Asymptotic regime

For the pseudo-binary mixture (a = 0.5) of sulfonate and nonylphenol with 30 E.O., figure 2 shows how the concentration of each of their monomer calculated by the RST theory (1), varies as a function of the overall surfactant concentration. It can be expected that the asymptotic regime in which monomer concentrations are stabilized will correspond to a plateau of the adsorption isotherm for the surfactant mixtures considered. 

Assuming that after their previous swap the two walks were sufficiently long to be in the asymptotic regime, this means that transient behavior has elapsed and the system has relaxed to equilibrium for the respective parameters. Then, the joint configurational probability density just before the current swap is simply... 

Figure 2. f inite-time Lyapunov exponents A( t J tor a driven I lulling oscillator with measurement strengths k = 5 x 10-4, 0.01, 10, averaged over 32 trajectories (linear scale in time, top, and logarithmic scale, bottom bands indicate the standard deviation over the 32 trajectories) (S. Habib et.al., 1998).The (analytic) 1/f fall-off at small k values (dashed red line), prior to the asymptotic regime, is evident in the bottom panel. 

Finally, we note that Eqs. (3.5-3.7) can also be used to extract the behavior of the xc-potential in the asymptotic regime of finite systems. Restricting the analysis to spherically averaged systems, a somewhat tedious analysis (the interested reader is referred to [19]) leads to the relativistic form of the Krieger-Li-lafrate identity for the highest occupied orbital (() , [56],... 

Note that these calculations complement well computer simulations done by Toussaint and Wilczek [27] clearly revealing the same transient process before reaching the asymptotical regime nA oc t x/2. 

The performed calculations demonstrate that a type of the asymptotic solution of a complete set of the kinetic equations is independent of the initial particle concentrations, iVa(0) and 7Vb(0). Variation of parameters a and (3 does not also result in new asymptotical regimes but just modifies there boundaries (in t and k). In the calculations presented below the parameters 7Va(0) = 7Vb(0) =0.1 and a = ft = 0.1 were chosen. The basic parameters of the diffusion-controlled Lotka-Volterra model are space dimension d and the ratio of diffusion coefficients k. The basic results of the developed stochastic model were presented in [21, 25-27],... 

Anisimov et al. [157] attributed this behavior to crossover with a rapidly shrinking asymptotic regime. There is, however, another possible mechanism... 

Under a continuous constant feed to the circuit there exists a steady-state (asymptotic) regime of the circuit operation. However, this solution cannot be obtained with the matrix (1) because it contains the absorbing state, the probability of which grows unlimitedly. In order to get round the difficulty, the absorbing cell can be presented not like the end of the chain, but as an intermediate element of the chain... 

On the other hand, the highest occupied eigenvalues which reflect the asymptotic form of the density, and thus even in the x-only limit should be close to the ionisation potential, are too small by roughly 50% for the DFS and RLDA and by 30- % for the RWDA . Thus while the latter improves on the asymptotic form of in principle, the effect of this formal improvement on the physically relevant part of the asymptotic regime is rather limited. Note further, that for Yb all three schemes incorrectly predict the 4F7/2-orbital to be most weakly bound instead of the 6S1/2-orbital. The same deficiency has been observed for Cr and Cu in the nonrelativistic case [59]. These difficulties to reproduce the size and the ordering of the outermost eigenvalues are well known from the nonrelativistic case and are not related to the relativistic corrections in Ex [n]-... 

The prime question is, how these various long-range Couiombic and short-range specific interactions affect the critical behavior. In general, liquid-gas and liquid-liquid coexistence curves in the asymptotic regime near the CP can be described by a power law of the form "... 

The rate of photon generation in the mth (odd) mode of the ID cavity is approximately m times less than in the principal mode with m = 1 (in the asymptotical regime). 

An experimental comparison of three phase estimations—NFM, unconstrained ML, and ML estimators—in the asymptotic regime is shown in Fig. 16. The experiment was done with photons. For comparison, the theoretical values of dispersions given in Table V are also shown. Several important... 

The PBE GGA [96] and the TPSS MGGA [2] (see below) may be partial exceptions [99, 100] because they work reasonably well near the equilibrium distance of the van der Waals bond, but they recover only the short-range behaviour and do not describe correctly the long-range asymptotic regime of the van der Waals interaction. 

When w decreases beyond wF, the exponent v(w) quickly diminishes and becomes even smaller than the value v = 2/3 which corresponds to a compact globule, but this fact is not significant it shows only that for the corresponding values of v(w), one remains far from the asymptotic regime. Besides, it is expected that in the asymptotic regime v(w) vary discontinuously and not continuously, with respect to w. 

Furthermore, by examining how an effective exponent varies with molecular mass and temperature, it is possible to determine whether one has to do with the intermediate or with the asymptotic regime of the polymer state in good solvent. 

Experiments made in this limit are crucial, because they determine universal constants related to the polymer state in good solvent. The interval z za defining the asymptotic regime depends on the required precision. We shall be interested in the situation where the exponent dln /d lnz has 3 significant... 

The interval z p z, also depends on the nature of the observable. Figure 15.11 shows, for instance, that the asymptotic behaviour of the osmotic parameter g(z) is practically reached when z P 10. For the end-to-end swelling, it is necessary to wait until zp 30 (see Fig. 15.12). We shall now examine to what extent we can observe the asymptotic regimes of the gyration swelling, of the second virial coefficient (for 2 P 1) and of the form function (for z p 1 and qRp l). 

Better experimental values of the exponent v are obtained from measurements of the radius of gyration of polymers and of the second virial coefficient, in dilute solutions. Indeed, when the number of links increases, these quantities reach the asymptotic regime much more rapidly. However, the realization of a true asymptotic state for these quantities remains the main difficulty to overcome. Some progress has recently been made thanks to the elaboration of samples of high molecular mass Mw = 5 x 107. 

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