Rescaled time

The trick involves using the correction factor g peq) to rescale time, pressure and viscosity. Specifically, we set... 

Figure 36. The scaled distributions of mean, P(H/Y ) (a), and Gaussian, P(K/Y]2) (b), curvatures scaled with the inteface area density, computed at several time intervals of the spindal decomposition of a symmetric blend. There is no scaling at the late times because the amplitude of the thermal undulations does not depend on the average growth of the domains, and therefore the scaled curvature distributions functions broaden with rescaled time.

Fig. 6.2 Single chain dynamic structure factors at Q=0.15 A vs. the rescaled time From...

Figure 3. Typical images of the surfaces obtained by the integration of the continuum equation. The rescaled time is shown below the images.

In any kinetic analysis, the time, t, comes naturally as an independent variable, but in Eq. (3) it is, in fact, a variable merely proportional to the real time. It can be regarded as a rescaled time with the scaling factors depending on the actual units of both coagulation kernel and concentration as well as on the type of... 

For practical reasons we used dimensionless units in our numerical calculations. The invariance of the governing equations under rescaling time, length, and mass allows us to choose three parameters in these equations to be equal to unity. We will set... 

In accordance with Table I, we will adopt from the beginning a dimensionless set of units. The symmetrized, rescaled time evolution operator for the model is then (compare with equation (1.30) for the three-body case)... 

The symmetrized and rescaled time evolution operator for the system described by the set (lij, simply defined adding to the two-body... 

Again we consider the symmetrized and rescaled time evolution operator, obtained by summation of the two rotational FPK operators for bodies 1 and 2, in the presence of the usual interaction potential. Since we suppose that both the bodies are spherical, no processional terms are present [6] [cf. Eq. (1.14) for the three-body case] ... 

Fig. 32. Phase diagram of the surface plotted in terms of the scaled variables h jy and g/y. For g/y < —2 one observes critical wetting and for g/y > —2 one observes first-order wetting. In the latter regime, mean field theory predicts melaslable wet and non-wet regions limited by the two surface spinodal lines ft, and ft( respectively. Also two quenching experiments arc indicated where starting at a rescaled time r = 0 from a stable state in the non-wet region one suddenly brings the system by a change of fti into the metastahle wet or unstable non-wet region, respectively. From Schmidt and Binder (1987).

Figure 10. Illustration of eight different predictions of photon antibunching of quantum fields, corresponding to the cases analyzed in Table III and Fig. 9. The two-time signal-mode correlation functions Agi(t,t + x) (dashed curves), Agn(t,t + t) (dot-dashed), and Agni(M + f) (solid) are plotted in their dependence on the rescaled time separation kt for fixed values of the evolution time (case 1) Kt = 2.8, (2) kt = 2.6, (3) kt = 0.1, (4) kt = 0.1, (5) Kt = 1.0, (6) kt = 2.3, (7) Kt = 0.7, and (8) Kt — 1.8. Signal and idler modes are initially in Fock states with Na — 3 and Nt, = 1 in cases 2 and 3, or with Na —2 and Nb — 1 in all other cases.

A key to solving (3-260) and (3-261), either exactly or for the large i asymptote, is to note that the velocity field will not generally be in-phase with the oscillating pressure gradient. We can see that this is true by a qualitative examination of the governing equation. It is convenient for this purpose to temporarily rescale time according to... 

To complete the analysis, we must obtain solutions for the amplitude and phase functions A(r) and [Pg.267]

We follow Tyson (1985) and bring the model to dimensionless form by rescaling time and concentrations ... 

Figure 7.3 The time dependence of the total concentration as a function of the rescaled time Dai, at several values of Da for the autocatalytic dynamics in the closed sine-flow of Eq. (2.66) Da increases from top to bottom, dashed line is the homogeneous result, which is approached as Da —> 0. The inset plots the evolution in terms of the unsealed time, showing that at any given time the amount of C is larger for larger Da, which corresponds to the most inhomogeneous configuration. Dashed line in the inset is the exponential growth of the length of a material line.

Fig. 8a. Scaling behavior of qm(t) vs the rescaled time f = tD q (0) = 2X(0) [XsM>o) — 3 for the polymer mixture as shown in Fig. 6a and a quench to T = 25 °C. Different symbols refer to different sample geometries. The solid curve is a fit to a formula ohtained by Furukawa [168] corresponding data for polyvinylmethylether (PVME)-polystyrene (PS), dash-dotted [158] and cyclohexane/meth-anol, dashed curve [169], are included. From Bates and Wiltzius [36]. b Coarsening behavior of mixtures of SBR (a random copolymer of styrene and polybutadiene) and polyisoprene (PI) at various compositions, at T = 60 °C. a shows qm(t) and b. corresponding intensity I (t), in arbitrary units, while arrows indicate the times where pinning (t,) or crossover (t ) from intermediate to late stages occurs. From Hashimoto et al. [173]...

The parameter K is the carrying capacity of the system, at which the growth rate vanishes. If we nondimensionalize the system by measuring the density in terms of the carrying capacity, p = p/K, and rescale time by the intrinsic growth rate, t = rt, then we obtain again KPP kinetics ... 

Figure 3. Comparison of double-minimum potentials V(fi) for the D — limit at i2 = 5 evaluated without (full curve) and with (dotted curve) rescaled time transformation.

On the next step of the proof Leontovich had evaluated the local map. She considered, in fact, the map from the cross-section Si y = d to 5o x = d, i.e. the inverse of the local map To in our notations. Note that by assumption of the theorem only the last saddle value an is bounded away from zero, whereas ai,.., an-i are small. Therefore, by rescaling time variable the system may... 

Remark 3. Let us draw more attention to the case where the unstable manifold of the saddle is one-dimensional. There is no v-variables here and i/ E R. By rescaling time, the system (13.8.1) can be reduced to the form... 

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