p, and f.i were valid all along the critical line. Data obtained for two critical points defined by Xc = 1.55 and Xc = 1.207 indicate that the values of the critical exponents and )U show an X dependence similar to that found for v and y. Furthermore, within the experimental accuracy, the obtained values of v, y, (i, and are in reasonable agreement with the theoretical predictions v = y 2/ = 3v (Table 2). 

Structural descriptors at the secondary level (mesoscale) are topology and domain size of polymeric aggregates (persistence lengths and radius), effective length and density of charged polymer sidechains on the surface, properties of the solution phase (percolation thresholds and critical exponents, water structure, proton distribution, proton mobility and water transport parameters). Moreover, -point correlation fimctions could be defined that statistically describe the structme, containing information about surface areas of interfaces, orientations, sizes, shapes and spatial distributions of the phase domains and their connectivity [65]. These properties could be... 

Clearly there is need for spin-glass theories beyond mean-field. One approach in this direction is presented by Malozemoff et al. (1983) and Malozemoff and Barbara (1985). They propose a critical fractal cluster model of spin glasses which is able to describe the essential features of the phenomena occurring near the spin-glass transition and to account for the static critical exponents. The basic assumption of this fractal model is the existence of a temperature- and magnetic-field-dependent characteristic cluster size on which all relevant physical quantities depend and which diverges at the transition temperature Tj. It is related to the correlation length and the cluster fractal dimension D by More... 

Other physical properties like correlation lengths and percolation probabilities follow as well power laws in the vicinity of pc, however with different critical exponents. Values of the critical exponent /r in Equation 3.102 are known in 2D and 3D from computer simulations (Isichenko, 1992). For lattice percolation in 2D, it is u. 1.3... 

Many important properties, such as critical dimensionality and critical exponents describing the divergence of correlation length and other quantities can thus be obtained from renormalization group analysis. It is worth noting that for X well below X only the quadratic terms of the potential contribute to the asymptotic properties of P, which reduces therefore to a multigaussian distribution in accordance with the central limit theorem [13]. There exists,however,a (frequently very narrow) vicinity of X ... 

In two dimensions, however, the situation will prove to be very different from what it is in three or more dimensions whereas van der Waals theory predicts o) = 1, which follows at once from the postulated identification a> = v, with v the critical exponent describing the divergence of the bulk correlation length and the exact... 

An important point here is that according to the theory of critical phenomena, the critical exponents take universal values essentially independent of the microscopic details of the system. The natural question then is whether the exponents characterizing the curve, the radiation scattering intensity, the correlation length, and the interfacial tension behavior in polymer-polymer-solvent systems are the same as for ternary mixtures of small molecules. It is also essential to study how the critical amplitudes depend on the molecular weight of chains. 

A first systematic study of such system was performed on the relatively large-molar-mass symmetric polyolefins PE and PEP and the corresponding diblock copolymer PE-PEP PE being polyethylene and PEP being poly(ethylene propylene). A mean-field Lifshitz like behavior was observed near the predicted isotropic Lifshitz critical point with the critical exponents y=l and v=0.25 of the susceptibility and correlation length, and the stmcture factor following the characteristic mean-field Lifshitz behavior according to S(Q)ocQ". Thermal composition fluctuations were apparently not so relevant as indicated by the observation of mean-field critical exponents. On the other hand, no Lifshitz critical point was observed and instead a one-phase channel of a polymeric bicontinuous miaoemulsion phase appeared. Equivalent one-phase channels were also observed in other systems. 

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