Asymptotic power laws

Several examples of the asymptotic power laws Eqs. (2.50) and (2.54) at small and large separations are collected in Table 1. Though we integrated pair interactions only in the investigations discussed here, setting up 

Separation Interacting particles Non-retarded limit Retarded limit 

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At higher Q, however, where the static structure factor reveals the asymptotic power law behavior S (Q, 0) Q 1/v, the assumption of ideal conformation clearly fails. In particular, this is evident for the core (sample 1) and shell contrast conditions (sample 2). 

Fig. 23. The intrinsic viscosity of several end-linked PS star molecules as a function ofM [95]. In the limit of low and high molar masses asymptotic power law behavior may be derived. That at low molar masses is widely controlled by the presence of non-reacted star molecules, that at high molar masses is expected from theory for randomly branched macromolecules. The exponents of the two asymptotic lines are a =0.49 0.08 for M <0.8x10 g/mol and a =0.18 0.05 for M >2.0xl0 g/mol. Reprinted with permission from [95]. Copyright [1997] American Society...

This relationship fulfills the condition of g =l at g=l and attains a power law proportional g =a for very small g. The exponent p defines how quickly the asymptotic power law is obtained. 

Des Cloizeaux [154] and de Gennes [4,176] suggested an asymptotic power law for linear chains when X 1... 

Fig. 3.21 Same data as shown in Fig. 3.20 in a representation of -6 n[Sseif(Q,0]/Q ) which is the mean-square displacement (r (t)) as long as the Gaussian approximation holds. Solid lines describe the asymptotic power laws Dotted lines prediction from the...

Expansion of the asymptotic power law for the order parameter in terms of a Wegner series (3) leads to [43]... 

Systems that display strange kinetics no longer fall into the basin of attraction of the central limit theorem, as can be anticipated from the anomalous form (1) of the mean squared displacement. Instead, they are connected with the Levy-Gnedenko generalized central limit theorem, and consequently with Levy distributions [43], The latter feature asymptotic power-law behaviors, and thus the asymptotic power-law form of the waiting time pdf, w(r) AaT /r1+a, may belong to the family of completely asymmetric or one-sided Levy distributions L+, that is,... 

We choose the representation in terms of a Levy distribution for convenience because it includes the Brownian limit. Indeed, any waiting time pdf w(t) with the asymptotic power-law trend following Eq. (6) leads to the same results as obtained in the following for 0[Pg.229]

The Mittag-Leffler function, or combinations thereof, has been obtained from fractional rheological models, and it convincingly describes the behavior of a number of rubbery and nonrubbery polymeric substances [79, 85]. The numerical behavior of the Mittag-Leffler function is equivalent to asymptotic power-law patterns that are often used to fit experimental data, see the comparative discussion of data from early events in peptide folding in Ref. 86, where the asymptotic power-law was confronted with the stretched exponential fit function. 

Fig. 13.2. Crossover diagram in the w — / plane (uncritical surface). The shaded parts indicate regions dominated by asymptotic power law behavior. Weak coupling region (/ < 1) Long dashes lines of fixed z (— 300 1, from above). Short dashes lines of fixed sfz (= 0,01 1, from above). Dotted lines of fixed s = (0-02 0,8, from above). In the strong coupling region a line of fixed z = 300 is shown...

An essential aspect, we want to demonstrate, is the fundamental importance of a (qualitatively) correct choice of the uncritical manifold. It is by this choice that our theory generates smooth crossover functions, which interpolate among asymptotic power law behavior as expected from scaling theory. It is most important that the correct behavior is found even in lowest order approximation. Otherwise higher order corrections, trying to reconstruct the correct asymptotics, must blow up. Then a one loop calculation cannot be reliable quantitatively... 

This result is in full accord with scaling theory (cf. Eq. (9,20), and recall IdA = CpN 2Ja ) As expected even the prefactor of the asymptotic, power law is independent of the overlap. 

Again we can easily calculate the full crossover. As an example Fig. 14.3 shows the scaling function V/s as function of s in the excluded volume limit. In unrenormalized tree approximation this ratio would be a constant proportional to the second virial coefficient. In renormalized theory we see a pronounced variation which rapidly approaches the asymptotic power law. 

Fig. 14.3. The thermodynamic scaling function in the excluded volume limit V (s), divided by the excluded volume overlap s. The broken line gives the asymptotic power law...

The first frequency response of the function I/Q (disk response) follows an asymptotic power law dependence I/Q [Pg.231]

The former parameter is the conventional factor of as5mimetry in the expansions truncated after linear terms. It can be used to introduce the presumed scaling relations at subcritical temperatures T = 1 -T/T >0 as well as to obtain the consistent description of stable phases, in which the asymptotic power laws are used. Unfortunately, the conventional analysis of the scaling consistency fails, often, even in the asymptotic range of temperatures T < 10 because the adjustable system-dependent amplitudes of the power laws are rather inaccurate. Besides, the implicit assumption of scaling, the parameter (pt Pg) to be the single factor of asymmetry, must be corroborated especially in the extended critical region. 

There is, of course, much that remains to be understood with regard to the physical interpretation. For example, the correlation-kinetic-energy field Z, (r) and potential W, (r) need to be investigated further. However, since accurate wavefunctions and the Kohn-Sham theory orbitals derived from the resulting density now exist for light atoms [40] and molecules [54], it is possible to determine, as for the Helium atom, the structure of the fields P(r), < P(r), and Zt (r), and the potentials WjP(r), W (r), W (r), and W (r) derived from them, respectively. A study of these results should lead to insights into the correlation and correlation-kinetic-energy components, and to the numerical determination of the asymptotic power-law structure of these fields and potentials. The analytical determination of the asymptotic structure of either [Z, (r), W, (r)] or [if (r), WP(r)] would then lead to the structure of the other. 

A simplistic view of the scattering features can be described as follows. The asymptotic power law at the low-Q is characteristic for sheet-like 2D structure, and the rapid fall of the intensity for about 2 orders of magnitude at ca. 0.03 A implies the loss of self-correlation beyond the lamellar thickness. Those are considered the form factor of individual sheets with a uniform thickness. On the other hand, the peaks at ca. 0.025, 0.05 and 0.08 A are the... 

In Table 2, typical values of the correlation length, and of the ratio of the compressibility to that of an ideal gas at the same density, are shown for carbon dioxide and water. Parameters and asymptotic power laws are those from Ref. [ 10. The correlation length is to be compared with a molecular dimension, which is 0.15 nm for carbon dioxide, and 0.13 nm for water, of the order of the average molecular radius. The second and third columns show how large the ratio becomes in fluids and the Ising model as the critical point is approached. In two dimensions, fluctuations are even more pronounced. 

The universal singular behavior of the thermodynamic properties near the critical point is associated with the presence of long-range fluctuations of the order parameter (density in a one-component fluid). The size of the fluctuations is called the correlation length which diverges at the critical isochore in accordance with the asymptotic power law... 

The validity of the asymptotic power laws is restricted to a very small region near the critical point. An approach to deal with the nonasymptotic behavior of fluids including the crossover from Ising-like behavior in the immediate vicinity of the critical point to classical behavior far away from the critical point has been developed by Chen et al. [3, 4]. This approach is based on earlier work of Nicoll and coworkers [14, 15] and it leads to a transformation of the classical Landau expansion of the free energy to incorporate the effect of critical fluctuations. This approach has a solid theoretical background, being based on the renormalization-group (RG) theory, and it has been applied successfully to the description of crossover critical phenomena in simple and complex fluids. 

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