Asymptotic laws

Asymptotic Laws and Preasymptotic Correction Formulas for the Relaxation Near Glass-Transition Singularities. 

Another very complicated problem where the approach to equilibrium with time after a quenching experiment is described by an asymptotic law is the owth of wetting layers, in a situation where thermal equilibrium would require the surface to be coated with a macroscopically thick film, but is initially nonwet. For a short-range surface potoitial as discussed in section 3.5, analytical theories predict for a non-conserved density a growth of the thicknm of the layer according to a law f(t) oc In t, and this has in fact been observed by simulations . In the case where the surface potential decays with stance z from the surface as z, the prediction for the thickness l(t) is for the nonconserved case and... 

Nearly the same limits of r exist in real solid state experiments. However, the relevant maximal time tm which could be achieved in such computer simulations (see equation (5.1,60)) for a given Tq, could turn out to be not long enough for determining the asymptotic laws under question. For instance, existence of so-called small critical exponents in physics of phase transitions [14] was not experimentally confirmed since to obtain these critical exponents, the process covering several orders of the parameter t — fo should be... 

For unequal concentrations, n (t) < ns(t), the reaction depth r < T0 = 3 reached in computer simulations is not enough for finding asymptotic laws but still permits to estimate qualitatively the accuracy of the superposition approximation. In Figs 5.7 and 5.8 numerical solution of the relevant kinetic equations is compared with computer simulations. To make situation more transparent, the linear approximation results are plotted in curves (d) for a single choice of initial concentrations only. 

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 = d/2. 

The value of R = 1 [by the definition of R and the dimensionless concentration n(R), such that n(0) = 1] corresponds to the case when the reaction has destroyed pairs AB separated by the relative distances r less or equal to the mean distance between particles. In other words, according to equation (6.1.73), a new asymptotic law with a = 1/2 occurs already at very small reaction depths, r 0.5 ... 

New reaction asymptotic law (2.1.78) emerges due to formation during the reaction course of a new spatial scale - the correlation length = Id- Similar to the case of immobile particles, we can expect here that at long times the coordinate r enters into the correlation function in a scaling form rj = r/Io, so that Y(r,t) —> Y(t, t), X (r,t) -> where the second variable... 

The Asymptotic Law of Heat Transfer at Small Velocities in the Finite Domain Problem ... 

The Asymptotic Laws of Freely-Ascending Convective Flows... 

Asymptotic laws are found for freely-ascending convective flows in the laminar and turbulent planar and radially-symmetric cases in the form u =... 

It should be mentioned that, as is often the case in a first draft, the author based his treatment of the turbulent convection on certain assumptions which in fact were not necessary, in particular the semi-empirical concept of L. Prandtl. Moreover, even in the analysis of laminar convection, the author [cf., for example the transition from equation (9) to equation (10)], to derive the asymptotic laws, resorts to simplifications of the equations which are really not necessary. Actually, it is possible to manage without these assumptions so that Zeldovich s asymptotic laws (8), (8a), (11), and (11a) may be obtained by simple dimensional analysis under the most general assumptions. 

Without considering transient regimes, we find the asymptotic laws which are satisfied by the wave and the state of gas compressed by it during the motion of the piston (which is long compared to the time of the chemical... 

Specifically, it has been shown that the successive bifurcation points X at which the solution switches from period 2 " to period 2k have an accumulation point X and obey the asymptotic law... 

Close to the crossover temperature Tc, the solution of the idealized MCT equations can be expanded around the nonergodicity parameter fc at Tc and some generic laws can be derived. These leading-order results establish the so-called asymptotic laws of MCT, which can be regarded as generic features of the MCT dynamics at T > Tc. Because almost all experimental tests analyzed the susceptibility spectra in this frame, we are going to briefly summarize the laws [16,17,19]. Examples of tests against experiment are found in Section IV.E. 

Figure 20. Comparison of light scattering (LS) and dielectric (DS) spectra of glycerol (top) and of propylene carbonate (bottom). Significant differences are observed among the spectra obtained by the different techniques. The spectra are interpolated using the Cole-Davidson susceptibility and a power-law contribution with exponents as indicated [Eq. (31)] in the case of the DS spectra the data are compatible with the asymptotic laws of mode couphng theory. (DS data compiled from Ref. [9], LS data compiled from Ref. 64.)...

Although in some cases a consistent analysis of LS or DS spectra was carried out by applying the asymptotic laws of MCT, there are strong indications that these features are not completely appropriate to quantitatively describe, for example, DS as well as LS spectra. As discussed above, this is by now well known for PC and glycerol, at least. In order to tackle the problem of different experimental probes in a more realistic fashion, several MCT approaches have been published [265,380,400]. In a two-correlator schematic model, in which the dynamics of some probe (e.g., molecular reorientation in a dielectric experiment) is coupled to the overall structural relaxation in a simple manner, a simultaneous description of LS, DS, and NS spectra was possible even below Tc. Some of the results are... 

Crossover Temperature for Various Glass Formers as Reported by the Different Methods From the Temperature Dependence of the Stretching Parameter y(T), Scaling the Time Constant xa — xa(r) [cf. Eq. (42)], Non-ergodicity Parameter 1 —f(T) Obtained from Spectra Analysis, Electron Paramagnetic Resonance (EPR), and from Tests of the Asymptotic Laws of Mode Coupling Theory ... 

Given that the asymptotic laws may fail, the only quantity expected to signal the crossover forecast by MCT is the anomaly of the nonergodicity parameter/. 

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